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Idempotent structures in optimization
Consider the set A = R āŖ {+ā} with the binary operations o1 = max
and o2 = + and denote by An the set of vectors v = (v1,...,vn) with entries
in A. Let the generalised sum u o1 v of two vectors denote the vector with
entries uj o1 vj , and the product a o2 v of an element a ā A and a vector
v ā An denote the vector with the entries a o2 vj . With these operations,
the set An provides the simplest example of an idempotent semimodule.
The study of idempotent semimodules and their morphisms is the subject
of idempotent linear algebra, which has been developing for about
40 years already as a useful tool in a number of problems of discrete optimisation.
Idempotent analysis studies infinite dimensional idempotent
semimodules and is aimed at the applications to the optimisations problems
with general (not necessarily finite) state spaces. We review here
the main facts of idempotent analysis and its major areas of applications
in optimisation theory, namely in multicriteria optimisation, in turnpike
theory and mathematical economics, in the theory of generalised solutions
of the Hamilton-Jacobi Bellman (HJB) equation, in the theory of games
and controlled Marcov processes, in financial mathematics
Note: Axiomatic Derivation of the Doppler Factor and Related Relativistic Laws
The formula for the relativistic Doppler effect is investigated in the
context of two compelling invariance axioms. The axioms are expressed in terms
of an abstract operation generalizing the relativistic addition of velocities.
We prove the following results. (1) If the standard representation for the
operation is not assumed a priori, then each of the two axioms is consistent
with both the relativistic Doppler effect formula and the Lorentz-Fitzgerald
Contraction. (2) If the standard representation for the operation is assumed,
then the two axioms are equivalent to each other and to the relativistic
Doppler effect formula. Thus, the axioms are inconsistent with the
Lorentz-FitzGerald Contraction in this case. (3) If the Lorentz-FitzGerald
Contraction is assumed, then the two axioms are equivalent to each other and to
a different mathematical representation for the operation which applies in the
case of perpendicular motions. The relativistic Doppler effect is derived up to
one positive exponent parameter (replacing the square root). We prove these
facts under regularity and other reasonable background conditions.Comment: 12 page
Approximation of Rough Functions
For given and , we establish
the existence and uniqueness of solutions , to the
equation where , , and . Solutions include well-known nowhere differentiable functions such as
those of Bolzano, Weierstrass, Hardy, and many others. Connections and
consequences in the theory of fractal interpolation, approximation theory, and
Fourier analysis are established.Comment: 16 pages, 3 figure
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