


Functional Equations and Their Applications
Org: Janos Aczel and CheTat Ng (Waterloo) [PDF]
 JOHN BAKER, University of Waterloo, 200 University Avenue W., Waterloo,
Ontario N2L 3G1
The Stability of a General Functional Equation
[PDF] 
Suppose that V is a vector space over Q,
R or C, the scalars a_{0},b_{0},...,a_{m},b_{m} are such that a_{j}b_{k}  a_{k}b_{j} ¹ 0 whenever 0 £ j < k £ m, B is a Banach space,
f_{k} : V ® B for 0 £ k £ m,
d ³ 0 and
 
m å
k=0

f_{k}(a_{k} x+b_{k} y)  £ d for all x,y Î V. 

Then, for each k = 0,1,...,m there exists c_{k} Î B and
a "generalized" polynomial function p_{k} : V® B of "degree" at most m1, such that
f_{k}(x)c_{k}p_{k}(x) £ 2^{m+1} d for all x Î V 

and

m å
k=0

p_{k} (a_{k} x+b_{k} y) = 0 for all x,y Î V. 

Moreover, if V = R^{n}, B = R or
C and, for some j, f_{j} is bounded on a set of positive
Lebesgue measure, then every p_{k} is a genuine polynomial function.
 WALTER BENZ, University of Hamburg, Department of Mathematics,
Bundesstrasse 55, D20146 Hamburg, Germany
Hyperbolic geometry via functional equations
[PDF] 
Let X be a real inner product space of (finite or infinite)
dimension at least 2. With implicit notions
(i) of translations t (of X along a fixed axis), and
(ii) of distances d satisfying essentially the functional
equations of translation, invariance and additivity,
exactly two geometries with translations t and distances d are
characterized up to isomorphism, namely euclidean and
BolyaiLobachevski geometry over X of dimension dimX.
The methods of the longer proof depend heavily on the theory of
functional equations, especially on results of J. Aczél and
Z. Daróczy. Our theory in question is part of a book "Geometry of
real inner product spaces", forthcoming.
 TOM DAVISON, Department of Mathematics, McMaster University, Hamilton,
Ontario
D'Alembert's Equation and the Binary Groups
[PDF] 
D'Alembert's equation f(xy) + f(xy^{1}) = 2f(x)f(y) is solved over
all finite groups. We introduce the notion of a basic
D'Alembert function: one for which f(xy) = f(x) for all x implies
that y=1. It is shown that every D'Alembert function factors
through a basic D'Alembert function. Then we show that the only
finite groups that support a basic D'Alembert function are the cyclic
groups (the classical case) and the binary groups:
á2,m,n ñ: = áR,S,T : R^{2} = S^{m} = T^{n} = RST ñ 

in Coxeter's notation. Conversely each of these groups supports a
nonclassical D'Alembert function.
 JEANCLAUDE FALMAGNE, University of California, Irvine
Functional Equations and Invariance in Scientific Laws
[PDF] 
Two fundamental invariance principles are formulated which enable the
derivation of some common physical laws via functional equation
techniques. The first invariance principle, called `meaningfulness',
is germane to the common practice requiring that the form of a
scientific law must not be altered by a change of the units of the
measurement scales. The second principle requires that the output
variable of the law be `orderinvariant' with respect to any
monotonic transformation (of one of the input variables) belonging to
a particular class of such transformations which is characteristic of
the law. These two principles are formulated as axioms of the
scientific theory. Three applications are mentioned, which involve:
the LorentzFitzerald Contraction, Beer's Law, and the Monomial Laws.
The first one is described in some details. If all scientific laws
should arguably be meaningful, not all of them are orderinvariant in
the sense of this work. An example is van der Waals' Equation. Open
problems are proposed.
 PAL FISCHER, University of Guelph, Guelph, Ontario N1G 2W1
On Schroder equations, linearizability and composition
square roots
[PDF] 
Results about linearizability and orientationreversing composition
square roots are presented. In addition, a sequence {f_{n}} of
strictly increasing and differentiable functions are constructed,
defined on an interval I of reals, containing 0 as an interior
point with the following properties:
(i) f_{n}(0)=0, f_{n}¢(0)=l, where 0 < l < 1;
(ii) there is no solution of the Schröder equation
on I such that F(0)=0 and F¢(0)=1 for any n positive
integer;
(iii) the sequence {f_{n}} converges uniformly on I to
lx.
 KONRAD HEUVERS, Michigan Technological University, 1400 Townsend Dr.,
Houghton, MI 499311295, USA
A third logarithmic functional equation and Pexider
generalizations (joint work with Palaniappan Kannappan)
[PDF] 
Let f : ] 0,¥[ ® R be a
real valued function on the set of positive reals. Then the
functional equations:

f(x+y)  f(xy) = f(1/x+ 1/y) 
 
 f(x+y)  f(x)  f(x) = f(1/x +1/y) 
 
and
are equivalent to each other.
If f,g,h : ] 0,¥[ ® R are
real valued functions on the set of positive reals then
f(x+y)  g(xy) = h(1/x +1/y) 

is the Pexider generalization of
f(x+y)  f(xy) = f(1/x + 1/y). 

We find the general solution to this Pexider equation.
If f,g,h,k : ] 0,¥[ ® R are
real valued functions on the set of positive reals then
f(x+y)  g(x)  h(y) = k(1/x +1/y) 

is the Pexider generalization of
f(x+y)  f(x)  f(y) = f(1/x + 1/y). 

We find the twice differentiable solution to this Pexider equation.
 CHETAT NG, University of Waterloo, Waterloo, Ontario N2L 3G1
A functional equation arising from the utility of gambling
[PDF] 
In joint work with D. Luce and A. A. J. Marley, on the utility of
gambling, an attempt was made to extend the approach of
J. R. Meginniss. A functional equation we came across is
f(a(r)x) + f(a(1r)x) = xm(r)+f(x) (r,x Î [0,1]) 

for some function m : [0,1] ® R. Here a is
a homeomorphism on [0,1] and we seek its continuous solution f.
We show the application of a uniqueness theorem with which the
equation is solved.
 PROBLEMS & REMARKS

 THOMAS RIEDEL, University of Louisville
Functional equations and an inequality on D^{+}
[PDF] 
Let D^{+} be the space of functions f : [0,¥]® [0,1] such that f(0)=0,f(¥)=1, and f is monotone
and leftcontinuous on (0,¥); this is the space of distance
distribution functions. D^{+} is a complete, completely
distributive lattice and hence, continuous lattice. We present our
earlier results for certain functional equations on this space and
present methods that have been used to solve such equations. We then
introduce the notion of a strict inequality defined via the waybelow
relation from the theory of continuous lattices, and investigate the
properties of certain functions on D^{+} in relation to this
inequality. It turns out that some of the methods for solving
functional equations can be applied to this question as well.
 DILIAN YANG, University of Waterloo
D'Alembert's functional equation on groups
[PDF] 
D'Alembert's functional equation is studied in detail and completely
solved on compact connected groups. Based on the structure theorem of
compact connected groups G, we prove that if G has no any direct
factors isomorphic to SU(2) then d'Alembert's equation has only
classical solutions; otherwise, nonclassical solutions exist and can
be factored through one of those direct factors. Our main tools are
from representation theory.

