A second step towards a stochastic mathematical description of human feelings

MATHEMATICAL AND COMPUTER MODELLING

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8CIENCIE ~,DIRBCTe ELSEVIER

Mathematical and Computer Modelling 41 (2005) 587--614 www.elsevier.com/locate/mcm

A Second Step Towards a S t o c h a s t i c M a t h e m a t i c a l D e s c r i p t i o n of H u m a n Feelings B . CARBONARO AND C . GIORDANO Dipartimento di Matematica della Seconda Universit~ degli Studi di Napoli v i a A. Vivaldi, 43 - 81100 C a s e r t a , I t a l y @unina2, it

(Received and accepted May 2003) A b s t r a c t - - T h i s paper deals with a mathematical description of h u m a n feelings (such as hostility, or indifference or love) in the trail of a previous one [1]. The discussion carried out there was acknowledged to be Somewhat oversimplified, so t h a t the present paper aims at deepening the views it relied upon and at widening the application field of the model, by taking into account the character, the tastes, and possibly the past experiences of each individual involved in a social relationship. Even the word "social" is here understood in a wider sense, ranging from the meaning of a "purely intellectual" and "mainly altruistic" relationship to t h a t of a "quite selfish physical (sexual)" one. From a mathematical viewpoint, the description proposed here still leads to a system of nonlinear integrodifferential stochastic equations. Existence and uniqueness of its solutions, as well as their stability and the possible occurrence of strong instability effects, are not discussed here, and postponed to a future work. The main feature of the paper lies in the explicit introduction of additional parameters related to behaviour, external influences of psychologic as well as practical kind, and the consideration of self-feelings. This leads to introduce and discuss additional relations t h a t are to be considered as "constitutive laws" to identify "psychological types" or "classes" of individuals. (~) 2005 Elsevier Ltd. All rights reserved.

Keywords--Dynamics of feelings, Stochastic dynamics, Stochastic integrodifferential equations.

1. I N T R O D U C T I O N As already observed in several recent papers and books [2-9], the development of statistical methods, based on the logical and mathematical tools provided by the theory of probability, has enabled mathematics to contribute formal and/or quantitative descriptions also for classes of phenomena falling in the realm of sciences traditionally lacking procedures of precise measurement, or for which even the notion of a "measure" seems to be quite meaningless. Accordingly, mathematical modeling has nowadays entered almost all the descriptions of world that aim at being considered as scientific, giving them a till now not recognized but certainly powerful tool to reach commonly accepted or at least comparable results, so that widely shared predictions about future phenomena can be drawn and compared with the outcomes of further experiences. As samples of the role played by mathematical modeling in natural as well as in the so-called human sciences, we may quote the mathematical models proposed in connection with special problems of geology (see, e.g., [10-12]), biology [3,5,13], traffic flow [14], and also economy and 0895-7177/05/$ - see front m a t t e r (~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2003.05.021

Typeset by fl,j ~ - T E X

588

B. CARBONARO AND C. GIORDANO

social sciences [15] (see all the above-quoted items for more complete references). And a young but already acknowledged as a promising tool for special psychologic and literary studies is "love dynamics", developed by using the schemes and the language of dynamical systems (see [1]). Now, in a previous paper [7], one of us has proposed an attempt to apply mathematical modeling to the description of dynamical features of human feelings (ranging from hate to indifference to love). The language and the logic of the description proposed there were the ones developed in a series of recent papers [2-7], dealing with a class of mathematical models called generalised kinetic (Boltzmann) models for the dynamics of a population with a large number of palrwise interacting individuals (see also [16]). As it was pointed out in the concluding remarks of the final section of that paper, the scheme adopted there was oversimplified under some respects. To explain this, let us recall in a few words the main features of the model. Roughly speaking, it is built on the assumption that to each human being a one-dimensional "state space" (which can be simply identified with the real axis ]~) may be associated, describing the "level" (a suitable measure) of its feelings towards any other human being (or even towards a nonhuman creature, or a political party, or a philosophical system, or a soccer team, and so on). It is agreed that a positive value of the state variable denotes positive feelings (sympathy, attraction, love), while a negative value denotes a corresponding degree of rejection; conventionally, the point 0 denotes the rather abstract state of complete indifference. Needless to say, the intensity of attraction or of rejection grows with the absolute value of the state variable. It is acknowledged that the feelings of an individual are not sharply identified by its behaviour, and above all the evolution of such feelings cannot be thought of as uniquely determined by their actual "value" and the interactions of the individual with their object, since it depends on several unobserved external conditions. Accordingly, the state of an individual is assumed to be not identified by a precise value of the coordinate in the state space, but by a probability density on R (a "state probability density"). Furthermore, the state changes due to interactions are ruled by a conditional probability density, the "transition probability density". Then, a forecast of the evolution of the feelings is obtained as a system of integrodifferential equations in the unknown "state probability densities", where the source terms depend on the unknowns via the transition probability densities (the details of this description, suitably modified in view of the developments we wish to propose, will be explained in Section 3). Many questions are raised at once by this kind of mathematical representation. First of all, is it correct to represent the state of feelings of an individual as a point in a one-dimensional space? Should not it be instead more correct to choose a higher-dimensional space? Next, it is quite evident that the transition probability density defines each individual, in that it expresses, at least in a stochastic sense, the characteristic way in which the individual reacts to interactions under assigned conditions. Thus, the transition probability density must be thought of as a "constitutive function", in the same way as the stress tensor in continuum mechanics, when described as a function of strain (see [17]). This suggests to examine with some care whether human beings could be divided into classes (analogous to the "classes of materials" of continuum mechanics), each described by a set of values of suitable characteristic parameters. In this connection, one should also remark that these parameters are in turn likely to change their values with time (thus representing the "growth" of an individuM, i.e., the evolution of its psychological structure). Moreover, this change could also depend on the history of the individual, according to a property to be suitably represented in mathematical terms, which we could call the "memory" of the subject (psychology is probably the context in which speaking about individuals with or without memory seems most appropriate). According to these preliminary remarks, the current paper aims at presenting one further step towards the use of mathematical models in psychology, with special regard to the dynamics of feelings. We hope to improve the description outlined in the previous paper [7], to enhance the generality and flexibility of the model, as well as the amount of information it could help psychologists to manage. Of course, only these latter may judge whether the model will turn

A Second Step

589

out to be useful in view of their aims. Still, a number of strictly statistical questions are raised, connected with the criteria, the methods, and the mathematical tools for the validation (or the rejection) of the model. As in the previous paper [7] we can only express our hope that professional psychologists could deem this paper worthy to be read, and find in it an effective tool to predict psychological evolutions. Notation

Throughout the paper, Latin letters will denote real numbers, either variable or fixed (but otherwise unspecified) constants, according to what will result from the context; upper-case Latin letters will be also used, at need, to denote sets. Boldface italic (either small or capital)letters will be used to denote vector variables. Lower-case Greek letters, unless otherwise explicitly stated, will be reserved to denote functions (in particular, probability density functions). A probability measure on the set R of real numbers will be always denoted by the boldface capital letter P, possibly affected by one or more indexes. Sans-s~rif upper-case letters (I, G, L, etc.) will stand for functionals or operators, i.e., mappings of the type F:S

>~

F : $1

> S~,

or of the type where S, $1, and $2 are sets (spaces) of functions. As usual, the symbol C(II) will denote the space of all continuous functions on the interval I C R, and the space of all functions that are continuous on ]I C_R together with their derivatives up to the order n inclusive will be referred to as C~(]I). For future purposes, it will be also convenient to use the space C~(I~) of the functions that are continuous on IR together with all their derivatives of any order, and the space C ~ (R) of the functions of C ~ (R) that have a compact support in R. As well known, this means that ~o e C ~ ( ~ ) if and only if ~ E C~(N[) and the set {x C R I ~(x) ~ 0) is contained in a bounded closed interval of R. We shall denote by 7) the space of distributions on R, i.e., the space of all continuous linear functionals F : ~o E C~(I~) - ~ (F, ~o) E ]R (for more details, the reader is referred to [18] and to any good textbook about functional analysis). In order to study some approximations of the model we want to propose here, and to discuss the meaning of some of its applications, we shall need in the sequel to use sometimes the Dirac ~ distribution at a point xo, defined by the condition -xo),

=

v

• cg

Consider now the space LI(A) of all functions ¢ defined on R for which I1¢111 - / f f

I¢(x)l dx < +c~,

and the space L~oc(R) of all functions • defined on R such that b

f I¢(x)ldx < +oe, for any - c ~ < a < b < +oc. It is well known that, for any ¢ • L~oc(R), the condition

(¢,

=

dx,

• Cg (R),

defines a continuous linear functional (distribution) on C ~ (R), still denoted by ¢, so that L~o~(R) turns out to be isomorphic to a subspace of :D. We shall occasionally use the notation LII(]~) for

590

B. CARBONARO AND C. GIOIIDANO

the set {¢ e LI(I~) I ]1¢[11 = 1}. And since L,I(R) C LI(R) C L~oc(R) and also Ck(]~) C L~o¢(X) for any k E l~, all the functions we shall have to deal with in the sequel will be indifferently treated as functions or as distributions. Finally, we shall denote by the symbol C k (0, T; S(R)) (where S(R) is any of the function spaces introduced above) the space of all functions ¢(x, t) which are continuous with their derivatives with respect to time up to the order k inclusive, and whose value at each instant is a function of the variable x belonging to the function space S(~). Some more symbols introduced ad hoc throughout the paper will be explained at their first occurrence.

2. A P R E L I M I N A R Y I N F O R M A L S T A T E M E N T OF T H E P R O B L E M This section is devoted to explain in more details the reasons why the description of feelings outlined in [7] seems to be unsatisfactory and unrealistic. The arguments we shall expose here are, of course, quite heuristic and mainly suggested by pure common sense, since we are not professional psychologists and, accordingly, dare not to propose any technical choice of the parameters describing with sufficient accuracy the relevant aspects of human psychology as well as an exhaustive list of the forms of feelings that may be considered as clearly distinguishable from each other by a psychologist. So, we shall simply collect a number of remarks that can address our research towards the most appropriate formal structure to represent human feelings. REMARK 2.1. Let A and B be two human beings. Of course, the feelings of A towards B may be either positive (suggesting a behaviour of protection or help or of simple search for the presence of B) or negative (leading A to avoid any interaction with B or even to attempt to destroy it). Moreover, they may be easily distinguished by their "magnitude" (this is the word we will use from now on to denote the intensity of feelings). This suggests that any particular feeling could be identified as a value of an ordered variable (and, more precisely, the real variable). But this simple representation is based upon deliberately disregarding the fact that A may nourish many different kinds of feelings towards B, being all positive (or negative) and equally strong. For instance, if A is a man and B is a woman, the feelings of A towards B may consist mainly of the desire of a physical contact (possibly reduced to gestures that can be acknowledged to express appreciation, or to give comfort and protection); but if A is an old professor and B is any one of his young pupils, then the feelings of A towards B will be probably much more altruistic, possibly purely intellectual. A clear perception of this difference should not be forbidden by the fact that altruistic behaviours are often suggested by feelings that, though positive, contain a high percentage of sense of possession. Several more examples could be now exhibited in the same vein (but we avoid to describe them here) to lead to the conclusion that feelings should be divided into kinds, and that each kind should be represented by the real variable: the sequence rejection --+ indifference ~ attraction may be reproduced in the intellectual as well as in the moral as well as in the purely physical sense (and so on: only a psychologist could indicate a plausibly complete list of kinds). The space of feelings as psychological states will be then better represented by an ]~n (n > 1) than by ]~. REMARK 2.2. It is quite obvious that, if we denote by C A B ( X A , XB, YA) the probability density that the feelings of A towards B change from the value YA to the value XA, when A meets B and B is in the "state of feelings" x s towards A, then the form of the function CAB is completely determined by what we could simply call the "character" of A, that is the complex of its psychological features. Nevertheless, we must observe at least two more aspects that are to be taken into account when we want to guess at least the general form of CAB. First, nobody is able to perceive directly the psychological state of another individual; so, what can affect the possible changes of the feelings of A towards B is not the true state of feelings of B towards A, but rather the behaviour of B during an interaction. Accordingly, the dependence of ?/JAB o n x B

A Second Step

591

should be replaced by a dependence on a number of "behavioural parameters" (b~, b ~ , . . . , b~). The values of these parameters will of course express some psychological state of B, so that they must depend on xB. Strictly speaking, we cannot probably expect that a behaviour be produced by a well-identified state, nor that any given state will always produce the same behaviour. Then, the link between the parameters (bB, 1 b2s , . . . ,b~) and the state xB should be expressed by a joint probability density. Nevertheless, for the sake of simplicity, one may assume that the vector bs -~ (b~, b ~ , . . . , b~) can be a prescribed function of xB (with the agreement, of course, that such a function is not invertible). The form of this function will obviously depend on the individual, and the way in which it is modified from an individual to another will be discussed later. In this connection, we want to consider now the second important feature of the function ~)AB, strictly related to its dependence on what we have called the "character" of A. We have two choices about this matter: (a) to take this dependence as a very definition of the function CAB, whose prescription will be considered as a complete characterisation of all the relevant psychological features of A; (b) to assume that the character of A is identified by the values of a number of "character parameters" (c~, c ~ , . . . , c~) (where the value of k is again to be determined by professional psychologists on the basis of experimental studies, maybe with the aid of statistical methods), so t h a t the probability (density) that the state of A changes from YA to XA is affected by both the form of CAB alld the values of the character parameters. In such a way, the form of the function CAB turns out to define a class of individuals, while the values of parameters CA, 1 CA,. 2 .., c~) "select" a particular individual in that class. Once the notion of "character parameters" has been introduced, one cannot help remarking that the link between the behavioural parameters and the states of B should in turn be influenced by the values (c~, c ~ , . . . , c ~ ) o f such parameters for B. Thus, the vector b of behavioural parameters must be a function of the vector cB ~ (c~,c2B,... ,c~) as well as of the state xB. REMARK 2.3. To develop further our discussion about the link between the values of behavioural parameters and the values of the state variables, and in accordance with the informal character of this discussion, we now also need to point out that the behavioural response of any individual must be thought of as depending on two more systems of conditions. First, we have to take into account the state of feelings of the individual towards itself; these will be here referred to as "self-feelings'. It is quite obvious that self-feelings strongly influence not only the behaviour, but also the attraction or the rejection of the considered individual towards another one. For instance, a man who is in love with a woman, when experiencing a behaviour on behalf of her which seems to hurt some aspects of his self-esteem, may react in three very different ways: (a) by simply reviewing his own self-feelings and trying to change to meet new criteria of self-esteem; (b) by stiffening his behaviours, to stop or at least to smoothen the behaviour of the woman, but without changing neither his self-feelings nor his love for her; (c) by a gradual cooling of his love, and a final complete detachment. The choice among responses (a)-(c) will then depend on character parameters. But what we want to communicate is that, to keep its self-feelings at a constant level, an individual may find itself in the need to change either the values of its behavioural parameters or even its feelings towards another individual. This suggests to introduce one more vector parameter s -----(s 1, s 2 , . . . , sS), which will be called the "self-feelings vector", and whose components will be called the "selffeelings parameters", such that both the variables b and x must depend on it (in a sense that will be discussed in the next section).

B. CARBONARO AND C. GIORDANO

592

REMARK 2.4. The second kind of conditions on which the behaviour of an individual must be acknowledged to depend consists of what we could call the "external conditions", grouping the whole set of (moral, social, economic) beliefs and rules it learns from parents, school, friends, and so on, and also the whole set of its feelings towards other individuals that, by habit or social or economic dependence, or some kind of established affection, have a leading role in its life. We cannot but acknowledge that the great deal of marriages forced or prevented by parents or by social or moral conventions are as many examples of behaviours induced by external "pressures". These "pressures" will be represented by a vector e of parameters (el, e 2 , . . . , ep). They depend on the individual, in the sense that different individuals usually have to suffer different kinds of pressures of different strength. It is questionable whether not only the behaviour, but also the very feelings of an individual may be affected by external influences. Though the last word in this connection is to be left to professional psychologists, we suspect this to be the case, so that, for the sake of generality, the state variables will be also assumed to depend on external conditions. This dependence will be expressed in mathematical terms in the next section: at this stage, we confine ourselves to conclude that the behavioural parameters depend on character, self-feelings, external conditions, and state of feelings: bz = bi(ai, si, ei, x i ) for any individual I on consideration. REMARK 2.5. As a final remark, we want to draw the attention of the reader to some features of the parameters c, s, and e introduced in the previous remarks. At a first glance, one could be led to consider them as real data of the problem, to be assigned once and for all at the very definition of the model. But a more careful look at their definition, and above all at their psychological meaning, shows that they must be understood as changing with time. Our everyday experience shows that the character of an individual is strongly influenced (at least for a reasonably long period from its birth) by external "pressures" as well as by the criteria of self-esteem these latter induce in it. Conversely, an individual may be able to modify the external pressures acting on it (by leaving its family or changing its job, say) according to its own criteria of self-esteem and to its character. And it is almost trivial that any human being experiences periodic variations in its self-esteem, depending on external pressures and on the stability of its character. Then, the parameters c, s, and e cannot be simply assigned. What can be assigned, and defines the psychological nature of the individual on consideration, is instead the mutual dependence of these parameters. This dependence may be expressed by a stochastic law, i.e., as a joint probability density function, or by a deterministic law, i.e., by means of a suitable number of constraints of the form

~ ( c , s, e) = 0, or by a mixed law, which, as we shall see in more detail in the sequel, is obtained when the number of deterministic constraints is strictly less than k + s + p and a suitable probability density function is defined on the set of points satisfying the constraints. In any case, we expect that the present character of an individual, as well as the variations of its self-feelings induced by external pressures and the way in which it is able to modify any external influence, depend on its past history. This will require the introduction of mathematical concepts that are well known and widely used in different contexts, and will be discussed in Section 4.

3. A N E W

MATHEMATICAL

DESCRIPTION

OF FEELINGS

This section is devoted to giving a precise definition of the problem we have in mind to tackle, and a description of the mathematical structures we plan to use in order to formulate it in accordance with the mainly informal discussion carried out in the previous section. Our aim is to draw a stochastic model that could enable us to give an at least probabilistic forecast of the evolution of mutual feelings between human individuals. As a first step, we shall confine ourselves to consider couples of individuals, disregarding psychological interactions with

A Second S t e p

593

other individuals. Nevertheless, for the sake of generality, we refer to a system ~ ---- {I1, I 2 , . . . , IN) of N human'individuals as the object to be described by our model. According to Remark 2.1, we shall consider the attitude of an human being toward another one as an n-tuple of ordered variables, each of them conventionally running from - o o (total rejection, or deadly hate) to +c~ (the deepest attraction). Thus, we are led to give the following. DEFINITION 3.1. A state or feelings of each individual li towards another individual Ij iS described by a point uij - ( u 1l j , u i2j , . . . ,u~.) of the space R n, i.e., by the values of n r e a / parameters £ (uij)l<~
V ~ (uij)l
/

~ll

U12

"'"

~IN

U21 •

U22 .

""' *..

U2N •

\ UN1

UN2

"""

~NN

= ( ij)l
of the space R N2n. The parameters uijk will be called, as usual in different context, the state variables of ~. The set R g2n in which the states of ~ take place will be called the state space of ~. REMARK 3.1. It has been already noticed (cf. [7]) that the introduction of measures ui~ is, at this stage, quite conventional. In this connection, it has also been observed that, in our everyday life, we all naturally take into account a number of clues, or signals, to acknowledge whether the feelings toward us of any person we meet are "positive" or "negative". According to Definition 3.1, for any i E {1, 2 , . . . , N ) , the n-dimensional vector uli -- (uil)l<~
U-

81 82

111 t 112

8iN

UN

.

,

where, for any i E {1, 2 , . . . , N}, ui - (u~)a<5
U -~-

ulj u2j

ulj u2j

UNj

tlNj ]

.

.

•

594

B. CARBONARO AND C. GIORDANO

In connection with Remark 3.1, and recalling Remark 2.1 of [7], we cannot but stress again that the feelings of an individual towards another can rather seldom be given a sharp measure: the feelings of a human being towards anybody else are in general a mixture of what we could call "pure" feelings. On the other hand, the feelings of an individual towards another are in general inferred by a number of experienced behaviours, that are often in contradiction to each other, so that our knowledge of the feelings originating them is always uncertain. So, we treat each state coordinate as a random variable, identified by a probability density function (p.d.f.)

R

e [0,

(3.1)

such that the probability Pijt(a, b) that the state coordinate corresponding to indexes i, j, and l takes a value in the real interval [a, b] is given by the well-known relation Pqt(a,b) =

7ri~ (u~j) dufj.

(3.2)

REMARK 3.2. As is well known, relation (3.2) implies

f-+5 (Wj)d% --Ik,'Jlll= 1,

(3.3)

so that ~ e L~(I~) or, more precisely, e

-1

(Wj) --IIf;Jllx IfffJ(Wj)I with f~ e L~(R). REMARK 3.3. In view of the psychologic significance of variables uqe, it seems quite reasonable to assume that, for any i and j,

lim Ja[b 7rij a,b-.-~+c~

a,

~b

and, for the sake of simplicity, this assumption will be replaced by the stronger requirement that

lim

=

o

with a suitable order. REMARK 3.4. The integration variable in the integral at the right-hand side of relation (3.2) should have been denoted by a nonindexed symbol, for two reasons: (a) to stress that all the state coordinates define probability distributions on the same set 1~; (b) to introduce a careful distinction between each random coordinate variable ui~ and its realizations (possible values). Nevertheless, from now on to the end of the paper, in order to avoid an excessive proliferation of symbols, which of course would be only confusing, we shall never use any symbol to denote the random variables we shall have to deal with: only possible realizations will be taken into account, and each of them will be always distinguished by the same indexes of the related variable, so that u~ej will always stand for a possible value of the random variable "state coordinate corresponding to indexes i, j, and l". According to our stochastic viewpoint, the measure of each mode of feelings of individual Ii towards individual Ij has the expected value

E,~

---- f + 5

e e UijTrij

(uej)

d u iej

(3.4)

(of course, 7r~ejis assumed to be defined in such a way that the integral at right-hand side of (3.4) has a finite vMue ). In view of the meaning we have in mind for the variables u~j, we are allowed to give the following definitions.

A Second Step

595

DEFINITION 3.4. For any g, the ~th mode of feelings of individual I~ towards individual Ij is mainly attraction (respectively, mainly rejection) if and only if the median of the state coordinate corresponding to indexes i, j, and t is positive (respectively, negative), i.e.,

F

7ri~ (uf~) d uiej <

oo

Jo

7riej (ufj) d u,~,

respectively~

jo

T(ij oo

7~ij

DEFINITION 3.5. For any ~, the ith mode of feelings of individual Ii towards individual mainly indifference if and only if

F

7Tij

-~"

oo

J0

"ffij

Ijis

duij"

of

DEFINITION 3.6. The ~ t h mode of feelings individual Ii towards individual Ij is attraction, rejection, or indifference in the mean according to whether E~j is positive, negative, or zero. Till now, we have discussed each mode of feelings separately. But a complete and synthetic picture of the psychologic attitude of an individual towards another must be obtained from a kind of comparison between the different modes of feelings. To this aim, we introduce one more random variable. DEFINITION 3.7. The arithmetic mean ?%

U i j ---- n A..a g=l

93

w~11be said the psychologic attitude of individual Ii towards individual Ij. Of course, the random variable fiij defines a probability density function ~rij, whose values, as is well known from the theory of probability, are linked to the ones of joint probability densities of all the modes of feelings. Once this density has been introduced, we are allowed to give the following. DEFINITION 3.8. The psychologic attitude of individual Ii towards individual Ij will be said to be a global attraction, global rejection, or global indifference in the mean according to whether

n

£=1

is positive, negative, or zero. It will be said to be mainly global attraction, mainly global rejection, or mainly global indifference according to whether

£=1

~_

+

7rij

u~. zj

du~j >

i~_7rij

z3. u ~.

dulj

+ 7rlj

or ~"

i.

+ qriJ

u l.. ~j

(D) 93. u ~.

du~j,

duij,

596

B. CAIIBONARO AND C. GIORDANO

where

n

~ - = { uij E R n

}

e=l

~,+ = {u~j

E R '~

__o

.

REMARK 3.5. As already observed in [1], the dispersion indexes, i.e., for m > 1, the moments [M'~]~eJ = f ? 5

( u ~ ) " 7r~e~(u[~) du[j

have a great statistical relevance, in that they give us additional information about the character of each mode of feelings of Ii towards Ij (e.g., the possible simultaneous presencc in a statistical sense--of aspects of strong rejection together with aspects of deep attraction. In particular, the second moment ( ( , j ) ) d(u~j)

(3.5)

may be taken as a measure of the "strength of reactivity" of individual Ii to the aspects of Ij switching the ~th mode of feelings. So, we are led to give some more definitions. DEFINITION 3.9. The expected value (3.5) will be said the energy of the ~th mode of feelings of individual I~ towards individual Ij. DEFINITION 3 . 1 0 .

The expected value 12 E ui~

E =

ue~j2

= E

e=l

will be said the total energy of feelings of individual I~ towards individual b. DEFINITION 3.11. The expected value =

will be said the mean energy of feelings of individual Ii towards individual Ij. REMARK 3.6. These definitions are motivated by remarking that in only one (unrealistic) case no psychologic energy is spent, namely, when an individual is completely (and "deterministically") indifferent to another. In other words, it seems reasonable to define an "energy" in such a way that its value is as higher as stronger are the feelings of an individual towards somebody or something. This should be thought of as an amount of energy to be spent in psychologic life. To conclude the statistical picture of the system Z, we state the following. STATISTICAL AXIOM 3.1. For any pair of triples (i, j, l), (i', f , ~t), such that at least one of the conditions i ¢ i', j 5~ f , ~ ¢ ~', (3.6)

is satisfied, the random variables ui~ and u~j, are statistically independent. REMARK 3.7. The above axiom simply means that the probability De e' l- b;c,d) that the modes of feelings corresponding to indexes (i,j, l) and ( i t , f , £~) take their values in the intervals (a, b) and (c, d), respectively, is given by the relation

P(a,b;c,d) =

f

lr,I (ufj) dufj

fL"

f"()

' i, uef, e' 7ri,

du~,'i,

JC

"(")

dui,i,,

A Second Step

597

for any pair of triples (i, j, l), (i', j', l') satisfying at least one of conditions (3.6). As a matter of fact, this condition should be thoroughly discussed, in connection with the influence the feelings of an individual may have on the feelings of another individual in relation with it, and above all with the influence a mode of feelings (e.g., intellectual admiration) may have on another mode of feelings (e.g., physical attraction) of the same individual. To solve this problem we need to take into account the variation of feelings with time. For the moment, in view of the above assumption, we feel allowed to introduce the joint probability density of all modes of feelings of an individual I~ towards another individual Ij simply as n

£=1

and the joint probability density of the feelings of Ii towards all the other individuals of system simply as N ~ri(ul) = H ~rij(uij), j=l j#i where ui is of course the two-dimensional matrix (u £ij)l<_£<_n,l~j
4. K I N E M A T I C S

OF F E E L I N G S

As is well known, a mathematical model for a system of empirical facts aims at being not only descriptive, but also, and mainly, normative. More precisely, we do not simply introduce a particular quantitative language to describe known facts in terms of arbitrarily invented properties, but aim at linking these properties by rules that could enable us to predict--from facts observed now and here---other facts that will happen in future times and elsewhere. A mathematical model for feelings is not an exception in this regard: at its very startpoint, it is acknowledged that the mutual feelings of individuals of a known system should vary in time according to either external circumstances or internal psychologic changes. In other words, the feelings between the individuals of a system always undergoes an evolution, and our aim is to predict such evolution in at least a finite time interval. Such an evolution of mutual feelings of a system E in a time interval [0, T], will be of course mathematically described by the N2n-tuple of random functions, or time-continuous stochastic processes

ui~: t E [0,T] --~ ufj(t) e ~, that to each instant t associate the N2n random state variables of the system at t. This means that to each t there corresponds the N2n-tuple (Trij(.,t)) of p.d.f, such that, for any couple (a, b) E ]~2, ~a b ~

Pfj(a,b) =

~,j (%,t) duff.

Since this is the whole information we have at our disposal in connection with both the experimental determination of the "initial" state of Z at a given instant and the theoretical deduction of the future states of the system, it is quite evident that our predictional model cannot be but stochastic and our aim will be achieved once the functions

7ri~: (ui~,t) E R x [O,T] ~

Iri~ (u~j,t) E [O,-t-cx~)

(4.1)

will have been determined. Therefore, by an evolution of mutual feelings of a system ~ in a time interval [0, T] we simply mean the set of functions (4.1), which will be said to be the components of the evolution. For this kind of psychological evolution a regularity problem arises analogous to the ones to be tackled in every mathematical model. In this connection, we give the following definitions.

B. CARBONARO AND C. GIORDANO

598

DEFINITION 4.1. For any triple (i,j,£), the corresponding component of the evolution will be

said to be regular in the time interval [0, T] if, for any ui~ E R, the partial derivative (rate of change of ~rij ~ (u~j, .) in time) is defined at each instant t E [0, T]. DEFINITION 4.2. Evolution (4.1) will be said to be regular in the time interval [0, T] if all of its

components are regular. As already done in [7], we agree to consider only regular evolutions with continuous time derivative. An interesting analogy may be observed, at least on a purely kinematical ground, between the model we are drawing and a particular problem that could arise in the framework of continuum mechanics. In this connection, consider a system of N2n continuous one-dimensional unbounded bodies ~Bi (i = 1, 2 , . . . , N2n), each constrained to shift on the corresponding coordinate axis. Let mi be the (finite) mass of ~Bi, and P~(Yi) its density (mass distribution) in a prescribed configuration. Also, let the motion of ~Bi on the i th axis be described by the one-parameter family xi = xi(yi,t) of invertible functions from R to R. Then, at each instant t a mass distribution pi(xi,t) is defined on the i th axis in such a way that

F~pi(x~,t)dxi

= fc~

oo

P*(Yi) dyi = mi,

oo

and, more in general, ~ b P~(Yi) dyi = f ~ ( b , t ) pi(xl, t) dxi, J ~ (a,t)

V (a, b) e ]~2.

As is well known, this means .

dxi

Now, if the ratios mi-1 Pi(Yi) . . . and m~lpi(xi,t) = 7ri(xi,t) are interpreted as the -- 7ri(Y=) above-introduced N2n probability densities, and these are assumed to be known, then the last relations may be used as independent ordinary differential equations to determine the functions xl = xi(yi, t), thus giving a different picture of the evolution of each mode of feelings. This kind of evolution will be said the semideterministic shift of feelings. Of course, the probability densities 7rij t (xij l , t) must be determined as solutions of a system of plausible evolution laws. These have been proposed in [7]. In the next section, we shall recall their general form. 5. G E N E R A L

DYNAMICS

OF

FEELINGS

As in [7], we shall assume throughout the paper that any change in (the probability distribution of) each mode of feelings of an individual towards another cannot but be due to some kind of interactions not only between the two individuals, but also of each individual with itself and with other individuals. This can seem a rather obvious assumption, b u t - - a t a deeper glance--it shows to be based on an assumption of "inertia", quite analogous to all the inertia assumptions the various branches of mechanics rely upon, i.e., the hypothesis that the feelings of a lonely individual towards any other individual must remain unchanged. This should of course be carefully discussed from a psychological viewpoint, but at this stage may be accepted as a mathematical startpoint. In mathematical terms, our assumption is expressed by the following. AXIOM 5.1. The rate of change of each ~r~j(uijl , .) in time is at each instant a function of the probability density functions ~r~k of a11 the state variables U~hk of system E

Ot (ui~'t) = J[j (u~j,t) - Ji~ (~r,, ~rj; uij~ , t) ,

(5.1)

A Second Step

599

for any (i,j) • {1,... , N } 2, where---for the sake of simplicity--we have set (for any h • {1,2,... , Y } ) ~rh =- ( "1"(hr k ( ' , ~ , ) ) l ~ k ~ _ N , l < r < n . -- -In other words, for any (i, j, g) E {1, 2 , . . . , N} 2 × { 1 , . . . , n}, a two-parameters family

of functionals

(

:

• [c 1 (0, T; q 1 (R))]

2Nn

( fj,t) •

exists such that the rate of change of the probability density function ~ri~j at the state ufj and at instant t equals the value of the corresponding functional J[j( .; uij, ~ t), evaluated at all the probability density functions of the state coordinates of the system at the same instant t. The right-hand side of equation (5.1) may be also envisaged as an operator I~ : (~,,Trj) • [C 1 (O,T;L~(~))] 2N~ ----* I ~ j ( ~ , , ~ j ) = J~ • C ( O , T ; L ~ ( R ) ) , which will be said the interaction operator of individuals Ii and Ij with respect to the gth mode of feelings. In the framework of this interpretation, we shall write

Axiom 5.1 states that the rate of variation of the feelings of each individual I~ towards another individual It is ruled at each instant t not only by the mutual feelings between Ii and lj at the same time, but also by the feelings between I~ and Ij and all the other members of the system, and the self-feelings of Ii and Ij. In order to give this statement a meaning such as to enable us to draw (probabilistic) predictions about the future states of C, i.e., the future reciprocal feelings of the members of the system, we need to give explicit constitutive laws, expressing the functional form of the interaction operator. This is a task which will be tackled in the next section. For the moment, we want to recall a particular case which will prove interesting in itself and to test a simplified version of the mathematical scheme, and is described by the following two definitions. DEFINITION 5.1. Let H = { h i , . . . , h M } ( M < N ) be any subset of{l, 2 , . . . , N}. The subsystem EH = {Ih~ , Ih~, . . . , IhM } Of ~ will be said to be an isolated subsystem i[ Ot

'

'

for any (i, j, g) • H 2 x { 1 , . . . , n}. DEFINITION 5.2. For any hi, h2 in {1,2,... ,N}, the subsystem ~(h~,h2) = {Ih~,Ih{} Of~ Wi1I be said to be an isolated couple if

at

' '

= Jh~h2 =

?rh~h~ i,je{1,2},l
'

(~ = 1, 2 , . . . , n).

(5.3)

Jhzhl

REMARK 5.1. As a matter of fact, according to what has been already observed in [7], the notion of an isolated system is rather redundant. Whenever a system of individuals is defined, it may be taken as "isolated", provided it contains a sufficient number of individuals. The notion of "isolated subsystem" is in fact a relative notion, connected to the possibility of excluding a number of members of a too crowded initial system. REMARK 5.0.. Much more interesting will prove the notion of an "isolated couple". Though rather unrealistic, this notion will be of help to reduce the mathematical and numerical complexity of the applications of the present description. This will be of special relevance in the current context, where a number of additional parameters will be used to obtain a detailed description of the differences between the individuals in consideration. Thus, we give up realism about the nature of the system, to obtain the most realistic description about the psychologic profiles of individuals.

600

B. CARBONARO AND C. GIORDANO

6. T H E

AXIOMS

OF

DYNAMICS

OF

FEELINGS

The task we want to tackle in the present as well as in the next section is the exploitation of the dependence of functionals J~j on all of its arguments 7rthk, by means of a critical discussion of the psychologic conditions influencing the change of the state of an individual with respect to another. In this connection, we start with the following general axiom. AXIOM 6.1. For each (i,j,£) e {1,... ,N} x {1,... ,N} x {1,... ,n}, the interaction operator

. j ) is the di ere.ce

two other operators

and

I. symbols,

where 1

G,~:(~r~,~r~)E[CI(O,T;LI(~))]

2Nn

~

)Go(1r~,lr~)=GoeC(O,T;L~(R)

),

k~j : (Tvi, Trj) e [C 1 (O,T;L~(~))] 2N" ----, I_ij(Iri,lrj)e = LU~ e C (0,T; L~(]~)) . For any t E R and t'or any assigned value uu, ~ the walue G U(uU, ~ t) is the rate at which the probability of a sut~ciently small interval surrounding ui~ is raised at instant t by interaction, while the value L~ (u~j, t) is the rate at which the probability of a sufficiently sinai! interval surrounding u~j is lowered at instant t by the same interaction. For this reason, Giej will be said the interaction gain operator and [-i~ will be said the interaction loss operator of individuals Ii and Ij with respect to the ~th mode of feelings. We may obviously write

for any pair (u~j, t). Roughly speaking, the operators just introduced have the same meaning as "force" terms. What we have to do now is to exploit the laws of force according to which these terms depend on the variables (probability density functions) they modify. The laws of dependence we are going to propose are under any respect the analogues of "constitutive relations" which in all branches of mechanics allow to trace unknown interaction terms (such as the stress tensor in continuum mechanics) back to the main unknown variables describing (in a suitable sense) the state of the system. The final form of these relations will be achieved by a number of steps, the first of which is the following. AXIOM 6.2. Each individual Ih of system ~. is identified at each instant by a (joint) probability density function Ph : (bh, Oh, eh, 8h, Uh, t) ~ R r+k+p+Nn+l ~ p h ( b h , Ch, eh, Sh, Uh, t) C [0, -~-OO), where the r-dimensional vector bh is said to be the behaviour vector, the k-dimensional vector Ch is said to be the character vector, and the p-dimensional vector e h is sMd to be the external pressure vector (recall that the n-dimensional vector sh is said the state of self-feelings). The r components (blh, . . . , b~h) of bh are said to be behavioural parameters, the k components (Clh, . . . , C~h) of Oh axe said to be character parameters, and the p components (elh,..., erh) of eh axe said to be external pressures. The meaning and the sense of this axiom require now some explanations. According to what has been laid out in Section 2, any individual is supposed to behave, in the same situation, in different ways obviously depending on its aptitudes (to be romantic, an individual who knows how to play the piano will play, while a professor of literature who has never touched any musical instrument will probably quote a poem), on the external influences acting on it (physical, of course, such as the presence or the absence of a piano, but also any kind of sickness and the police at door, but mainly psychological: education, moral convictions, fear to be hurt in selfesteem, self-conservation instinct, etc.), as well as on its self-feelings and on its feelings towards

A Second Step

601

any other individual of the system. Several examples could be exhibited to illustrate more clearly the different ways the above-introduced parameters, when properly interpreted, could in fact influence the behaviour of an individual, but this is hoped to be in the future the task of a much more lengthy and complete work on the subject tackled here and in [7], which should also be developed with the help of a psychologist. For the moment, we confine ourselves to these few remarks, simply adding that there is no loss of generality in considering only feelings towards other members of the system, since (as we plan to show in more detail in a future paper) any system may be reasonably taken as isolated, unless we consider what can be called "birth" and "death" phenomena (an individual of E meets by chance an individual which does not belong to the system and starts loving--or hating--it for some reason, and introduces it to a number of other members of E; or a member of the system actually dies or simply leaves the system and, what is more important, is forgotten by all the remaining individuals of E). We conclude by only one more remark about our use of the notion of "character". By this word one usually means (in rational mechanics, with particular reference to continuous media, as well as in our everyday life) a particular link between stimulus and response, which is typical of a single kind of material system: corresponding to the same strain, a viscous fluid produces a stress field different from the one produced by a perfect fluid under the same strain; when tackling a problem, an aggressive man behaves in a very different way with respect to a mild, cautious man. And the fact that the joint probability densities Ph are indexed (i.e., each individual has in principle a different density function) seems in itself sufficient to express this feature, so that the introduction of character parameters seems to be redundant. But, at a deeper sight, it is clear that they allow to take into account subtle distinctions between otherwise very similar correspondences stimulus-response. Some examples to show how reasonable these distinctions may be will be discussed elsewhere. Before going on, we need now to introduce a number of further definitions and assumptions that will enable us to handle the parameters introduced by Axiom 6.2, and to link them with the state variables, as to obtain a closed system of evolution laws for their absolute probability densities. In this connection, we must indeed observe first of all that the probability densities 7r~j are in fact absolute probability densities, that is to say independent of any additional information about individuals Ii and Ij. But we have just introduced a number of parameters that actually give additional information about the individuals of the system, which we consider as relevant to obtain a realistic and accurate description of the evolution of mutual feelings of individuals in E. Therefore, we shall have often to deal not only with the conditional probability densities of the state variables as well as of behavioural parameters, character parameters, and external pressures, but also with the marginal densities associated with the basic constitutive density Ph. In connection with these latter ones, for any h E {1, 2 , . . . , N}, we shall set

(6.2a)

qh(Ch, eh, Sh, Uh; t) = _/~r ph(bh, Ch, eh, 8h, Uh; t) dbh, qhk(Ch, eh, 8h, Vhk; t) = f

JR(N--2)n

qh(Ch, eh, Sh, Uh; t) dVhk ,

wh ( bh, Oh, eh; t) -~ ~t¢~ ph (bh' c'h' eh, 8h, Uh; t) dshdUh-

~/k # h,

(6.2b) (6.2c)

For convenience, we also introduce, for any triple (i,j, g) C { 1 , . . . , N} × { 1 , . . . , N} × {1,..., n} g (with i # j) and for any assigned vq E ](, the (n - 1)-dimensional hyperplane

Moreover, we set D,j

(vS)

We are now in a position to assert some further basic assumptions.

B.

602

CARBONARO AND C. GIORDANO

AXIOM 6.3. For any i , j C {1, 2 , . . . , N}, there exist: (1) a continuous, nonnegative, and bounded function a j : (u~, ci, e~, s~) ~ ]~(N-1),~ x ~k x R p x R ~

, ~j(u~, c-,, e~, s,) ~ [0, +~¢),

(6.3)

called self-interaction rate for individual Ii with respect to its feelings towards individual Ij ; (2) a continuous, nonnegative, and bounded function , , j : (u,,

sj)

o ×

×

×

(6

- - , ~j(u~, c-~, e~, s,, uj, c-j, ej, sj) e [0, +o~),

called interaction rate of individuals I, and Ij, (3) n continuous, nonnegative functions R" ×

~ij :

×

×

~,j (y,~ I~j, ~,, e,, s,) C [0, +co)

×

(I < ~ < n)

(6.5)

called self-interaction transition probability densities of individual Ii with respect to its feelings towards individual lj, (4) n continuous, nonnegative functions

~)i~ : (llij,ci, ei, si, bj,cJ,eJ,Yi~) E ~2(n+k+p)+r-bX

(1 <

' ¢,~ (yi~j [ u O, ei, ei, si, bj, cj, ej) C [0, +oo),

£ < n)

(6.6)

called interaction transition probability densities of individual Ii with respect to its feelings towards individual Ij; such that

xq, (c-,, e~, s,, vi; t) dc, de~ ds, dv~ (6.7)

+ [ 0,j (v,, c-,, e,, s~, vj, cj, ej, ,-qj) d~ 2(kq-pq-Nn)

×py (bj, c-j, ej, sj, vj; t) dvi dc, de, dsi dvj dcj dej dsj dbj, and

D

£

£

× q, (c-,, e,, sl, u,; t) dui dci de, dsidve~j + /Di~

(~,~) ×~+~+(~+1)~+.+~

l]ij (U/, C.i, el, 8,, Uj, C-j, e j, 8j)

(6.8)

xpi (bj, cj, ej, sj, uj; t) du~ dc, dcj dei de~ duj dsi dsj dbj dv~j. Relations (6.7),(6.8) are of course quite technical and rather hard to be interpreted; as a matter of fact, as already remarked in [7], Axiom 6.3 as a whole is a very formal expression of a number of reasonable heuristic assumptions that now require to be explained. We shall now try to illustrate the psychological meaning of the functions there involved.

A Second Step

603

REMARK 6.1. The functions uij, ~Tq are interaction frequencies (with dimension [T]-I), i.e., the former is the number of interactions (per unit time) of individual Ii with itself in connection with its feelings towards Ij (in other words, it is the number of times in which li thinks about its feelings towards Ij), while the latter is the number of interactions (per unit time) between individuals Ii and Ij. Of course, they are to be meant as mean frequencies. Their arguments express the psychological features that are supposed to influence the number of interactions: let us comment on these influences on the frequency vii, which will be called self-interaction frequency. To start with, pq is taken depending on the whole vector vi, for it is reasonable to assume that Ii will meditate about its own feelings towards Ij more or less according to its feelings not only towards Ij, but also towards all the other members of the system, in the sense that a strong affection (either positive or negative) towards another individual may have the effect to distract the mind of Ii from Ij; in the same way, a superficial character will lead Ii to very little reflections about its feelings toward whoever; as far as ei is concerned, we may for instance observe that external pressures of moral type could prevent Ii from thinking of its hate toward (say) a brother; and analogously~ a little self-estimate could exclude any hope (hence any thought) about a love. This stated, it seems quite obvious that two individuals may decide to meet or not to meet according to the same influences affecting the self-interactions of each of them. And, provided they meet, this does not necessarily mean that they interact at a psychological level. If at least one of them is distracted by thoughts about its own self-acceptance or about other individuals, then the mere physical encounter can be not an interaction (or rather, it is an interaction in a sense we shall discuss later). And these few remarks could be of help to understand the choice of the arguments of function ~q. REMARK 6.2. As already suggested in [7], in view of their meaning of frequencies, the interaction rates are reasonably assumed to be positive and bounded. REMARK 6.3. In view of their meaning, for any i, j E {1, 2 , . . . , N} the functions 7/ij and yji were assumed in [7] to satisfy the relation

This relation may be still accepted in the present context, but its meaning must be carefully discussed. As a matter of fact, it depends on what we actually mean by an "interaction" between two individuals. Once the reflections of each of them about the behaviours and possible feelings of the other have been excluded and classified as self-interactions, what remains are physical encounters and exchanges of words and gestures. But it is a matter of discussion whether any dialogue between two individuals is perceived as an interaction by both of them. Assume an individual Ii to be talking with an individual Ij about some very serious topic, e.g., about reciprocal behaviours. Of course, we may assume that Ii perceives such a talk as an effective interaction, but Ij may be bored by the talk and lost after different thoughts. This absent-minded condition is in turn perceived by an interaction by Ii, whose psychologic condition will be influenced by the behaviour of Ij, but according to the feelings and the psychologic conditions of Ij the whole encounter is not an interaction with I~. In conclusion, this is a case in which ~ j should be acknowledged to count one interaction more than yj~. REMARK 6.4. Now, strictly speaking, the above-introduced functions ~ij(Y~j~ e [ u~j, ei, e~, si), and ¢ij~ (Yijt l uij, ci, el, si, bj, cj, e j) are conditional probability densities, namely, for any a, b E R, the integral P,j,(a, b

=

L

expresses the probability that, after a self-interaction, the £th mode of feelings of individual I~ towards individual Iy has a value in the interval (a, b), provided before the self-interaction its state of feelings, its character, the external pressures acting on it, and the state of its self-feelings

B.

604

CARBONARO AND

C.

GIORDANO

were measured by the (vector) values uij, ci, ei, si; analogously, the integral

Pul(a, b luij , ci, ei,ss, bj,cj,ej) =

f

Cs~ (Yi~ l uij, ci, es, si, bj , cj,ej) d Y i~j

expresses the probability that, after an interaction with Ij, the £th mode of feelings of individual Is towards individual lj has a value in the interval (a, b), provided before the interaction its state of feelings, its character, the external pressures acting on it, the state of its self-feelings, the character of lj, and the external pressures acting on the latter, were measured by the (vector) values usj, cs, es, ss, cj, ej, and the behaviour of Ij- during the interaction is measured by the vector bj. So, each function ~U~(Ysj~ I usj, ci, es, s~) is called a self-interaction transition probability density and each function ¢~j(Yij ~ I usj, c~, e~, ss, bj, cj, ej) is called an interaction transition probability

density. REMARK 6.5. The dependence of each function Cs~ on the behaviour of the object Ij of possibly changing feelings instead of its state of feelings towards Is deserves some words of explanation. The observation on which is based the choice of such a dependence is simply that the external pressures acting on an individual and its character features may be inferred from a sufficient experience of its environment, of the ideas expressed by other individuals having influence on it, and of its behaviour in situations that may be taken as easy to be checked and to be described in details. The same is not true for the feelings of an individual towards anyone, that can only be smelled, or conjectured, on the basis of its behaviour. The interpretation of behaviours is a matter of a purely probabilistic reasoning, as we shall discuss at a wider extent in the next section. For the moment, we should only add that the feelings of an individual Ii towards another individual Ij are very seldom (and slightly) modified by the feelings of Ij towards I~. They are much more influenced by the behaviour of Ij and by a certain knowledge of its character. A man does not stop loving a woman who does not love him, not just because she does not love him. He stops loving her only when her behaviour towards him is in conflict with his self-estimate, thus leading him to suspect her character to be at least partly psychologically cruel. REMARK 6.6. In view of the meaning of functions ~tj and Cs~, it is quite clear that [G~j(Tri, ~rj)] • (u~.j, t) is a term which measures the probability (density) that Is falls into the value u~ of the / th mode of its feelings towards Ij after a (possible) interaction at instant t. Thus, it has the effect of increasing the denmty " ~rij ~ (uu, t), so that it is called the gain term or the strengthening interaction

probability density. REMARK 6.7. It is also obvious that [l_~j(~r~, 7rj)](u~j, t) is a term which measures the probability (density) that the l th mode of feelings of Ii towards Ij leaves the value us~. after a (possible) interaction at instant t. It will then decrease the density ~rsj(usj ~ , t), and will be called the loss term or the weakening interaction probability density. REMARK 6.8. Only for the sake of completeness, we want to recall some language conventions. Functions (6.5),(6.6) act on suitable systems of realizations of random variables. Now let us fix our attention on functions (6.6). If, for any pair (I~, b) of individuals and for any ~ E {1,..., n}, the dependence of the output value of the ~th mode of feelings of Is towards Ij on the input variables is deterministic, i.e., N2n functions Yij --0jlsj (uij, cs, e s, si, bj, c j, ej ) are assigned, then functions (6.6) act only on the input random variables, and the evolution is called simply stochastic, while, in the general case, it is called doubly stochastic. It is quite obvious that, when the output state is assumed to be a deterministic consequence of the input variables, then all the functions ¢ ~ must have the functional form of a Dirac delta distribution

A Second Step

605

REMARK 6.9. The presence of the integrals related to self-interactions in equations (6.7),(6.8) cleaxly corresponds to a stochastic description of the inner evolution of each individual. Assume instead that such an inner evolution is deterministic for at least one of individuals Ii and B, i.e., for any t, a one-to-one mapping ~. e,,e,, ~0 = ~j£ u~ = ~,~,, ( U £,~, u,~,

£ ~,,e,, (u;~,£ u,~,

~,,t)

(6.9)

from R onto itself is given (just for instance) for individual Ii, so that the values of the state variable ui~ may be envisaged as the images at instant t of the values of a reference state variable U~. Then, assuming the function ~ j to be continuous with at least its first time-derivative with respect to both of its arguments, we may define an inner evolution rate v~j for individual Ii, given by ---- ~

ei~

•

This rate must depend on the state U~ as well as on the time and on the variables U~, u~j, c~, ei, s~, since it must carry all the information that, in the stochastic scheme, is furnished by ~3 and by the condition

e

e

si,t)),

i.e., for any state, it must depend on the frequency at which the individual A thinks of B or about its own feelings towards B, and on the effects its thoughts have on its feelings. We shall set

Analogously to what happens in the mechanics of continuous systems [1], in view of our interpretation of function (6.9) as the law of the inner evolution of the e ta mode of feelings of individual Ii toward individual Ij in the absence of interactions with individual Ij, and in accordance with what has been observed at the end of Section 4, the p.d.f. 7r~ must satisfy at each instant the continuity equation 0~r~ 0 e e at + ~ (c~r~) = o. Ouij It is readily seen that, when some interactions between A and B take place, then the above relation no longer holds, and must be replaced by the relation

0-7- +

(CA-A) = J~)(-A,-~),

(6.10)

where J~j(zr,, rcj;u,~, t) is furnished by relation (6.1) and [G~(Iri,Trj)](u~j, t) and [L,~.(~-,, 7rj)] • (u~j, t) are simply the last integrals at right-hand sides of relations (6.7),(6.8). If the inner evolution of all the modes of feelings of any individual Ij toward any other individual is assumed to be deterministic, then lr~j will obey the equation

Ore -

at + ~ -

0

e e

(c,j.,j) = [Gf~(~,,~j)] (urn,t) - [L,~(~,, ~ ) ] (u,~,t),

(6.11)

fora any i, j, e, with obvious meanings of c~j, [G~j(zr,,zcj)](u~j,t), [L~j(zr,,Trj)](u$j,t). Such an evolution will be named semistochastic. The appropriate scheme to be used to describe a real situation should be determined by psychologists on the basis of a suitable preliminary experimental analysis, since it will probably differ from case to case.

606

B. CARBONARO AND C. GIORDANO

7. THE INITIAL-VALUE P R O B L E M A N D THE N E E D FOR C O N S T I T U T I V E RELATIONS According to what has been assumed and discussed in the previous section, the equations governing the evolution of mutual feelings of the N members of a system ~ are (5.1)-(6.1)-(6.2a)(6.7)-(6.8). For convenience of the reader, we write down again the whole system of equations,

07r~Jot (u~j,t) = J~ (u~j,t) = Ji~ (Tri,~rj,"u,j,t)t

,

(7.1a)

= [Gi~(Tr,,~rj)] (ufj,t) -[L,~(wi, ~'j)] (ufj,t) ,

(7.1b)

b'ij(vi, ci, ei, si)~i~ k+p+Nn x q,(ci, e,, s,, vi; t) dcc de, ds~ dvi

+ / ~ j (v~, c~, e~, si, vj, cj, e j, s~) J~ 2(k4-p4-Nn) ×

I v,j,,:,, e.

bj,

e,,

x pj(bj,cj,ej,sj,vj;t)dvidcideidsidvjdcjdejdsjdbj,

(7.1c)

× qi(ci, e~, s~, u~; t) dui dci de~ ds~ dv~j p

+ x ¢~ (v,~ [ u~,c,, e , s~, bj)q~(e~,e,, s,,u,;t)

x p~(b~, c3, e~, sj, uj; t) dud dc~ de~ de~ de~ duj ds~ dsj db~ dvej, (7.1d) qh(Ch,

eh,

Sh, uu; t)

= J~. ph(bh, Ch, eh, Sh, Uh; t) dbh

(7.1e)

(with 1 < i, j < N, 1 < g < n). The (stochastic) evolution of all the modes of reciprocal feelings of the members of system • in a time-interval [0, T] should be completely determined by this system of equations, provided the initial conditions 7Tij

(4j,o)

~ 7rij 0

(7.2)

are given. The solution of system (7.1),(7.2) is the initial-value problem associated with the

evolution equations of feelings. As we have repeatedly pointed out in the previous section, in system (7.1) the functions vii, ~ij, ~ , and ¢i~ are assigned, and express the ways in which both the psychological features of individuals I~ and Ij and the external conditions could either favour or block their encounters and influence the changes in their mutual feelings. As far as they depend on the psychological features of the considered individuals, they can (or rather must) be considered as constitutive functions. The explicit assumption of their forms will be then expressed by a system of constitutive relations. With a procedure quite analogous to the one adopted in continuum mechanics, we feel then allowed to define (at the moment, independently of the existence of any empirical counterpart) a number of psychologic constitutive classes of individuals. The empirical sense of these classes should be checked by psychologists in a future steps, after a more precise definition of the ranges of values of parameters bj, c j, e j, sj. Before showing a sample of possible constitutive classes, we want to suggest some general rules to assign at least the encounter rates, lying on a plausible general principle, according to

A Second Step

607

which any individual Ii, whatever its constitutive class may be, has for any other individual I~ a self-interaction rate ~,ij which is an increasing function of the intensity (absolute value) of feelings. Of course, the upper and lower bounds of the self-interaction rate, as well as its growth rate, depend on external conditions and on character and self-feelings parameters, as well as on possible interferences between feelings towards different individuals. At this first stage of study, we simply propose for v~j the constraints

Ou~j

>

0,

vj e {1,2,...,Y},

Vr E { 1 , 2 , . . . , n } .

As far as the interaction rate r]/~ of Ii with another individual Ij is concerned, it will obviously also depend on the psychologic state (and class) of Ij, so that we cannot but confine ourselves to assume Orl~j (u~, e~, e~, 8 i ' uj, ej, ej, sj) > - - O,

OU~j

Y j ~ {1, 2 , . ." , N},

Yr ~ {1, 2, " " " ' n},

without any other assumption depending on a knowledge of external (psychologic and practical) conditions and of parameters describing Ij. On the other hand, it is not among the scopes of this paper a precise definition of the "score scales" describing as parameters the character features. A deeper study of these scales will be carried out in a future work.

I. Stable Normal Individuals Much more interesting could be an analysis of the form of functions ~j(v~3 ~ e I ui3, c~, e~, s~) and ¢~j(v~ej I u~j, ci, ei, s~, bj), expressing probability distributions whose parameters, such as mean value and standard deviation, must depend on the conditions before the interaction. The individual I~ will be said to be of normal class if both qoiej and ¢ie are Gaussian, i.e.,

e

~

1

{

[v,~-~'j(uij, ci, e,,si)] 2 } -e b ,)] 2 [v,je - vij(u,j,e,,e,,si,

1

where, as is wen known, ~(u~j, c~, e~, s~, b~) (respectively, ~ej(u~j, ei, e,, s~)) is th~ expected value of the £ta mode of feelings of h towards b after an interaction (respectively, after a selfinteraction), and a(u~j, c~, e,, si, b~) (respectively, a(u~j, c~, e~, s~) is the standard deviation of ~ej (respectively, of ¢ej). Now, the individual h will be said to be of stable normal class if, in the above Gaussian forms, v~-e = u~je for any set of values of its arguments, and a is suitably small. This last condition should be precise, but to this aim we would need also to precise the ranges of parameters on which ~ depends, which cannot be done in the present paper. Thus, we must postpone an exhaustive discussion of such a condition to a future paper. II. N o n s y r n m e t r i c A t t r a c t i o n - O r i e n t e d Normal Individuals An individual I~ will be said to be of nonsymmetric attraction-oriented normal class if the functions ~o~j and ¢~. are of the form

qo~j (vii I u~j, ei, ei, si) =

e

~ exp

2cr~jlx/21r

+ 2a~j2v/~~ exp

~ 2 2a~j I l 2

20"ij 2

B. CARBONAKO AND C. GIORDANO

608

2a,t

_ _ exp

i j l g27r

(

9,v/~ 2 •.,v i j l

+ - ,e 1

_ exp { Z(Y ij2VZTr

where, of course, #ijl = Pijl (vii

Uij, Ci, el, 8i) ,

[zij 2 = ~tij 2 (vii

Ui]~ Ci, ei, 8i)

(7ij I .~- (7ij 1 (Vi i

U i j , Ci, e i , 8i) ,

g g (Yij2 = (7ij2 (vii

uij~ Ci, ei, 8i) ,

ij2 = ~'ij2 (v{~ uij, el, ei, si, bj), Cr i j l "--" (T ijl (Vi~

uij,

ei, sit

Ui/, Ci~ e l , 8i, b i )

and, assuming for definiteness lZijl e < [Zij2 t and Iz,lijl < l2 ,eij2, ~

~

e

le

~ij2 -- Uij > Uij -- ~ i j l ,

l

~

re

~ i]2 -- Uij > Uij -- ~ i j l ,

and aij 2~ > a~j 1~ and a'eij2 > a'e{51 (an example of this kind of transition probability density is shown in Figure 1, where ~ j~z = u iej = - 1 , [zlj e 2 = 1, °'ij~ 1 = 1, (~ij 2 = 2. The reason for the name given to this class of individuals is the obvious fact t h a t the median and the expected value of a random variable distributed according to one of the above densities are b o t h lower t h a n its mode, so t h a t for any mode of feelings of an individual of this class the probability of finding, after an interaction or a self-interaction, a greater value t h a n the previous one is obviously greater t h a n the probability of finding a lower value (see Figure 1).

2

0.8 0.6 0.4 0.2 I

-6

-4

-2

2

4

I

6

Figure 1. The diagram of a (self-interaction) transition density for an attractionoriented individual. Here ~ j l = U~j ~-----1, /Zi£j2 = 1, ai~jl ----1, ai~ 2 ----2,

A Second Step

609

IIIo N o n s y m m e t r i c R e j e c t i o n - O r i e n t e d N o r m a l Individuals

An individual Ii will be said to be of nonsymmetric rejection-oriented normal class if the e and ¢ qe are of the form functions ~ij e e s~) = 1 { qOij (Vii I ~£ij, el, gi, 20.fjl V / ~ exp ,, - - ~ 2crijl +

1

e ~ exp 2ff ij 2 ~/27r

{ - - -g1 2 (x -- #ij2) ~ 2~

]

2¢7ii2 1

~,e

/ ,e ~ 2 ] 2~ x-''j')

•"-" ij 1

1 ( 9,.¢;e 2 x

+ 2a,fj2~-ff~ exp

,e ~ 2 }

- # q2)

~'" ij2

where, of course, g g e , i j l = ~zql (vq I u q , c~, ei, s 0 , g = ~q2 e (vq I u q , c~, e~, s~), /zij2 g e e

ff ijl : ff ijl (Vii I Uij, ai, ei, 8i) , o.i5 2 =

e,j2

I

c,,

sO,

,e

I"z ijl "~" I~'ijl (Vi~" I~ij,c-i, ei, si, b j ) , tg # ,j2 = #'ij2 (v,~ l u i j , c i , e , , s i , b j ) ,

crtgql _crte qx ( v , 5 1 u q , e . e . s . at e

_¢rt

bj,

e

and, assuming for definiteness /z~jl e < #~j2 a n d / z ,eijl < # ,eq2, e e e g ~zij2 - uij < uij - ~ i j l ,

/e g g ig ~ ij2 - uij < uij - ~ ijl,

and vri~j2 < aiS.1 and a,eij2 < a'eij1 (such a transition probability density is plotted in Figure 2, which, for the sake of simplicity, has been obtained only exchanging the parameters of Figure 1).

1.2 1 0 0 0

i

-6

,

-4

I

-2

,

i

2

4

6

Figure 2. The diagram of a (self-interaction) transition density for an rejectionoriented individual. Here/,el = --1, /~q2t = u~j = 1, al/lt = 2, alj2t = 1.

610

B. CARBONARO AND C. GIORDANO

This class is clearly defined starting from the previous one by symmetry with respect to the density axis, so that for any mode of feelings of an individual of this class the probability of finding, after an interaction or a self-interaction, a lower value than the previous one is obviously greater than the probability of finding a greater value (see Figure 2). T h a t is the reason why the individuals of this class are said to be rejection-oriented. IV. U n s t a b l e B i m o d a l I n d i v i d u a l s Just in order to give one more sample of constitutive class of individuals with respect to feelings, and with no claim to completeness or even to accuracy or likelihood in connection with real psychologic kinds, which however should be checked by psychologists, we suggest that an individual could be said to be of unstable bimodal class if the functions ~o~ and C ej are of the form ~

1[{1

(v,~ ]u~j, e~,

2aij + exp

~2 2a~

2O"~ij%/27~ ~ + exp

,

exp

,.~ ,~

2a,~2 x - # ij2

2

Z u ij

where, of course, E

g

E

g ~ g ~ i j 2 = t~ij2 ( v i i Cri~ ~

£

~$ij~ Ci, ei~ 8i) , ?-gij~ Ci~ ei~ 8 i ) , ?'$ij, Ci, e l , 8i)

u~j, ci, e~, st, b j ) , u

u i j , ci, ei, si, b j ) ,

=

,E ,E {T ij = G i j l (Vi~

u i j , ci, ei, si, bj ) ,

and, assuming for definiteness #ijl ~ < #ij2 ~ and # '~ijl < # 'gij2, E

E

E

~

~ i j 2 -- •ij : Uij -- ]~ijl ~--- ~, 2

~2

,g

E

~

tg

~ ij2 -- Uij = Uij -- ~ i j l -~" 6',

_,~ 2

and a~j < and o ~j < ~,2 (a transition probability density of this form is plotted in Figure 3, e = - 4 , #~j2 g = 4). with u~ = 0, a~j = 2, #ijl This is obviously only one possible ehoiee of the constraints to be imposed on #~jl, #ij2, # ijl, '~ij2, a igj , and O"'~ij in order to obtain a bimodal transition distribution. We have chosen these particular forms to keep the expected value of any mode of feelings of each individual toward another unchanged by interaction and, at the same time, to suggest the maximum uncertainty between attraction and rejection. We may now conclude this section with a remark which is in fact the most important one concerning the structure and the significance of system (7.1). The reader should have noticed that the right-hand sides of equations (7.1c),(7.1d) do not contain explicitly the unknown functions 7rij (uij, ~ t), but only--beside the assigned functions uij, ~]ij, ~0ij, ~ and ¢ ~ - - t h e joint probability

A Second Step

611

0.2

•

5

,

-I0

,

,

1

1

,

5

-5

,

,

,

i

i0

Figure 3. The diagram of the transition density for an unstable bimodal individual. = 0, = = - 4 , #i j2 = 4.

Here

densities qi(ci, ei, si, ui; t) and pi(bi, ci, ei, si, ui; t). Of course, the former can be derived from the latter by using relation (7.1e). But, as regards the probability density function pi, Axiom 6.2 may be somehow misleading, in that it requires that this function characterises individual Ih, thus suggesting that for any i, Pi must be assigned a priori in the same way as ~ij, ~]ij, ~oi~, and ¢i~ must be assigned for any pair (i,j). But there are at least two reasons why we may understand that this is not the case. The first reason is mathematical: should pi be assigned, relations (7.1c),(7.1d) would not be equations, but simply postulated functional dependences of the probability densities of the modes of feelings on all the other parameters introduced in the previous sections. On the other hand, the theory of probability shows that, setting uij - (uij, ui~) r £ uij, £. t) (which is always (i.e., ui~ - (uij)r¢e) and pi(bi, ci, ei, si, ul; t) _-- pi(bi, ci, ei, sl, uij, uij, possible in view of our notation conventions), it results

7ri~ (ui~,t) =

JfRrq-k+pq-N~--I

pi (b~,ci, ei, si, u~j, uij,uij,t) db~dcide~dsiduijdui~, ~

~"

which leads at once to a system of N 2 n compatibility conditions (equations in the unknown functions Pi~ deserving of course to be studied separately, but which will be not treated here) that prevent to assign any pi arbitrarily. The second reason why the functions pi are not to be assigned a priori is heuristic and involves their psychologic meaning. It must be clear that the knowledge of a given function Pi is the same as to establish the probability of coexistence of some kind of behaviours with some character data, some external (mainly psychologic) pressures and some levels of self-acceptance or self-rejection, as well as with the whole complex of feelings of individual h towards all the other individuals of system ~. But we must think of character data, external pressures, and self-feelings as built with previous experiences and feelings toward other individuals, and capable to be modified by actual experiences and states of feelings. Therefore, what we can reasonably state is that the probability (density) of any set of values for these parameters depends not only on actual experiences and feeling, but also on the whole past history of experiences and feelings. Since this past history may involve individuals not belonging to E (unless E is defined in such a complex way as to catch all mankind and even the whole body of ideas produced by mankind through the centuries, which would be unrealistic), it must be assumed as a datum. And at least the initial values of character, external pressures, and self-feelings parameters should be assumed as a stochastic consequence of this datum. This problem will be discussed in some details in future works: at the moment, we shall simply introduce the random variables c °, e °, s ° expressing the values of character, external pressures, and self-feelings parameters for each individual Ih at the initial instant t -- 0,

612

B. CARBONARO AND C. GIORDANO

and write the (joint) absolute probability density fh(Ch, eh, 8h; t) of character, external pressures, and self-feelings parameters of Ih at each instant t of the considered evolution time-interval [0, T] by means of the theorem of alternatives

A(ch, eh, s~; t)

= 9fR

k+p+N~+Nn

/h (oh,

I cL

d,b, u )gh 0 0 0h,b, u ;t) dc°

dd dbdu ,

where fh(Ch, eh, Sh [ C°, e °, s °, b, uh) has the obvious meaning of the conditional probability of (Ch, eh, Sh) under the hypothesis (Ch, 0 eh, 0 Sh, 0 b, Uh), b is a vector variable collecting the behaviour parameters of all members of ~ (in other words, b - (bl, b2,..., bN)) and gh is the joint probability density of (c °, eh, 0 Sh, 0 b, Uh). It is to be carefully noted that fh(Ch, eh, 8h [ ~h,"0~h,'~0°h,O0b, Uh) is assumed not to depend on time. Under this assumption is the principle according to which the same initial conditions and the same behaviours and the same states always produce the same changes in character, conditioning beliefs and attitudes and self-feelings. In view of the well-known meaning of conditional probability, we are allowed to write

gh (c°, e°, s ° , b , uh;t) = g~ (c°, e°, S0h l b, uh) dh(b, Uh;t), where, again, gh(Ch, * 0 eh, 0 Sh0 i b, Uh) is the probability density of (ca, 0 eh, 0 Sh) 0 under the assumption that the behaviour of all members of ~ and the states of feelings of Ih toward the other individuals are expressed by the couple (b, Uh), and dh(b, Uh) is the joint (absolute) density of the same couple at instant t. Now, the whole scheme here proposed is based upon the implicit assumptions that feelings may be modified by behaviours rather than by other feelings; in other words, your love or your hate toward a person will be turned into a milder or into a sharper feeling by his or her behaviours, not by his or her feelings toward you, that however you cannot know but through behaviours. On the other hand, it is common knowledge that feelings influence behaviours: if an individual is in anger toward another one, then its behaviour, though possibly not aggressive but rather kind, will have probably at least a tone of coldness. Nevertheless, it is not difficult to imagine a number of examples in which the behaviour of an individual is not only influenced by its feelings towards a person or a party or a philosophy, but also by the behaviours of all members of the system ~ to which the individual itself belongs, even when its feelings toward the most part of these members are to be labelled as "indifference". The most evident (and widely known, in our culture) example in this connection is Saint Peter's denial after the capture of Jesus Christ, in front of the aggressive behaviour of Jews. There is no matter to doubt that Saint Peter's feelings towards Jesus Christ were of fondness, but his behaviour was guided by the one of the people who had captured his Lord. This stated, in a quite formal fashion, let us write

j=l ~=1 j#i

From a probabilistic (or statistical) viewpoint, the knowledge of the feelings of individual Ih toward all other members of ~ and of the behaviours of the latter allows us to draw a conditional probability distribution over the possible behaviour of Ih

dh(b I Uh) = dh(bh [ bh, Uh)

(7.3)

(where bh ----(bi)~#h). And conversely, via the relation

dh(b [ Uh) = dh(bh [ bh, Uh),

(7.4)

A Second Step

613

the knowledge of the feelings of individual Ih toward all other members of E and of its behaviours allows us to draw a joint conditional probability distribution over the possible behaviours of all the other individuals. Of course, the derivation of both these probability distributions depends on an a priori statistical model on the link between behaviours and states of feelings for each individual. And the key concluding observation we want to propose in this section is just that, in view of the mutual statistical dependence of behaviours (or rather, of the knowledge of behaviours), as it results from relations (7.3),(7.4), this statistical model, which is equivalent to assuming the form and the statistical parameters of the distribution dh(b I Uh), is not only meaningful but also, together with the above assumption about the joint density of external, character, and selffeelings parameters, the most appropriate to recover the form of the joint probability Ph, which, as we have seen, cannot be assigned. 8. O P E N

QUESTIONS

AND

PERSPECTIVES

Because of its nature of initial plot of a mathematical model for the dynamics of feelings, the discussion presented here has left of course a considerable number of open questions, some of conceptual character, some of a specifically mathematical, more technical character. In these short concluding notes, we shall list some of them, at least the ones we believe worthy to be considered as first. NOTE 1. Above all, a deeper discussion of the principles the definition of constitutive classes of individuals must rely upon is required. The need for statistical models of the dependence of behaviour on the remaining parameters and on the state variables needs to be carefully analysed, and some statistical models should be checked on the basis of general experimental plausibility criteria. This is hoped to be the next step of the research started in [7] and here. NOTE 2. A more specifically psychologic analysis of both the statistical models underlying the formulation of all constitutive constraints and the mathematical parameters introduced to describe the causes determining an evolution of feelings will then be necessary. This step should be logically the fourth one, but it requires the cooperation of professional psychologists, so we hope to be able to perform such an analysis in the future. NOTE 3. The self-consistency and the effectiveness of the present model should also be checked from a purely mathematical viewpoint, at least starting with the simplest case of an isolated system of only two individuals, which would offer the advantage of reducing considerably the number of variables, hence of equations, involved in the model; existence, uniqueness, and stability of solutions will be studied in future papers. It is well known that the problem of stability is hard to tackle for various technical reasons, and wiU probably require lengthy and deep studies to give results. NOTE 4. An immediate step to check the real power of the model to give a plausible forecast as well as a manageable description of feeling evolution should be the discretization of the evolution equation with respect to both the time and the state variables and the psychologic parameters. This would enable us, on one hand, to obtain simulated solutions by methods of the same kind as the ones used in [14]; on the other hand, it would also allow to replace equations by maps, and probably to use the statistical description in terms of Markov chains. Of course, the above-listed research lines are only a little sample of the problems spontaneously arising from the model proposed here. But they seem to be the most urgent ones, to obtain at least a first check of a possible reliability of the model for applications. REFERENCES 1. s. Rinaldi, Laura and Petrarch: An intriguingcase of cyclical love dynamics, SIAM J. App£ Math. 52 (4), 1442-1468, (1998). 2. L. Arlotti, N. Bellomo and K. Latrach, From the Jager and Segel model to kinetic population dynamics: Nonlinear evolutionproblems and application, Mathl. Comput. Modelling 30 (1/2), 15-40, (1999).

614

B. CARBONAROAND C. GIORDANO

3. M.L. Bertotti and M. Delitala, From discrete kinetic and stochastic game theory to modelling complex systems in applied sciences, Math. Models Meth. Appl. Sci. 14 (7), 1061-1084, (2004). 4. N. Bellomo, A. Bellouquid and M. Delitala, Mathematical topics on the modelling complex multicellular systems and immune competition, Math. Models Meth. Appl. Sci. 14~ 1683-1733, (2004). 5. E. De Angelis and P.E. Jabin, Qualitative analysis of a mean field model of tumor immune system competition, Math. Models Meth. Appl. Sci. 19, 187-206, (2003). 6. N. Bellomo and M. Pulvirenti, Special Issue on the Modeling in Applied Sciences by Methods of Transpor~ and Kinetic Theory 12, 909-990, (2002). 7. B. Carbonaro and N. Serra, Towards mathematical models in psychology: A stochastic description of human feelings, Math. Model Meth. Appl. Sci. 12, 1463-1498, (2002). 8. N. Bellomo and M. Lo Schiavo, Lecture Notes on the Generalized Boltzmann Equation, World Sci., London, (2000). 9. N. Bellomo and M. Pulvirenti, Modelling in Applied Sciences--A Kinetic Theory Approach, Birkh~iuser, Boston, (2000). 10. C. Godano and V. Caruso, Multifractal analysis of earthquake catalogues, Geophys. J. Int. 121, 385-392, (1995). 11. C. Godano, P. Tosi, V. Derubeis and P. Augliera, Scaling properties of the spatio-temporal distribution of earthquakes: A multifractal approach applied to a Californian catalogue, Geophys. J. Int. 136, 99-108, (1999). 12. Z. Juming, L. Chengzou and S. Huiwen, Three-dimensional mathematical models of geological bodies and their graphical display, In Mathematical Geology and Geoinformatics, Proceedings of the Thirtieth International Congress, Vol. 25, (1997). 13. E. Jager and L. Segel, On the distribution of dominance in a population of interacting anonymous organisms, SIAM J. Appl Math. 52, 1442-1468, (1992). 14. N. Bellomo, V. Coscia and M. Delitala, On the mathematical theory of vehicular traffic flow I: Fluid dynamic and kinetic modelling, Math. Models Meth. Appl. Sci. 12, 1801-1844, (2002). 15. A.K. Guts, V.V. Korobitsin, A.A. Laptev, L.A. Pautova and J.V. Frolova, Mathematical Models o/Social Systems, Omsk State University, (2000). 16. L. Arlotti, N. Bellomo and E. De Angelis, Generalized kinetic (Boltzmann) models--Mathematical structures and applications, Math. Models Meth. Appl. Sci. 12, 567-592, (2002). 17. C. Truesdell, Introduction ~ la Mdcanique Rationnelle des Milieux Continus, Dunod, Paris, (1993). 18. A.N. Kolmogorov and S.V. Fomin, Introductory Real Analysis, Dover, New York, (1970).