80,047 research outputs found

### Partial orderings for hesitant fuzzy sets

New partial orderings =o=o, =p=p and =H=H are defined, studied and compared on the set HH of finite subsets of the unit interval with special emphasis on the last one. Since comparing two sets of the same cardinality is a simple issue, the idea for comparing two sets A and B of different cardinalities n and m respectively using =H=H is repeating their elements in order to obtain two series with the same length. If lcm(n,m)lcm(n,m) is the least common multiple of n and m we can repeat every element of A lcm(n,m)/mlcm(n,m)/m times and every element of B lcm(n,m)/nlcm(n,m)/n times to obtain such series and compare them (Definition 2.2).
(H,=H)(H,=H) is a bounded partially ordered set but not a lattice. Nevertheless, it will be shown that some interesting subsets of (H,=H)(H,=H) have a lattice structure. Moreover in the set BB of finite bags or multisets (i.e. allowing repetition of objects) of the unit interval a preorder =B=B can be defined in a similar way as =H=H in HH and considering the quotient set View the MathML sourceBÂż=B/~ of BB by the equivalence relation ~ defined by A~BA~B when A=BBA=BB and B=BAB=BA, View the MathML source(BÂż,=B) is a lattice and (H,=H)(H,=H) can be naturally embedded into it.Peer ReviewedPostprint (author's final draft

### Brouwerâ€™s Fan Theorem and Convexity

In the framework of Bishopâ€™s constructive mathematics we introduce co-convexity as a property of subsets B of , the set of finite binary sequences, and prove that co-convex bars are uniform. Moreover, we establish a canonical correspondence between detachable subsets B of and uniformly continuous functions f defined on the unit interval such that B is a bar if and only if the corresponding function f is positive-valued, B is a uniform bar if and only if f has positive infimum, and B is co-convex if and only if f satisfies a weak convexity condition

### The principle of inclusion-exclusion and mĂ¶bius function as counting techniques in finite fuzzy subsets

The broad goal in this thesis is to enumerate elements and fuzzy subsets of a finite set enjoying some useful properties through the well-known counting technique of the principle of inclusion-exclusion. We consider the set of membership values to be finite and uniformly spaced in the real unit interval. Further we define an equivalence relation with regards to the cardinalities of fuzzy subsets providing the MĂ¶bius function and MĂ¶bius inversion in that context

### Brittleness of Bayesian inference and new Selberg formulas

The incorporation of priors in the Optimal Uncertainty Quantification (OUQ)
framework \cite{OSSMO:2011} reveals brittleness in Bayesian inference; a model
may share an arbitrarily large number of finite-dimensional marginals with, or
be arbitrarily close (in Prokhorov or total variation metrics) to, the
data-generating distribution and still make the largest possible prediction
error after conditioning on an arbitrarily large number of samples. The initial
purpose of this paper is to unwrap this brittleness mechanism by providing (i)
a quantitative version of the Brittleness Theorem of \cite{BayesOUQ} and (ii) a
detailed and comprehensive analysis of its application to the revealing example
of estimating the mean of a random variable on the unit interval $[0,1]$ using
priors that exactly capture the distribution of an arbitrarily large number of
Hausdorff moments.
However, in doing so, we discovered that the free parameter associated with
Markov and Kre\u{\i}n's canonical representations of truncated Hausdorff
moments generates reproducing kernel identities corresponding to reproducing
kernel Hilbert spaces of polynomials.
Furthermore, these reproducing identities lead to biorthogonal systems of
Selberg integral formulas.
This process of discovery appears to be generic: whereas Karlin and Shapley
used Selberg's integral formula to first compute the volume of the Hausdorff
moment space (the polytope defined by the first $n$ moments of a probability
measure on the interval $[0,1]$), we observe that the computation of that
volume along with higher order moments of the uniform measure on the moment
space, using different finite-dimensional representations of subsets of the
infinite-dimensional set of probability measures on $[0,1]$ representing the
first $n$ moments, leads to families of equalities corresponding to classical
and new Selberg identities.Comment: 73 pages. Keywords: Bayesian inference, misspecification, robustness,
uncertainty quantification, optimal uncertainty quantification, reproducing
kernel Hilbert spaces (RKHS), Selberg integral formula

### Densities for sets of natural numbers vanishing on a given family

Abstract upper densities are monotone and subadditive functions from the
power set of positive integers into the unit real interval that generalize the
upper densities used in number theory, including the upper asymptotic density,
the upper Banach density, and the upper logarithmic density.
At the open problem session of the Workshop ``Densities and their
application'', held at St. \'{E}tienne in July 2013, G. Grekos asked a question
whether there is a ``nice'' abstract upper density, whose the family of null
sets is precisely a given ideal of subsets of $\mathbb{N}$, where ``nice''
would mean the properties of the familiar densities consider in number theory.
In 2018, M. Di Nasso and R. Jin (Acta Arith. 185 (2018), no. 4) showed that
the answer is positive for the summable ideals (for instance, the family of
finite sets and the family of sequences whose series of reciprocals converge)
when ``nice'' density means translation invariant and rich density (i.e.
density which is onto the unit interval).
In this paper we extend their result to all ideals with the Baire property

### Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees

We study the computational difficulty of the problem of finding fixed points
of nonexpansive mappings in uniformly convex Banach spaces. We show that the
fixed point sets of computable nonexpansive self-maps of a nonempty, computably
weakly closed, convex and bounded subset of a computable real Hilbert space are
precisely the nonempty, co-r.e. weakly closed, convex subsets of the domain. A
uniform version of this result allows us to determine the Weihrauch degree of
the Browder-Goehde-Kirk theorem in computable real Hilbert space: it is
equivalent to a closed choice principle, which receives as input a closed,
convex and bounded set via negative information in the weak topology and
outputs a point in the set, represented in the strong topology. While in finite
dimensional uniformly convex Banach spaces, computable nonexpansive mappings
always have computable fixed points, on the unit ball in infinite-dimensional
separable Hilbert space the Browder-Goehde-Kirk theorem becomes
Weihrauch-equivalent to the limit operator, and on the Hilbert cube it is
equivalent to Weak Koenig's Lemma. In particular, computable nonexpansive
mappings may not have any computable fixed points in infinite dimension. We
also study the computational difficulty of the problem of finding rates of
convergence for a large class of fixed point iterations, which generalise both
Halpern- and Mann-iterations, and prove that the problem of finding rates of
convergence already on the unit interval is equivalent to the limit operator.Comment: 44 page

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