NONSTANDARD APPROACH TO POSSIBILITY MEASURES

1996 ◽  
Vol 04 (03) ◽  
pp. 275-301 ◽  
Author(s):  
IVAN KRAMOSIL
Keyword(s):  
Random Variable ◽  
Unit Interval ◽  
Partial Ordering ◽  
Linear Ordering ◽  
Possibility Measure ◽  
Real Numbers ◽  
Standard Linear ◽  
General Abstract ◽  
Nonstandard Approach ◽  
Abstract Probability

Possibility measures have been conceived by Zadeh [1], and developed by, e. g., Dubois, Prade and others, as very simple, if compared with, e.g., probability theory, but still nontrivial and reasonable uncertainty measures. The relatively poor descriptional and operational abilities of possibility measures seem to be closely related to the standard linear ordering relation and the corresponding supremum and infimum operations defined over the unit interval of real numbers. Having discussed, very briefly, the possibilities how to overcome these limitations, we propose and investigate possibility measures taking their values in the well-known Cantor subset of the unit interval but defined with respect to a nonstandard operation of supremum resulting from a nonstandard partial ordering of real numbers from the Cantor set. Such a nonstandard possibility measure is proved to be a sufficient tool to define an infinite sequence of probability measures defined over countable sets, consequently, the distribution of each continuous real-valued random variable defined over a general abstract probability space can be defined by an appropriate nonstandard possibility measure.

A Model of the Real Numbers

1963 ◽  
Vol 6 (2) ◽  
pp. 239-255
Author(s):  
Stanton M. Trott
Keyword(s):  
Irrational Number ◽  
Additive Group ◽  
Linear Ordering ◽  
Real Numbers ◽  
Ordered Group ◽  
The Real ◽  
Linear Orderings ◽  
Positive Elements ◽  
The Law ◽  
Ordered Pairs

The model of the real numbers described below was suggested by the fact that each irrational number ρ determines a linear ordering of J2, the additive group of ordered pairs of integers. To obtain the ordering, we define (m, n) ≤ (m', n') to mean that (m'- m)ρ ≤ n' - n. This order is invariant with group translations, and hence is called a "group linear ordering". It is completely determined by the set of its "positive" elements, in this case, by the set of integer pairs (m, n) such that (0, 0) ≤ (m, n), or, equivalently, mρ < n. The law of trichotomy for linear orderings dictates that only the zero of an ordered group can be both positive and negative.

scholarly journals SOME GROUP THEORETIC ASPECTS OF t-NORMS

2000 ◽  
Vol 08 (01) ◽  
pp. 1-6 ◽  
Author(s):  
FRED RICHMAN ◽  
ELBERT WALKER
Keyword(s):  
Automorphism Group ◽  
Multiplicative Group ◽  
Order Relation ◽  
Unit Interval ◽  
Real Numbers ◽  
Positive Real ◽  
Explicit Formulas ◽  
One To One ◽  
The Right ◽  
Usual Order

Let A be the automorphism group of the unit interval with its usual order relation, and let ℝ+ be the embedding of the multiplicative group of positive real numbers into A given by exponentiation. Strict t-norms are in one-to-one correspondence with the right cosets of ℝ+ in A. Here, we identify the normalizer of ℝ+ in A and give explicit formulas for the corresponding set of t-norms.

scholarly journals A Nonuniform Bound to an Independent Test in High Dimensional Data Analysis via Stein’s Method

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Nahathai Rerkruthairat
Keyword(s):  
Normal Approximation ◽  
Correlation Coefficients ◽  
Random Variable ◽  
Stein’S Method ◽  
High Dimensional ◽  
Stein's Method ◽  
Real Numbers ◽  
Complete Independence ◽  
Independent Test ◽  
Sample Correlation

The Berry-Esseen bound for the random variable based on the sum of squared sample correlation coefficients and used to test the complete independence in high diemensions is shown by Stein’s method. Although the Berry-Esseen bound can be applied to all real numbers in R, a nonuniform bound at a real number z usually provides a sharper bound if z is fixed. In this paper, we present the first version of a nonuniform bound on a normal approximation for this random variable with an optimal rate of 1/0.5+|z|·O1/m by using Stein’s method.

A Mathematical Setting for Fuzzy Logics

1997 ◽  
Vol 05 (03) ◽  
pp. 223-238 ◽  
Author(s):  
Mai Gehrke ◽  
Carol Walker ◽  
Elbert Walker
Keyword(s):  
Fuzzy Logic ◽  
Propositional Logic ◽  
Normal Forms ◽  
Unit Interval ◽  
Fuzzy Logics ◽  
Real Numbers ◽  
Truth Values ◽  
Finite Algebras ◽  
Finite Algorithms ◽  
Interval Valued

The setup of a mathematical propositional logic is given in algebraic terms, describing exactly when two choices of truth value algebras give the same logic. The propositional logic obtained when the algebra of truth values is the real numbers in the unit interval equipped with minimum, maximum and -x=1-x for conjunction, disjunction and negation, respectively, is the standard propositional fuzzy logic. This is shown to be the same as three-valued logic. The propositional logic obtained when the algebra of truth values is the set {(a, b)|a≤ b and a,b∈[0,1]} of subintervals of the unit interval with component-wise operations, is propositional interval-valued fuzzy logic. This is shown to be the same as the logic given by a certain four element lattice of truth values. Since both of these logics are equivalent to ones given by finite algebras, it follows that there are finite algorithms for determining when two statements are logically equivalent within either of these logics. On this topic, normal forms are discussed for both of these logics.

A NEW STATISTICAL METHOD FOR FILTERING AND ENTROPY ESTIMATION OF A CHAOTIC MAP FROM NOISY DATA

2004 ◽  
Vol 14 (11) ◽  
pp. 3989-3994 ◽  
Author(s):  
G. JOGESH BABU ◽  
ABRAHAM BOYARSKY ◽  
YOGENDRA P. CHAUBEY ◽  
PAWEL GÓRA
Keyword(s):  
Statistical Method ◽  
Chaotic Map ◽  
Noisy Data ◽  
Random Variable ◽  
Unit Interval ◽  
Noise Amplitude ◽  
Small Noise ◽  
Chaotic Dynamical System ◽  
Entropy Estimation ◽  
Noise Contamination

We consider a discrete time deterministic chaotic dynamical system, xn+1=τ(xn), where τ is a nonlinear map of the unit interval into itself. We assume that τ is piecewise expanding and piecewise C2. The effects of noise contamination are modeled by xn+1=τ(xn)+ξn, where ξn is an independent random variable with small noise amplitude. A new statistical method is presented for filtering τ and estimating the metric entropy of τ from observed noisy data.

PRESERVATION OF LOG-CONCAVITY AND LOG-CONVEXITY UNDER OPERATORS

2020 ◽  
pp. 1-14
Author(s):  
Wanwan Xia ◽  
Tiantian Mao ◽  
Taizhong Hu
Keyword(s):  
Distribution Function ◽  
Operations Research ◽  
Random Variable ◽  
Group Property ◽  
Semi Group ◽  
Real Numbers ◽  
Bernstein Type ◽  
Stochastic Comparisons ◽  
Weighted Distributions ◽  
The Family

Log-concavity [log-convexity] and their various properties play an increasingly important role in probability, statistics, operations research and other fields. In this paper, we first establish general preservation theorems of log-concavity and log-convexity under operator $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$ , θ ∈ Θ, where Θ is an interval of real numbers or an interval of integers, and the random variable $X_\theta$ has a distribution function belonging to the family $\{F_\theta , \theta \in \Theta \}$ possessing the semi-group property. The proofs are based on the theory of stochastic comparisons and weighted distributions. The main results are applied to some special operators, for example, operators occurring in reliability, Bernstein-type operators and Beta-type operators. Several known results in the literature are recovered.

A High–Low search game on the unit interval

1985 ◽  
Vol 97 (2) ◽  
pp. 345-348 ◽  
Author(s):  
V. J. Baston ◽  
F. A. Bostock
Keyword(s):  
Real Number ◽  
Closed Interval ◽  
Unit Interval ◽  
Real Numbers ◽  
Search Game ◽  
Zero Sum

We consider the following two-person zero-sum game on the closed interval [0,1]. The hider chooses any real number h in [0,1]. The searcher successively chooses real numbers ξ1ξ2… in [0,1], where at each choice ξi he is told whether h = ξih < ξ1 or h > ξi and he may choose ξi+1 in the light of this information. The payoff (to the hider) is the sum of the distances of the searcher from the hider at each of the moves, that is .

scholarly journals Degrees of monotone complexity

2006 ◽  
Vol 71 (4) ◽  
pp. 1327-1341 ◽  
Author(s):  
William C. Calhoun
Keyword(s):  
Kolmogorov Complexity ◽  
Minimal Pair ◽  
Partial Ordering ◽  
Binary String ◽  
Linear Ordering ◽  
Relative Complexity ◽  
Computably Enumerable

AbstractLevin and Schnorr (independently) introduced the monotone complexity, Km (α), of a binary string α. We use monotone complexity to define the relative complexity (or relative randomness) of reals. We define a partial ordering ≤Km on 2ω by α ≤Km β iff there is a constant c such that Km(α | n) ≤ Km(β | n)+ c for all n. The monotone degree of α is the set of all β such that α Km β and β Km α. We show the monotone degrees contain an antichain of size , a countable dense linear ordering (of degrees of cardinality ), and a minimal pair.Downey, Hirschfeldt, LaForte, Nies and others have studied a similar structure, the K-degrees, where K is the prefix-free Kolmogorov complexity. A minimal pair of K-degrees was constructed by Csima and Montalban. Of particular interest are the noncomputable trivial reals, first constructed by Solovay. We defineareal to be (Km,K)-trivial if for some constant c, Km(α | n) ≤ K(n) + c for all n. It is not known whether there is a Km-minimal real, but we show that any such real must be (Km,K)-trivial.Finally, we consider the monotone degrees of the computably enumerable (c.e.) and strongly computably enumerable (s.c.e.) reals. We show there is no minimal c.e. monotone degree and that Solovay reducibility does not imply monotone reducibility on the c.e. reals. We also show the s.c.e. monotone degrees contain an infinite antichain and a countable dense linear ordering.

scholarly journals TOPOLOGICAL FULL GROUPS OF -ALGEBRAS ARISING FROM -EXPANSIONS

2014 ◽  
Vol 97 (2) ◽  
pp. 257-287 ◽  
Author(s):  
KENGO MATSUMOTO ◽  
HIROKI MATUI
Keyword(s):  
Real Line ◽  
Piecewise Linear ◽  
Unit Interval ◽  
Linear Functions ◽  
Discrete Groups ◽  
Piecewise Linear Functions ◽  
Real Numbers ◽  
The Real ◽  
Theoretical Property ◽  
Thompson Groups

AbstractWe introduce a family of infinite nonamenable discrete groups as an interpolation of the Higman–Thompson groups by using the topological full groups of the groupoids defined by $\beta $-expansions of real numbers. They are regarded as full groups of certain interpolated Cuntz algebras, and realized as groups of piecewise-linear functions on the unit interval in the real line if the $\beta $-expansion of $1$ is finite or ultimately periodic. We also classify them by a number-theoretical property of $\beta $.

scholarly journals Complete moment convergence of moving average processes for m-WOD sequence

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lihong Guan ◽  
Yushan Xiao ◽  
Yanan Zhao
Keyword(s):  
Moving Average ◽  
Random Variables ◽  
Random Variable ◽  
Complete Moment Convergence ◽  
Real Numbers ◽  
Mild Conditions ◽  
Moment Convergence ◽  
Summable Sequence ◽  
Moving Average Processes ◽  
Widely Orthant Dependent

AbstractIn this paper, the complete moment convergence for the partial sum of moving average processes $\{X_{n}=\sum_{i=-\infty }^{\infty }a_{i}Y_{i+n},n\geq 1\}$ { X n = ∑ i = − ∞ ∞ a i Y i + n , n ≥ 1 } is established under some mild conditions, where $\{Y_{i},-\infty < i<\infty \}$ { Y i , − ∞ < i < ∞ } is a sequence of m-widely orthant dependent (m-WOD, for short) random variables which is stochastically dominated by a random variable Y, and $\{a_{i},-\infty < i<\infty \}$ { a i , − ∞ < i < ∞ } is an absolutely summable sequence of real numbers. These conclusions promote and improve the corresponding results from m-extended negatively dependent (m-END, for short) sequences to m-WOD sequences.

Sign in / Sign up

Export Citation Format

Share Document