On a class of meaningful permutable laws

Transcription

1 On a lass of meaningful permutable laws Jean-Claude Falmagne University of California, Irvine September 2, 202 Abstrat The permutability equation G(G(x, y), z) = G(G(x, z), y) is satisfied by many sientifi and geometri laws. A few examples among many are: The Lorentz- FitzGerald Contration, Beer s Law, the Pythagorean Theorem, and the formula for omputing the volume of a ylinder. If we required that a permutable law be meaningful, the possible forms of a law are onsiderably restrited. The lass of examples desribed here ontains the Pythagorean theorem. The mathematial expression of a sientifi law typially does not depend on the units of measurement. The most important rationale for this onvention is that measurement units do not appear in nature. Thus, any mathematial model or law whose form would be fundamentally altered by a hange of units would be a poor representation of the empirial world. As far as I know, however, there is no agreed upon formalization of this type of invariane of the form of sientifi laws, even though there has been some proposals (see Falmagne and Narens, 983; Narens, 2002; Falmagne, 2004). Expanding on the just ited referenes, I propose here a general ondition of meaningfulness onstraining a priori the form of any funtion desribing a sientifi or geometri law expressed in terms of ratio sales variables suh as mass, length, or time 2. We define this meaningfulness ondition in the seond setion of this paper. In this definition, all the units of the variables are expliitly speified by the notation, as opposed to being impliitly embedded in the onepts of quantities and dimensions of dimensional analysis (.f. for example Sedov, 943, 956). The interest of suh a meaningfulness ondition from a philosophy of siene standpoint is that, in its ontext, general abstrat onstraints on the funtion, formalizing gedanken experiments, may yield the exat possible forms of a law, possibly up to some real valued parameters. An example of suh a onstraint is the ondition below, whih applies to a real, positive valued funtion G of two real positive variables. It is formalized by the equation G(G(y, r), t) = G(G(y, t), r), () where G is stritly monotoni and ontinuous in both real variables. An interpretation of G(y, r) in Equation () is that the seond variable r in modifies the state of the first variable y, reating an effet evaluated by G(y, r) in the same measurement variable as y. The left hand side of () represents a one-step iteration of this phenomenon, in that G(y, r) is then modified by t, resulting in the effet G(G(y, r), t). Equation (), The only exeption is the ounting measure, as in the ase of the Avogadro number. 2 The results an be extended to other ases, in partiular interval sales.

2 whih is referred to as the permutability ondition by Azél (966), formalizes the onept that the order of the two modifiers r and t is irrelevant. Many, and various, sientifi laws are permutable in the sense of Equation (). Some examples of permutable laws are the Lorentz-FitzGerald Contration, Beer s law, the formula for omputing the volume of a ylinder, and the Pythagorean theorem. For the Lorentz-FitzGerald Contration, for example, written in the form ( v ) 2 L(l, v) = l in whih is the speed of light, we have ( s ) 2 ( v ) 2 ( s ) 2 L(L(l, v), s) = L(l, v) = = L(L(l, s), v). Not all sientifi laws are permutable. Van der Walls Law, for instane is not: see the Counterexample 4(f). Under fairly general onditions of ontinuity and solvability making empirial sense, the permutability ondition () implies the existene of a general representation G(y, r) = f (f(y) + g(r)), (2) where f and g are real valued, stritly monotoni ontinuous funtions. This is stated preisely in our Lemma 7, whih is due Falmagne (202), and generalizes results of Hosszú (962a,b,) (f. also Azél, 966). It is easily shown that the representation (2) implies the permutability ondition (): we have G(G(y, r), t) = f (f(g(y, r)) + g(t)) (by (2)) = f (f(f (f(y) + g(r))) + g(t)) (by (2) again) = f (f(y) + g(r) + g(t)) (simplifying) = f (f(y) + g(t) + g(r)) (by ommutativity) = G(G(y, t), r) (by symmetry). We will also use a more general ondition, alled quasi permutability, whih is defined by the equation and lead to the representation M(G(y, r), t) = M(G(y, t), r) (3) M(y, r) = m((f(y) + g(r)) (4) (see also Falmagne, 202, and our Theorem 7 here). The ombined onsequenes of permutability or quasi permutability and meaningfulness are powerful ones. For instane, suppose that meaningfulness holds, that the funtion G is symmetri, and that it also satisfies a ertain quasi permutability ondition. Our Theorem 9 states that, under reasonable solvability onditions, there are only two possible forms for G, whih are:. G(y, x) = θyx (θ > 0) (5) 2. G(y, x) = ( y θ + x θ) θ (θ > 0). (6) 2

3 With θ =, the first equation is the formula for the area of a retangle. With θ = 2, the seond one is the Pythagorean Theorem. In our first setion, we state some basi definitions and we desribe a few examples of laws, taken from physis and geometry, in whih the permutability ondition applies. We also give one example, van der Waals Equation, whih is not permutable. The seond setion is devoted to meaningfulness and anillary onepts. The third setion ontains preparatory lemmas. The last setion ontain the main results of the paper. Basi Conepts and Examples Definition. We write R + and R ++ for the nonnegative and the positive reals, respetively. For some positive integer n 2, let I, I 2,..., I n+ be n + non negative real intervals of positive length. A n-dimensional (numerial) ode, or an n-ode fort short, is a funtion M : I... I n I n+ (7) that is stritly monotoni and ontinuous in its n arguments, and stritly inreasing in its first argument. As we mostly deal with 2-odes in this paper, we simplify our language and just write ode to mean 2-ode. A 2-ode M is solvable if it satisfies the following two onditions. [S] If M(x, t) < p H, there exists w J suh that M(w, t) = p. [S2] The funtion M is -point right solvable, that is, there exists a point x 0 J suh that for every p H, there is v J satisfying M(x 0, v) = p. In suh a ase, we may say that M is x 0 -solvable. By the strit monotoniity of M, the points w and v of [S] and [S2] are unique. Two funtions M : J J H and G : J J H are omonotoni if M(x, s) M(y, t) G(x, s) G(y, t), (x, y J; s, t J ). (8) In suh a ase, the equation F (M(x, s)) = G(x, s) (x J; s J ) (9) defines a stritly inreasing ontinuous funtion F : H that G is F -omonotoni with M. onto H. We may say then We turn to the key ondition of this paper. 2 Definition. A funtion M : J J H is quasi permutable if there exists a funtion G : J J J o-monotoni with M suh that M(G(x, s), t) = M(G(x, t), s) (x, y J; s, t J ). (0) We say in suh a ase that M is permutable with respet to G, or G-permutable for short. When M is permutable with respet to itself, we simply say that M is permutable, a terminology onsistent with Azél (966, Chapter 6, p. 270). 3

4 3 Lemma. A funtion M : J J H is G-permutable only if G is permutable. This is straightforward. Suppose that G is F -omonotoni with M. For any x J and s, t J, we get G(G(x, s), t) = F (M(G(x, s), t)) = F (M(G(x, t), s)) = G(G(x, t), s). Many sientifi laws embody permutable or quasi permutable numerial 2-odes, and hene an be written in the form of Equation (2). We give four quite different examples below. In eah ase, we derive the forms of the funtions f and g in the representation equation (2). 4 Four Examples and One Counterexample. (a) The Lorentz-FitzGerald Contration. This term denotes a phenomenon in speial relativity, aording to whih the apparent length of a rod measured by an observer moving at the speed v with respet to that rod is a dereasing funtion of v, vanishing as v approahes the speed of light. This funtion is speified by the formula ( v ) 2, L(l, v) = l () in whih > 0 denotes the speed of light, l is the atual length of the rod (for an onto observer at rest with respet to the rod), and L : R + [0, [ R + is the length of the rod measured by the moving observer. The funtion L is a permutable ode. Indeed, L satisfies the strit monotoniity and ontinuity requirements, and we have ( ( v ) ) 2 ( 2 ( w ) ) 2 2 L(L(p, v), w) = p = L(L(p, w), v). (2) Solving the funtional equation ( v ) 2 l = f (f(l) + g(v)) (3) leads to the Pexider equation (.f. Azél, 966, pages 4-65) with f(ly) = f(l) + k(y) (4) ( v ) 2 y = and k(y) = g ( ) y 2. As the bakground onditions (monotoniity and domains of the funtions 3 ) are satisfied, the unique forms of f and k in (4) are determined. They are: with ξ > 0, f(l) = ξ ln l + θ (5) k(y) = ξ ln y. So, we get for the funtion g in (3): 3 Note that the standard solutions for Pexider equations are valid when the domain of the equation is an open onneted subset of R 2 rather than R 2 itself. Indeed, Azél (987, see also Azél, 2005, Chudziak and Tabor, 2008, and Radó and Baker, 987) has shown that, in suh ases, this equation an be extended to the real plane. 4

5 ( ) ( v ) 2 g(v) = ξ ln. (6) (b) Beer s Law. This law applies in a lass of empirial situations where an inident radiation traverses some absorbing medium, so that only a fration of the radiation goes through. In our notation, the expression of the law is I(x, y) = x e y, (x, y R+, R ++ onstant) (7) in whih x denotes the intensity of the inident light, y is the onentration of the absorbing medium, is a referene level, and I(x, y) is the intensity of the transmitted radiation. The form of this law is similar to that of the Lorentz-FitzGerald Contration onto and the same arguments apply. Thus, the funtion I : R + R + R + is also a permutable ode. The solution of the funtional equation x e y = f (f(x) + g(y)) follows a pattern idential to that of Equation (3) for the Lorentz-FitzGerald Contration. The only differene lies in the definition of the funtion g, whih is here g(y) = ξ y. The definition of f is the same, namely (5). So, we get ( I(x, y) = f (f(x) + g(y)) = exp ξ (ξ ln x + θ ξ y ) θ = x e y. () The volume of a ylinder. The permutability equation applies not only to many physial laws, but also to some fundamental formulas of geometry, suh as the volume C(l, r) of a ylinder of radius r and height l, for example. In this ase, we have whih is permutable. We have Solving the funtional equation yields the solution C(l, r) = lπr 2, (8) C(C(l, r), v) = C(lπr 2, v) = lπr 2 πv 2 = C(C(l, v), r). lπr 2 = f (f(l) + g(r)) f(l) = ξ ln l + θ (again, the funtion f is the same as in the two preeding examples), and g(r) = ξ ln ( πr 2), 5

6 with ( ( ( f (f(l) + g(r)) = exp ) ξ ln l + θ + ξ ln πr 2 θ )) = lπr 2. ξ We give another geometri example below, in whih the form of f is different. (d) The Pythagorean Theorem. The funtion P (x, y) = x 2 + y 2 (x, y R ++ ), (9) representing the length of the hypothenuse of a right triangle in terms of the lengths of its sides, is a permutable ode. We have indeed P (P (x, y), z) = P (x, y) 2 + z 2 = x 2 + y 2 + z 2 = P (P (x, z), y). The other onditions are learly satisfied, and so is Condition [S]. Condition [S2] would be ahieved by taking an appropriate restrition of the funtion P as in the ase of Examples 4(a) and (b). Notie that the ode P is a symmetri funtion: we have P (x, y) = P (y, x) for all x, y R ++ So, we must solve the equation x2 + y 2 = f (f(x) + f(y)) or, equivalently, ( ) f x2 + y 2 = f(x) + f(y). (20) With z = x 2, w = y 2, and defining the funtion h(z) = f h(z + w) = h(z) + h(w), ( ) z 2, Equation (20) beomes a Cauhy equation on the positive reals, with h stritly inreasing. It has the unique solution h(z) = ξ z, for some positive real number ξ (.f. Azél, 966, page 3). So, we get f(x) = ξx 2 and f (f(x) + f(y)) = ( ( ξx 2 + ξy 2)) 2 = x2 + y ξ 2. (f) The Counterexample: van der Waals Equation. One form of this equation is T (p, v) = K (p + a ) (v b), (2) v 2 in whih p is the pressure of a fluid, v is the volume of the ontainer, T is the temperature, and a, b and K are onstants; K is the reiproal of the Boltzmann onstant. It is easily shown that the funtion T in (2) is not permutable. 6

7 Meaningful Colletion of Codes One of our goals in this paper is to axiomatize a partiular type of invariane that must hold for all sientifi or geometri laws. The onsequene of this axiomatization should be that the form of an expression representing a sientifi law should not be altered by hanging the units of the variables. The next definition, whih generalizes that used by Falmagne (2004) (see also Falmagne and Narens, 983; Narens, 2002) applies to odes regarded as funtions of n real ratio sale variables. We illustrate the definition by our Example 4(a) involving the Lorentz-FitzGerald Contration, whih we expressed by the equation ( v ) 2. L(l, v) = l (22) The trouble with this notation is its ambiguity: the units of l, whih denotes the length of the rod, and of v, for the speed of the observer, are not speified. Writing L(70, 3) has no empirial meaning if one does not speify, for example, that the pair (70, 3) refers to 70 meters and 3 kilometers per seond, respetively. Suh a parenthetial referene is standard in a sientifi ontext, but is not instrumental for our purpose, whih is to express, formally, an invariane with respet to any hange in the units 4. To retify the ambiguity, we propose to interpret L(l, v) as a shorthand notation for L, (l, v), in whih l and L on the one hand, and v on the other hand, are measured in terms of two partiular initial or anhor units fixed arbitrarily. Suh units ould be m (meter) and km/se, if one wishes. The, index of L, signifies these initial units. Desribing the phenomenon in terms of other units amounts to multiply l and v in any pair (l, v) by some positive onstants and β, respetively. At the same time, L also gets to be multiplied by, and the speed of ligh by β. Doing so defines a new funtion L,β, whih is different from L = L, if either or β (or both), but arries the same information from an empirial standpoint. For example, if our new units are km and m/se, then the two expressions L 0 3,0 3(.007, 3000) and L(70, 3) = L,,(70, 3), while numerially not equal, should desribe the same empirial situation. This points to the appropriate definition of L,β. We should write: ( ) 2 v L,β (l, v) = l. (23) β The onnetion between L and L,β is thus ( ) ( ) 2 βv L,β(l, βv) = l (24) β ( v ) 2 = l = L(l, v). 4 A relevant point is made by Suppes (2002, see Why the Fundamental Equations of Physial Theories Are not Invariant, p. 20). 7

8 This implies, for any, β, ν and µ in R ++, L,β(l, βv) = ν L ν,µ(νl, µv), (25) whih is the invariane equation we were looking for, in this ase, and whih is generalized as Equation (27) in the next definition. Note that the range of the seond variable of the funtion L,β is now [0, β[ instead of ]0, [. The range of the first variable of L is the non negative reals and so did not hange. It is lear from our disussion of this example and from Equation (25) that the definition of meaningfulness must apply to a olletion of odes, eah of whih orresponds to another hoie of units, that is, the hoie of (, β) and (ν, µ) in the ase of Equation (25). We formulate the definition in the general ase of a family of n-odes. 5 Definition. Let [a, a [,..., [a n+, a n+[ be n + half open intervals, and let M = {M = (..., n ) R n ++} (26) be a olletion of n-odes, with for the initial ode M M = M,..., : [a, a [... [a n, a n[ [a n+, a n+[. }{{} n indies Eah of the terms,..., n in represents a hange of the unit of one of the measurement sale. We will speify the domain and range of any ode M in a moment. Let δ,..., δ n be a finite sequene of rational numbers. The olletion of n-odes M is (δ,..., δ n )-meaningful if for any vetor (x,..., x n ) R n + and any pair of vetors = (,..., n ) R n ++ and µ = (µ,..., µ n ) R n ++, the following equality holds: M δ ( x,..., n x n ) = M... n δn µ δ µ (µ x,..., µ n x n ), (27)... µ δn n whih implies M δ ( x,..., n x n ) = M(x,..., x n ). (28)... n δn Aordingly, any ode M in a (δ,..., δ n )-meaningful family M satisfies M : [ a, a [... [ n a n, n a n[ [ ( ) δ... n δn an+, ( ) [ δ... n δn a n+. The exponents δ i s are alled the markers of the family M. The next onept will also play an important role. A olletion of n-odes M is self-transforming, or an ST-olletion, if for all odes M in the olletion, the measurement unit of the output of the ode M is the same as the measurement unit of its first variable. In other words, if for every vetor = (..., n ) R n ++, we have δ... δn n =. (29) 8

9 In the rest of this paper, we apply these onepts to the ase of a olletion of 2-odes M = {M,β, β R ++ }. For [a, a [, [b, b [ and ]d, d [, three real non negative intervals, we have and so M,β : [a, a [ [βb, βb [ M = M, : [a, a [ [b, b [ onto [d, d [, onto [ δ β δ 2 d, δ β δ 2 d [, (, β R ++ ). Aordingly, the meaningfulness equation (27) speializes into δ β δ 2 M,β (x, βr) = µ δ ν δ 2 M µ,ν (µx, νr), (x [a, a [ ; r [b, b [ ). (30) Let us exerise this definition in the ase of our four examples. We will see that in one ase the Pythagorean Theorem the exponents δ s in (30) are not integers. 6 Examples. (a) The Lorentz-FitzGerald Contration. We have a olletion L = {L,β (, β) R 2 ++} of odes. We require that the olletion L be (, 0)- meaningful. This implies that and so β 0 L,β(l, βv) = ( ) l L,β(l, βv) = ν L ν,µ(νl, µv), ( ) 2 βv ( v ) 2 = l = L(l, v) β whih is our equation (25) and is a speial ase of (27) and (30). Clearly, the family L is self-transforming. So is the family of our next example. (b) Beer s Law. The form of this law is similar to the preeding one. We have a olletion I = {I,β (, β) R 2 ++} of odes, whih is also (, 0)-meaningful. This gives yielding another speial ase of (27)-(30). β I,β(x, βy) = βy (x) 0 e β I(x, βy) = I(νx, µu), ν = xe y () The volume of a ylinder. This example is quite different. We have a olletion of odes C = {C,β (, β) R 2 ++}, whih must be (, 2)-meaningful. We get and so β C,β(x, βy) = 2 β (l) π 2 (βr)2 = lπr 2, β 2 C,β(x, βy) = ν µ 2 C ν,µ(νx, µy). 9

10 The olletion C is not an ST-olletion sine β 2 if β. (d) The Pythagorean Theorem. Here, we have only one measurement sale, whih is the same for the two input variables and for the output variable. We require the olletion of odes P = {P, R ++ } to be (, )-meaningful. We obtain 2 2 P 2, (x, y) = (x) 2 + (y) 2 = x 2 + y 2. (3) Preparatory Lemmas We reall a reent result (see Falmagne, 202), whih generalizes Hosszú (962a,b,) (f. also Azél, 966). 7 Lemma. (i) A solvable ode M : J J H is quasi permutable if and only if there exists three ontinuous funtions m : {f(y) + g(r) x J, r J } H, f : J R, and g : J R, with m and f stritly inreasing and g stritly monotoni, suh that M(y, r) = m(f(y) + g(r)). (32) (ii) A solvable ode G : J J J is a permutable ode if and only if, with f and g as above, we have G(y, r) = f (f(y) + g(r)). (33) (iii) If a solvable ode G : J J J is a symmetri funtion that is, G(x, y) = G(y, x) for all x, y J then G is permutable if and only if there exists a stritly inreasing and ontinuous funtion f : J J satisfying G(x, y) = f (f(x) + f(y)). (34) The meaningfulness ondition introdued in Definition 5 and Equation (30) is a powerful one. In partiular, it enables some properties of one of the odes in M to extend to all the others odes in that olletion. The next lemma illustrates this point. 8 Lemma. Let M be a (δ, δ 2 )-meaningful olletion of odes so all the odes in M are funtions of two variables. Suppose that some ode M,β in the olletion M-satisfies any of the following five properties: (i) M,β is solvable; (ii) M,β is differentiable in both variables; (iii) M,β is quasi permutable; (iv) M,β is a symmetri funtion, with = β; (v) M is a self-transforming olletion and M,β is permutable. 0

11 Then all the odes in M satisfy the same property. Moreover, if M,β = M is solvable and permutable, so that M(x, r) = f (f(x)+g(r)) by Lemma 7(ii), then for any ode M µ,η in the olletion M, we have M µ,η (x, r) = µ δ ν δ 2 f (f ( ) x + g µ ( )) r. (35) η Proof. Without loss of generality, we suppose that = β =, with M = M,. As the family M is (δ, δ 2 )-meaningful, we have, for all positive real numbers µ and ν and writing η = µ δ ν δ 2 for simpliity: ( x M µ,ν (x, r) = ηm µ, u ) (x [a, a [ ; r [βb, βb [ ). (36) ν (i) Suppose that the ode M is solvable. If M µ,ν (x, r) < p, for some ode M µ,ν in M, then M( x, u) < p follows from (36). As the ode M satisfies [S], there must be µ ν η some w [b, b [ suh that M( x, w) = p. Defining t = νw, we get µ η ( x M µ,ν (x, t) = ηm µ, t ) = p. ν Thus, the ode M µ,ν also satisfies [S]. Sine M satisfies [S2], there exists some x 0 in [a, a [ suh that M is x 0 -solvable. Define y 0 = µx 0 [µa, µa [ and take any q in the range of the funtion M µ,ν. This implies that q is in the range of M, and by [S2] η applied to M, there is some w suh that M(x 0, w) = q or, equivalently with v = βw, η ( y0 q = ηm µ, v ) = M µ,ν (y 0, v), ν by the meaningfulness of the family M. Thus, M µ,ν is y 0 -solvable. (ii) The differentiability of M µ,ν,η results from that of M via (36). (iii) Suppose now that M is quasi permutable. (We do not assume here that M is a self-transforming family.) Thus, there exists a ode G : [a, a [ [b, b [ [a, a [ o-monotoni with M suh that M(G(x, s), t) = M(G(x, t), s) (x, y [a, a [ ; s, t [b, b [ ). (37) For any pair of parameters (µ, ν), define the funtion G µ,ν : [a, a [ [b, b [ [a, a [ by the equation ( x G µ,ν (x, r) = µg µ, u ). (38) ν

12 Thus, G µ,ν is omonotoni with M µ,ν and we have suessively ( M µ,ν (G µ,ν (x, r), v) = ηm µ G µ,ν(x, r), v ) (by (δ, δ 2 )-meaningfulness) ν ( ( x = ηm G µ, r ), v ) (by the definition of G µ,ν ) ν ν ( ( x = ηm G µ, v ), r ) (by the permutability of G) ν ν = M µ,ν (G µ,ν (x, v), r) (by symmetry). Consequently, any ode M µ,ν,η is G µ,ν -permutable. (iv) This follows from the definition of the (δ, δ 2 )-meaningfulness of the olletion. (v) Suppose that M is a self-transforming olletion and that M is permutable. We have thus, for any M µ,ν, ( µ M µ,ν (M µ,ν (x, r), v) = M µ M µ,ν(x, r), v ) (by (δ, δ 2 )-meaningfulness) ν ( ( x = M M µ, r ), v ) (by (δ, δ 2 )-meaningfulness) ν ν ( ( x = M M µ, v ), r ) (by the permutability of M) ν ν = µ M µ,ν (M µ,ν (x, v), u) (by symmetry). We have thus M µ,ν (M µ,ν (x, r), v) = M µ,ν (M µ,ν (x, v), r) and so M µ,ν is permutable. Equation (35) results from Equation (33) of Theorem 7(ii) and Equation (36). Main Result 9 Theorem. Let G = {G,β, β R ++ } be a 2-meaningful olletion of odes, with onto G,β : R ++ R ++ R ++ for all, β R ++. Moreover, suppose that one of these odes, say the ode G,β, is solvable, stritly inreasing in both variables, and permutable with respet to the initial ode G. (i) Then the initial ode G = G, must have the following form G(y, r) = y r θ (39) for some positive onstant θ. This implies that, for all G,β G: G,β (y, r) = β θ y rθ. (40) 2

13 (ii) If, in addition, the ode G,β is a symmetri funtion, with = β and G = G,, then we have two possible ases.. Case. G(y, x) = θyx. (θ > 0) (4) with for all G G G (y, x) = θ yx. (42) 2 2. Case 2. G(y, x) = ( y θ + x θ) θ (θ > 0) (43) with for all G G G (y, x) = ( y θ + x θ) θ. (44) Proof. (i) By Lemma 8, all the odes in G are solvable, permutable, and stritly inreasing in both variables, and we get for all G,β in G: ( ( )) y r G,β (y, r) = f (f ) + g, (45) β with in partiular We get suessively G,β (G(y, r), s) G(y, r) = f (f(y) + f(x)). (46) = G,β (f (f(y) + g(r)), s) (by Lemma 7(iii)) (47) ( = G f (f(y) + g(r)), s ) (by 2-meaningfulness) (48) β ( ) ( )) s = f (f f (f(y) + g(r)) + g (by Lemma 7(iii)) (49) β ( ) ( )) r = f (f f (f(y) + g(s)) + g (by quasi permutability). (50) β Equating the last two r.h. sides and applying the funtion f on both sides, we get ( ) ( ) ( ) ( ) s r f f (f(y) + g(r)) + g = f β f (f(y) + g(s)) + g. (5) β Setting s = f(y), t = g(r), fixing s =, and fixing also temporarily ν = and η = β, transform Equation (5) into f ( νf (s + t) ) + g (η) = f ( νf (s + g()) ) + g ( ηg (t) ). (52) Defining the funtions h ν = f νf, k ν (s) = h ν (s + f()), p η (t) = g ( ηg (t) ) g (η), 3

14 (62) beomes h ν (s + t) = k ν (s) + p η (t), a Pexider equation. Its solution is h ν (s) = a s + b(ν) +, (53) k ν (s) = a s + b(ν), (54) p η (s) = a s +, (55) for some onstants a and and some funtion b whih vary with ν but not η. Indeed, the funtion k ν does not depend upon η and the funtion p η does not depend upon ν, so a and must be onstants. We first rewrite Equation (55) to get the form of the funtion g. We get As g (t) = r, the seond equation gives p η (t) = g ( ηg (t) ) g (η) = a t + g(ηr) = ag(r) + g(η) + (56) With η =, we get yielding g(r) = ag(r) + g() +, (57) g(r)( a) = g() +. as g is stritly monotoni, we must have a = and g() + = 0. Equation (56) simplifies into g(ηr) = g(r) + g(η) +, the only solution of whih is, for some onstant φ and with = ψ, g(r) = φ ln r + ψ. We now rewrite Equation (53) in terms of the funtions f. As s = f(y), we get h ν (s) = (f νf )(s) = f(νy). Equation (53) beomes with = ψ and a =, f(νy) = f(y) + b(ν) ψ again, a Pexider equation. Its unique solutions for the funtions f and m are f(y) = φ 2 ln y + ψ 2 b(y) = φ 2 ln y + ψ. 4

15 We finally obtain G(y, r) = f (f(y) + g(r)) = e φ 2 (f(y)+g(r) ψ 2 ) = e φ 2 (φ 2 ln y+ψ 2 +φ ln r+ψ ψ 2 ) = e φ 2 (ln y φ 2 +ln r φ +ψ ) = e ln y+ln r φ φ 2 + ψ φ 2. So, with θ = e ψ φ 2 and φ = φ φ 2, we finally get G(y, r) = θ y r φ. (58) ( ) y Equation (40) follows from (58) and 2-meaningfulness: G,β (y, r) = G, r. β Proof of (ii). Suppose now that G,β = G is a symmetri funtion, with = β. By Lemma 8 (iii), all the odes in G are symmetri, and we have ( y ( x G,β (y, x) = f (f ) + f (59) )) replaing Equation (45), with in partiular G(y, r) = f (f(y) + f(x)). (60) Applying the same derivation as in asymmetri ase, namely Eqs. (47)-(50), we get ( ) ( ( ) z ( x f f (f(y) + f(x)) + f = f ) f (f(y) + f(z)) + f, (6) ) instead of Equation (5). We then proeed as in our proof of (i). (We will, however, end up with a different funtional equation, whih will give us two possible solutions.) Setting s = f(y), t = f(x), and fixing z =, and fixing also temporarily β =, Equation (5) beomes f ( βf (s + t) ) + f (β) = f ( βf (s + f()) ) + f ( βf (t) ). (62) Defining the funtions h β = f βf, and k β : s h β (s + f()) f(β), (62) yields h β (s + t) = k β (s) + h β (t), a Pexider equation. Beause the funtions h β and κ β are defined on the reals and are stritly monotoni, its solution is h β (s) = w(β)s + v(β) (63) k β (s) = w(β)s, (64) 5

16 for some onstants w(β) and v(β) whih may, hovever, depends on β. Rewriting now (63) in terms of the funtion f, we get f(βy) = w(β)f(y) + v(β), (65) another standard funtional equation (Azél, 966). We thus have two possible solutions for the funtion f. Case. With w a onstant funtion in (65): f(y) = φ ln y + ψ (φ > 0). (66) Replaing f in the representation equation (60) by its form in (66) leads for the ode G to the equation G(y, x) = θyx, (67) with θ = e ψ φ. With G (y, x) = G ( y, x ), we get Case 2. With w not onstant in (65): G (y, x) = θ 2 yx. Replaing f in (60) by its form in (68) leads to f(y) = ψy θ (ψθ > 0). (68) G(y, x) = ( y θ + x θ) θ (θ > 0). (69) This implies ( y G (y, x) = G ), x = ( y θ + x θ) θ. The Pythagorean Theorem With θ = 2, Case 2 of Theorem 9 is the formula for the Pythagorean Theorem. In fat, Theorem 9 an provide us with still another proof of the Pythagorean theorem, to be added to the several hundreds that already exists. We suppose that the length P (x, y) of the hypotenuse of a right triangle with leg lengths x > x 0 and y > x 0 (for some x 0 > 0) is a symmetri solvable ode 5 ; thus P : [x 0, [ R + [x 0, [. We take the funtion P to be the initial ode of a family of odes {P }. We establish the permutability and the quasi permutability of the ode P with respet to P, for any > 0, by an elementary geometri argument. 5 Cf. our disussion of Condition [S2] in the ontext of Example 4(e). 6

17 0 The Permutability of P. A right triangle ABC with leg lengths x and y and hypothenuse of length P (x, y) is represented in Figure A. Thus AB = x, BC = y and P (x, y) = AC. Another right triangle ACD is defined by the segment CD of length z, whih is perpendiular to the plane of ABC. The length of the hypothenuse AD of ACD is thus P (P (x, y), z) = AD. Still another right triangle EAB is defined by the perpendiular AE to the plane of ABC. We hoose E suh that AE = z = CD; we have thus EB = P (x, z). Sine AE is perpendiular to the plane of ABC and ABC is a right triangle, EB is perpendiular to BC. The lines BC and BE are perpendiular. (Indeed, the perpendiular L at the point B to the plane of triangle ABC is oplanar with AE. So, as BC is perpendiular to both AE and L, it must be perpendiular to to the plane of EAB, and so it must be perpendiular to EB.) Aordingly, EC = P (P (x, z), y) is the length of the hypothenuse of the right triangle EBC. It is lear that, by onstrution, the four points A, C, D and E are oplanar. They define a retangle whose diagonals AD and EC must be equal. So, we must have P (P (x, y), z) = P (P (x, z), y), establishing the permutability of the ode P. The Quasi Permutability of P. For any positive real number, the triangle A B C pitured in Figure B, with C =, A ollinear with A C, B ollinear with B C, and A B = x, B C = y and A C = P (x,y), is similar to the triangle ABC also represented in Figure B. So, we have ( x P ), y = P (x, y). (70) The funtion P is the initial ode of the meaningful family of odes {P }. For the ode P in that family, we get ( P (x, y) P (P (x, y), z) = P, z ) (by ( 2, ) 2 -meaningfulness) ( ( x = P P ), y, z ) (by Equation (70)) ( ( x = P P ), z, y ) (by the permutability of P ) ( P (x, z) = P, y ) (by Equation (70)) = P (P (x, z), y) (by ( 2, ) 2 -meaningfulness). We onlude that any ode P in the family {P } is quasi permutable with respet to the initial ode P. Referenes J. Azél. Utility of extension of funtional equations when possible. Journal of Mathematial Psyhology, 49: ,

18 L D E P(P(x,y),z) z z P(x,z) P(P(x,z),y) P(x,y) C A x B y A. Permutability D E z P (P(x,z),y) P(, ) = P(x, z)/ x z x y P(, ) = P(x, y)/ E z P(x,z) P (P(x,y),z) P(P(x,z),y) P(P(x,y),z) P(x,y) y A x B L z D z C= C A y x B B. Quasi permutability Figure : The upper graph A illustrates the permutability ondition formalized by the equation P (P (x, y), z) = P (P (x, z), y). The lower graph B shows that the quasi permutability ondition formalized by the equation P (P (x, y), z) = P (P (x, z), y) only involves a resaling of all the variables pitured in Figure A, resulting in a similar figure, with the retangle A B C D similar to the retangle ABCD. The measures of the two diagonals of the retangle A B C D are P (P (x, y), z) and P (P (x, z), y). J. Azél. Letures on Funtional Equations and their Appliations. Aademi Press, New York and San Diego, 966. J. Azél. A short ourse on funtional equations based on reent appliations to the soial and behavioral sienes. Reidel/Kluwer, Dordreht and Boston, 987. J.-Cl. Falmagne. Meaningfulness and order invariane: two fundamental priniples for sientifi laws. Foundations of Physis, 9:34 384, J.-Cl. Falmagne. A set of independent axioms for positive Hölder systems. Philosophy of Siene, 42(2):37 5, 975. J.-Cl. Falmagne. On a bounded version of Holder s Theorem and an appliation to the permutability equation. Submitted to a volume honoring Patrik Suppes on the oasion of his 90 th birthday. arxiv: ,

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