14,687 research outputs found
A decomposition theorem for maxitive measures
A maxitive measure is the analogue of a finitely additive measure or charge,
in which the usual addition is replaced by the supremum operation. Contrarily
to charges, maxitive measures often have a density. We show that maxitive
measures can be decomposed as the supremum of a maxitive measure with density,
and a residual maxitive measure that is null on compact sets under specific
conditions.Comment: 11 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
Axiomatizations of quasi-polynomial functions on bounded chains
Two emergent properties in aggregation theory are investigated, namely
horizontal maxitivity and comonotonic maxitivity (as well as their dual
counterparts) which are commonly defined by means of certain functional
equations. We completely describe the function classes axiomatized by each of
these properties, up to weak versions of monotonicity in the cases of
horizontal maxitivity and minitivity. While studying the classes axiomatized by
combinations of these properties, we introduce the concept of quasi-polynomial
function which appears as a natural extension of the well-established notion of
polynomial function. We give further axiomatizations for this class both in
terms of functional equations and natural relaxations of homogeneity and median
decomposability. As noteworthy particular cases, we investigate those
subclasses of quasi-term functions and quasi-weighted maximum and minimum
functions, and provide characterizations accordingly
Belief functions on lattices
We extend the notion of belief function to the case where the underlying
structure is no more the Boolean lattice of subsets of some universal set, but
any lattice, which we will endow with a minimal set of properties according to
our needs. We show that all classical constructions and definitions (e.g., mass
allocation, commonality function, plausibility functions, necessity measures
with nested focal elements, possibility distributions, Dempster rule of
combination, decomposition w.r.t. simple support functions, etc.) remain valid
in this general setting. Moreover, our proof of decomposition of belief
functions into simple support functions is much simpler and general than the
original one by Shafer
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
Phase transitions and quantum effects in anharmonic crystals
The most important recent results in the theory of phase transitions and
quantum effects in quantum anharmonic crystals are presented and discussed. In
particular, necessary and sufficient conditions for a phase transition to occur
at some temperature are given in the form of simple inequalities involving the
interaction strength and the parameters describing a single oscillator. The
main characteristic feature of the theory is that both mentioned phenomena are
described in one and the same setting, in which thermodynamic phases of the
model appear as probability measures on path spaces. Then the possibility of a
phase transition to occur is related to the existence of multiple phases at the
same values of the relevant parameters. Other definitions of phase transitions,
based on the non-differentiability of the free energy density and on the
appearance of ordering, are also discussed
Integrals and Valuations
We construct a homeomorphism between the compact regular locale of integrals
on a Riesz space and the locale of (valuations) on its spectrum. In fact, we
construct two geometric theories and show that they are biinterpretable. The
constructions are elementary and tightly connected to the Riesz space
structure.Comment: Submitted for publication 15/05/0
The four dimensional site-diluted Ising model: a finite-size scaling study
Using finite-size scaling techniques, we study the critical properties of the
site-diluted Ising model in four dimensions. We carry out a high statistics
Monte Carlo simulation for several values of the dilution. The results support
the perturbative scenario: there is only the Ising fixed point with large
logarithmic scaling corrections. We obtain, using the Perturbative
Renormalization Group, functional forms for the scaling of several observables
that are in agreement with the numerical data.Comment: 30 pages, 8 postscript figure
How regular can maxitive measures be?
We examine domain-valued maxitive measures defined on the Borel subsets of a
topological space. Several characterizations of regularity of maxitive measures
are proved, depending on the structure of the topological space. Since every
regular maxitive measure is completely maxitive, this yields sufficient
conditions for the existence of a cardinal density. We also show that every
outer-continuous maxitive measure can be decomposed as the supremum of a
regular maxitive measure and a maxitive measure that vanishes on compact
subsets under appropriate conditions.Comment: 24 page
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