Aczel, J., "Lectures on Functional equations and their Applications", Academic Press, New York, 1966.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... equation to be homogeneous, namely, to have the property that if u is an eigenfunction, then cu is an eigenfunction for all c, then we are led to the functional equation (c 1 u) c 2 (u) hence the Cauchy functional equation (x) y) xy) 1) Thus (u) must be either or u (see [1]) It turns out that this choice is mandatory also at some steps of our method of proof. As sign changes of solutions play an important role, it is natural to demand that sgn (u) sgn u. 2. Notation and tools The linear eigenvalue problem (1.2) 1.4) is equivalent to an integral equation ....

J. Aczel. Lectures on functional equations and their applications (Academic, 1966).

....an identity element e 2 S such that 8u 2 S; F(u; e) F(e; u) u; 8u 2 S, there exists an inverse element u 2 S such that F(u; u ) F(u ; u) e. In other words, denoting u v = F(u; v) these conditions imply that the set (S; is an Abelian group. Under this condition, Aczel [41] proved that there exists a monotonic and continuous function f : R [a; b] such that f(x y) F(f(x) f(y) f(x) f(y) 22) Clearly, applying f (which exists since f is monotonic) to the above equation leads to x y = f (F(f(x) f(y) f (f(x) f(y) 23) Using the above ....

J. Aczel, Lectures on Functional Equations and Their Applications. New-York: Academic Press, 1966.

....For x (x(i) i 1, n) R and p= p(i) li = 1, n) c R s, we define IIXlI1 y]in l Ix(i)l, x(X) y]ixX(i) X N) p,x) l P(i)x(i) For a C R t: d R with a , we define the intervs [a, 7gl a7 d(a, 7gl a 7 . A set S R is said to be co.vex if (1 d a [0, 1], and a polshedro if there exist some p = R ) d i i=1 0) such that S = x R [ i, x) 5 i (Vi) Let f: R R be a function. The effective domain d the epigraph e given by domf: x e gl f(x) epif = x,a) R R]a k f(x) We denote the set of minimizers of f by gminf = x f(y) Vy ....

.... f(x) epif = x,a) R R]a k f(x) We denote the set of minimizers of f by gminf = x f(y) Vy R) which c be the empty set. A function f is said to be conv if epif is a convex set. If f , then f is convex if d only if f(ax (1 a)y) af(x) 1 a)f(y) 2. 1) for all x,y domf d a [0, 1]. The inequity (2.1) for a = 1 2 is called mid point conveery. For continuous function, the mid point convexity is equivalent to the convexity. Remark 2.1. Mid point convexity does not imply convexity in general. It is known that there exists a discontinuous d nonconvex function : R R satisfying ....

[Article contains additional citation context not shown here]

J. Aczel, Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966.

....exists a c(r) such that f(rN) r = c(r)f(N) for all N . PROPOSITION 1.Every natural description of experts belief has the form f(N) dN for some ff 2 [0; 1] and d 0. Comments. 1. The proof of this Proposition easily follows from the known properties of functional equations (see, e.g. [1]) 2. Due to this Proposition, the functions dN are the best approximation for Y (S; N ) Therefore, we arrive at the following procedure of describing the degree of belief in 2 D terms: C. How to Describe the Degree of Belief To describe the degree of belief in a statement S, we do the ....

J. Aczel, Lectures on functional equations and their applications. Academic Press, NY-London, 1966.

.... f (a; b) by f a (b) Then, due to (I1) the function f a satisfies the property f a (b Delta c) f a (b) Delta f a (c) Since f is continuous, the function f a is also continuous, and therefore, the solution of this functional equation is f a (b) b for some p depending on a (see, e.g. [1]) So, f (a; b) b . Since f is continuous, the function p(a) is also continuous. From (I2) we can now conclude that (c p(b) p(a) Deltap(b) p(a Deltab) for all c, hence, p(a Delta b) p(a) Delta p(b) This is the same functional equation as before, so, we know that p(a) ....

J. Aczel, Lectures on functional equations and their applications, Academic Press, NY-London,

.... For every a 2 (0; 1) let us denote f (a; b) by f a (b) Then, due to (I1) the function f a satisfies the property f a (b Delta c) f a (b) Delta f a (c) Since f is continuous, the function f a is also continuous, and therefore, the solution of this functional equation is (see, e.g. [1]) ffl either f a (b) 0, ffl or f a (b) b for some p depending on a. Let us show that the case f a (b) f (a; b) 0 for all a; b 2 (0; 1) is impossible. Indeed, in this case, from the assumed continuity of f , we would get f (0; 1) lim 0 f ( 1 Gamma ) lim 0 = 0; ....

J. Aczel, Lectures on Functional Equations and Their Applications (Academic Press, NYLondon, 1966).

....notations, to f(fi) So, the above equality takes the form f(ff Delta fi) f(ff) Delta f(fi) In other words, we get a functional equation for the function f . This particular functional equation is well known: it has been first solved in [28] and its most general monotonic solution is [1], Section 3.1.1: f(ff) ff for some real number q. 17 To complete the proof of the theorem, we must show that q = 1. To do that, let us now consider another particular case of the consistency requirement: p [0;1] 0:5; 0:75] p [0;1] 0:5; 1] Delta p [0:5;1] 0:5; 0:75] 5:1 ....

.... interval [c; d] 0; 1] we have p [0;1] c; d] p [0;1] 0; d] Gamma p [0;1] 0; c] d Gamma c = 1 ( c; d] The probability measure p [0;1] is localized on [0; 1] therefore, the measure of any other other interval [c; d] is determined only by this interval s intersection with [0,1]. Hence, for an arbitrary interval [c; d] we have p [0;1] c; d] 1 ( 0; 1] c; d] 1 ( 0; 1] c; d] 1 ( 0; 1] So, we have proved the desired formula for [a; b] 0; 1] The formula for other intervals [a; b] follows from the one that we have just proved, if we apply ....

[Article contains additional citation context not shown here]

J. Acz'el, Lectures on functional equations and their applications, Academic Press, N.Y., London, 1966.

....Luce, leaving the others amilable upon request from the authors. We need the following preliminary Lemm. If Siota I holds, then S( Alog for A O. If Siota 2 holds, then for O O, S( o. The proofs of these two Lemma s are given as the first pt of Luce s major proofs and follow from Aczel [1]. The follong Lemma is our own, given to simplify the major proofs to follow. Lema 8. If iom 3 holds then ; T[f]dy= T [f(x) dx where y is induc by multiplying aH outcomes x by the constt . Proof. f(x) and dx Simil Lemmas exist for the other assumptions on the function form, but these ....

.... Deviation (4) is Kijima and Ohshini s first measure (27) is R[oo, c, 1] O Furthermore, to generalise Stone s measure, we can now also include The Variance (5) is R[c, 2,1,1] The Semi Variance (6) is The Probability of Making a Loss (20) is Lower Partial Moments (21) are R[0,0,0,1,1] R[ c, 1] Fishburns c t measures (7) are [t,t, 1,1] So far, we have not utilised the W( function in (44) By abstracting in this direc tion, we capture the following additional measures, The Probability Weighted Moments (22) are t[o 3, 0, 0, 1, r(y)r(1 r(y) s] The Extended ....

J. Aczel. Lectures on Functional Equations and Their Applications. New York, 1966.

....a b means that an event with plausibility value a is not more plausible than one with the value b. A system of conditional propositions or events with plausibility values is a plausibility model. Standard probability is one possible plausibility measure, where the domain is the real numbers in [0, 1] and the functions F and G are multiplication and addition, respectively. Extended probability is standard probability extended with infinitesimal probabilities. We will motivate a definition, prove a theorem and justify a claim: Definition 1 A proper plausibility measure is a seven tuple ....

....A B the plausibility of A given that we know B to be true. This is the model in which foundational studies are usually made, and which we use here. Cox[12] assumes that the domain of plausibility values is an interval of real values, which without loss of generality can be assumed to be [0, 1], with 0 for falsity and 1 for truth. He furthermore finds that two functions F and S must exist such that A B C = F (A B C,B C) and A C = S(A C) With a number of regularity assumptions, he shows that the domain of plausibility values can be rescaled so that F is tranformed to ....

[Article contains additional citation context not shown here]

J. Aczel. Lectures on Functional Equations and their Applications. Academic Press, 1966.

....functions can be obtained from each other by a linear transformation Cf(u) x 0 , therefore, f(cu) Cf(u) x 0 for some C and x 0 . These values C and x 0 depend on c. So we arrive at the following functional equation for f(u) f(cu) C(c)f(u) x 0 (c) In the survey on functional equations [A66] the solutions of this equation are not explicitly given, but a for a similar functional equation f(x y) f(x)h(y) k(y) all solutions are enumerated in Corollary 1 to Theorem 1, Section 3.1.2 of [A66] they are f(x) flx ff and f(x) flexp(cx) ff, where fl 6= 0, c 6= 0 and ff are ....

....equation for f(u) f(cu) C(c)f(u) x 0 (c) In the survey on functional equations [A66] the solutions of this equation are not explicitly given, but a for a similar functional equation f(x y) f(x)h(y) k(y) all solutions are enumerated in Corollary 1 to Theorem 1, Section 3.1. 2 of [A66] they are f(x) flx ff and f(x) flexp(cx) ff, where fl 6= 0, c 6= 0 and ff are arbitrary constants. So, let us reduce our equation to the one with known solutions. The only difference between these two equations is that we have a product, and we need a sum. There is a well known way to ....

J. Aczel. Lectures on functional equations and their applications. Academic Press, NY--London, 1966.

.... on this formula, we can compute the corresponding odds O( Delta r) first, we compute the value 1 Gamma P ( Delta r) b( a( Delta O(r) b( 14) and then divide (13) by (14) resulting in: O( Delta r) c( Delta O(r) 15) where we denoted c( a( b( It is known (see, e.g. [1,9]) that all monotonic solutions of the functional equation (15) are of the form O(r) C Delta r ff . Therefore, we can reconstruct the probability P (r) as P (r) O(r) O(r) 1 = C Delta r ff C Delta r ff 1 : Dividing both the numerator and the denominator of the right hand side ....

Aczel, J. Lectures on functional equations and their applications (Academic Press, New York, London, 1966).

....of conjunctions, so it is similar to well known CNF and DNF forms of a propositional expression. 10 Proofs: Main Ideas Proof of Proposition 2. For every a, the function f a (b) def = a b is additive f a (b) f a (c) f a (b c) Since it is also monotonic, we conclude (see, e.g. [1]) that this function is linear, i.e. a b = f a (b) C(a) Delta b. Commutativity implies that C(a) Delta b = C(b) Delta a for all a and b, hence, for b = 1, C(a) Delta 1 = C(1) Delta a, and a b = C(a) Delta b = C Delta a Delta b. From monotonicity, we conclude that C 0. Proof of ....

....= 1, C(a) Delta 1 = C(1) Delta a, and a b = C(a) Delta b = C Delta a Delta b. From monotonicity, we conclude that C 0. Proof of Proposition 5. De Morgan condition leads to the functional equation : a b) a) Delta : b) All monotonic solutions to this equations are known; see, e.g. [1]. Acknowledgments This work was supported in part by NASA under cooperative agreement NCC5 209 and grant NCC 21232, by NSF grants CDA 9522207, ERA 0112968 and 9710940 Mexico Conacyt, by Future Aerospace Science and Technology Program (FAST) Center for Structural Integrity of Aerospace Systems, ....

J. Aczel, Lectures on functional equations and their applications, Academic Press, New York, London, 1966.

....So, we arrive at a functional equation for f . Let us reduce this equation to a one with a known solution. For that purpose, let us use the fact that fractionally linear transformations are projective transformations of a line, and for such transformations the cross ratio is preserved (see, e.g. [2], Section 2.3) i.e. if g(x) A B Delta f(x) C D Delta f(x) then g(x 1 ) Gamma g(x 3 ) g(x 2 ) Gamma g(x 3 ) Delta g(x 2 ) Gamma g(x 4 ) g(x 1 ) Gamma g(x 4 ) f(x 1 ) Gamma f(x 3 ) f(x 2 ) Gamma f(x 3 ) Delta f(x 2 ) Gamma f(x 4 ) f(x 1 ) Gamma f(x 4 ) for all x ....

.... Gamma f(x 3 c) Delta f(x 2 c) Gamma f(x 4 c) f(x 1 c) Gamma f(x 4 c) f(x 1 ) Gamma f(x 3 ) f(x 2 ) Gamma f(x 3 ) Delta f(x 2 ) Gamma f(x 4 ) f(x 1 ) Gamma f(x 4 ) 6 The most general continuous solutions of this functional equation are given by Theorem 2.3. 2 from [2]: either f is fractionally linear, or f(x) a b Delta tan(k Delta x) c d Delta tan(k Delta x) for some a, b, c, d, and k, or f(x) a b Delta tanh(k Delta x) c d tanh(k Delta x) Fractionally linear functions can be excluded because they do not map the real line into a ....

J. Aczel, Lectures on functional equations and their applications, Academic Press, N.Y.--London, 1966.

.... Delta r) and F opt consists of all the functions of the type C Delta f(r) Thus, we conclude that for every 0, there exists a C 0 (depending on ) for which, for every r, we have f( Delta r) C( Delta f(r) 1) All decreasing solutions of this functional equations are known (see, e.g. [1]) these solutions are f(r) A Delta r Gammaff . Thus, the theorem is proven. For readers convenience, let us describe how this can be proven when f(r) is a differentiable function. In this case, f( Delta r) is a differentiable function as well, hence their ratio C( f( Delta r) f(r) ....

J. Aczel, Lectures on functional equations and their applications, Academic Press, N.Y.-- London, 1966.

No context found.

Aczel, J., "Lectures on Functional equations and their Applications", Academic Press, New York, 1966.

No context found.

Acz'el, J., "Lectures on Functional equations and their Applications", Academic Press, New York, 1966.

No context found.

J. Acz'el, Lectures on Functional Equations and Their Applications, Academic Press, New York, 1966.

No context found.

J. Aczel, \Lectures on functional equations and their applications," Academic Press, New York, 1966.

No context found.

J. Aczel, Lectures on functional equations and their applications, Academic Press, New York, 1966.

No context found.

J. Aczel, Lectures on functional equations and their applications, Academic Press, New York, 1966.

No context found.

Aczel, J. Lectures on Functional Equations and Their Applications Hansjorg Oser, Academic Press, 1966, New York, 38--39. 13

No context found.

J. Aczel. Lectures on functional equations and their applications. Academic Press, N.Y. and London, 1966.

No context found.

J. Aczel. Lectures on functional equations and their applications (Academic, 1966).

No context found.

Aczel, J.(1966): Lectures on Functional Equations and Their Applications. Academic Press, New York

First 50 documents  Next 50

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC