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



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.
Let X be a real inner product space of (finite or infinite) dimension at least 2. With implicit notions
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.
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:

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.
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:

Let f : ] 0,¥[ ® R be a
real valued function on the set of positive reals. Then the
functional equations:
 


If f,g,h : ] 0,¥[ ® R are
real valued functions on the set of positive reals then


If f,g,h,k : ] 0,¥[ ® R are
real valued functions on the set of positive reals then


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

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.
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.