2,143 research outputs found
The idempotent Radon--Nikodym theorem has a converse statement
Idempotent integration is an analogue of the Lebesgue integration where
-additive measures are replaced by -maxitive measures. It has
proved useful in many areas of mathematics such as fuzzy set theory,
optimization, idempotent analysis, large deviation theory, or extreme value
theory. Existence of Radon--Nikodym derivatives, which turns out to be crucial
in all of these applications, was proved by Sugeno and Murofushi. Here we show
a converse statement to this idempotent version of the Radon--Nikodym theorem,
i.e. we characterize the -maxitive measures that have the
Radon--Nikodym property.Comment: 13 page
Representation of maxitive measures: an overview
Idempotent integration is an analogue of Lebesgue integration where
-maxitive measures replace -additive measures. In addition to
reviewing and unifying several Radon--Nikodym like theorems proven in the
literature for the idempotent integral, we also prove new results of the same
kind.Comment: 40 page
Integral Concentration of idempotent trigonometric polynomials with gaps
We prove that for all p>1/2 there exists a constant such that,
for any symmetric measurable set of positive measure E\subset \TT and for any
, there is an idempotent trigonometrical polynomial f
satisfying \int_E |f|^p > \gamma \int_{\TT} |f|^p. This disproves a
conjecture of Anderson, Ash, Jones, Rider and Saffari, who proved the existence
of for p>1 and conjectured that it does not exists for p=1.
Furthermore, we prove that one can take when p>1 is not an even
integer, and that polynomials f can be chosen with arbitrarily large gaps when
. This shows striking differences with the case p=2, for which the
best constant is strictly smaller than 1/2, as it has been known for twenty
years, and for which having arbitrarily large gaps with such concentration of
the integral is not possible, according to a classical theorem of Wiener.
We find sharper results for when we restrict to open sets, or
when we enlarge the class of idempotent trigonometric polynomials to all
positive definite ones.Comment: 43 pages; to appear in Amer. J. Mat
Three-term idempotent counterexamples in the Hardy-Littlewood majorant problem
The Hardy-Littlewood majorant problem was raised in the 30's and it can be
formulated as the question whether whenever
. It has a positive answer only for exponents which are
even integers. Montgomery conjectured that even among the idempotent
polynomials there must exist some counterexamples, i.e. there exists some
finite set of exponentials and some signs with which the signed
exponential sum has larger norm than the idempotent obtained with
all the signs chosen + in the exponential sum. That conjecture was proved
recently by Mockenhaupt and Schlag. \comment{Their construction was used by
Bonami and R\'ev\'esz to find analogous examples among bivariate idempotents,
which were in turn used to show integral concentration properties of univariate
idempotents.}However, a natural question is if even the classical three-term exponential sums, used for and
already by Hardy and Littlewood, should work in this respect. That remained
unproved, as the construction of Mockenhaupt and Schlag works with four-term
idempotents. We investigate the sharpened question and show that at least in
certain cases there indeed exist three-term idempotent counterexamples in the
Hardy-Littlewood majorant problem; that is we have for 0
. The proof combines delicate calculus with numerical integration and precise error estimates.Comment: 19 pages,1 figur
Spin Path Integrals and Generations
The spin of a free electron is stable but its position is not. Recent quantum
information research by G. Svetlichny, J. Tolar, and G. Chadzitaskos have shown
that the Feynman \emph{position} path integral can be mathematically defined as
a product of incompatible states; that is, as a product of mutually unbiased
bases (MUBs). Since the more common use of MUBs is in finite dimensional
Hilbert spaces, this raises the question "what happens when \emph{spin} path
integrals are computed over products of MUBs?" Such an assumption makes spin no
longer stable. We show that the usual spin-1/2 is obtained in the long-time
limit in three orthogonal solutions that we associate with the three elementary
particle generations. We give applications to the masses of the elementary
leptons.Comment: 20 pages, 2 figures, accepted at Foundations of Physic
On maxitive integration
A functional is said to be maxitive if it commutes with the (pointwise) supremum operation. Such functionals find application in particular in decision theory and related fields. In the present paper, maxitive functionals are characterized as integrals with respect to maxitive measures (also known as possibility measures or idempotent measures). These maxitive integrals are then compared with the usual additive and nonadditive integrals on the basis of some important properties, such as convexity, subadditivity, and the law of iterated expectations
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