scholarly journals Primes in Intervals and Semicircular Elements Induced by p-Adic Number Fields Qp over Primes p

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 199 ◽  
Author(s):  
Ilwoo Cho ◽  
Palle Jorgensen
Keyword(s):  
Probability Space ◽  
Free Probability ◽  
Number Fields ◽  
Linear Functionals ◽  
Real Numbers ◽  
Probabilistic Information ◽  
Closed Intervals ◽  
Measurable Functions ◽  
Semicircular Elements

In this paper, we study free probability on (weighted-)semicircular elements in a certain Banach *-probability space ( LS , τ 0 ) induced by measurable functions on p-adic number fields Q p over primes p . In particular, we are interested in the cases where such free-probabilistic information is affected by primes in given closed intervals of the set R of real numbers by defining suitable “truncated” linear functionals on LS .

scholarly journals Deformation of semicircular and circular laws via p-adic number fields and sampling of primes

2019 ◽  
Vol 39 (6) ◽  
pp. 773-813
Author(s):  
Ilwoo Cho ◽  
Palle E. T. Jorgensen
Keyword(s):  
Banach Algebra ◽  
Probability Space ◽  
Number Fields ◽  
Linear Functionals ◽  
Real Numbers ◽  
Circular Law ◽  
Semicircular Law ◽  
Different Types ◽  
Semicircular Elements

In this paper, we study semicircular elements and circular elements in a certain Banach \(*\)-probability space \((\mathfrak{LS},\tau ^{0})\) induced by analysis on the \(p\)-adic number fields \(\mathbb{Q}_{p}\) over primes \(p\). In particular, by truncating the set \(\mathcal{P}\) of all primes for given suitable real numbers \(t\lt s\) in \(\mathbb{R}\), two different types of truncated linear functionals \(\tau_{t_{1}\lt t_{2}}\), and \(\tau_{t_{1}\lt t_{2}}^{+}\) are constructed on the Banach \(*\)-algebra \(\mathfrak{LS}\). We show how original free distributional data (with respect to \(\tau ^{0}\)) are distorted by the truncations on \(\mathcal{P}\) (with respect to \(\tau_{t\lt s}\), and \(\tau_{t\lt s}^{+}\)). As application, distorted free distributions of the semicircular law, and those of the circular law are characterized up to truncation.

scholarly journals Banach-Space Operators Acting on Semicircular Elements Induced by p-Adic Number Fields over Primes p

Mathematics ◽  
2019 ◽  
Vol 7 (6) ◽  
pp. 498
Author(s):  
Ilwoo Cho
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Number Fields ◽  
Semicircular Elements ◽  
Banach Space Operators

In this paper, we study certain Banach-space operators acting on the Banach *-probability space ( LS , τ 0 ) generated by semicircular elements Θ p , j induced by p-adic number fields Q p over the set P of all primes p. Our main results characterize the operator-theoretic properties of such operators, and then study how ( LS , τ 0 ).

Dempster Combination Rule for Signed Belief Functions

1998 ◽  
Vol 06 (01) ◽  
pp. 79-102 ◽  
Author(s):  
Ivan Kramosil
Keyword(s):  
Large Class ◽  
Probability Space ◽  
Binary Operation ◽  
Random Variables ◽  
Belief Function ◽  
Measurable Space ◽  
Point Of View ◽  
Belief Functions ◽  
Combination Rule ◽  
Real Numbers

A possibility to define a binary operation over the space of pairs of belief functions, inverse or dual to the well-known Dempster combination rule in the same sense in which substraction is dual with respect to the addition operation in the space of real numbers, can be taken as an important problem for the purely algebraic as well as from the application point of view. Or, it offers a way how to eliminate the modification of a belief function obtained when combining this original belief function with other pieces of information, later proved not to be reliable. In the space of classical belief functions definable by set-valued (generalized) random variables defined on a probability space, the invertibility problem for belief functions, resulting from the above mentioned problem of "dual" combination rule, can be proved to be unsolvable up to trivial cases. However, when generalizing the notion of belief functions in such a way that probability space is replaced by more general measurable space with signed measure, inverse belief functions can be defined for a large class of belief functions generalized in the corresponding way. "Dual" combination rule is then defined by the application of the Dempster rule to the inverse belief functions.

Extension of linear functionals in Banach spaces of measurable functions

1976 ◽  
Vol 20 (5) ◽  
pp. 969-973
Author(s):  
M. S. Braverman ◽  
G. Ya. Lozanovskii
Keyword(s):  
Banach Spaces ◽  
Linear Functionals ◽  
Measurable Functions

Large correlated families of positive random variables

1988 ◽  
Vol 103 (1) ◽  
pp. 147-162 ◽  
Author(s):  
D. H. Fremlin
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Characteristic Functions ◽  
Measurable Functions ◽  
Uniformly Bounded

S. Argyros and N. Kalamidas([l], repeated in [2], Theorem 6·15) proved the following. If κ is a cardinal of uncountable cofinality, and 〈Eξ〉ξ<κ is a family of measurable sets in a probability space (X, μ) such that infξ<κμEξ = δ, and if n ≥ 1, , then there is a set Γ ⊆ κ such that #(Γ) = κ and μ(∩ξ∈IEξ) ≥ γ whenever I ⊆ ξ has n members. In Proposition 7 below I refine this result by (i) taking any γ < δn (which is best possible) and (ii) extending the result to infinite cardinals of countable cofinality, thereby removing what turns out to be an irrelevant restriction. The proof makes it natural to perform a further extension to general uniformly bounded families of non-negative measurable functions in place of characteristic functions.

Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

scholarly journals Discrete temperature values in the sintering process as a BaTiO3-ceramics properties parameter

2017 ◽  
Vol 49 (4) ◽  
pp. 469-477
Author(s):  
Z.B. Vosika ◽  
V.V. Mitic ◽  
G.M. Lazovic ◽  
Lj. Kocic
Keyword(s):  
Time Scale ◽  
Finite Union ◽  
Sintering Process ◽  
Temperature Step ◽  
Real Numbers ◽  
Closed Intervals ◽  
Complex Process ◽  
Crystal Growth Kinetics ◽  
Basic Temperature ◽  
Kinetics Of

In this paper, we develop the new physical-mathematical time scale approach-model applied to BaTiO3-ceramics. At the beginning, a time scale is defined to be an arbitrary closed subset of the real numbers R, with the standard inherited topology. The time scale mathematical examples include real numbers R, natural numbers N, integers Z, the Cantor set (i.e. fractals), and any finite union of closed intervals of R. Calculus on time scales (TSC) was established in 1988 by Stefan Hilger. TSC, by construction, is used to describe the complex process. This method may be utilized for a description of physical, material (crystal growth kinetics, physical chemistry kinetics - for example, kinetics of barium-titanate synthesis), bio-chemical or similar systems and represents a major challenge for nowadays contemporary scientists. Generally speaking, such processes may be described by a discrete time scale. Reasonably it could be assumed that such a ?scenario? is possible for discrete temperature values as a consolidation parameter which is the basic ceramics description properties. In this work, BaTiO3-ceramics discrete temperature as thermodynamics parameter with temperature step h and the basic temperature point a is investigated. Instead of derivations, it is used backward differences with respect to temperature. The main conclusion is made towards ceramics materials temperature as description parameter.

On the Equivalence of Modes of Convergence(1)

1973 ◽  
Vol 16 (4) ◽  
pp. 571-575 ◽  
Author(s):  
R. J. Tomkins
Keyword(s):  
Probability Space ◽  
Random Variables ◽  
Real Numbers ◽  
Probability Zero

Let (Ω,ℱ, P) be a probability space. Let R denote the set of real numbers and the set of all random variables defined on Ω. Throughout this work, random variables which differ only on a set of probability zero will be considered identical. EX represents, as usual, the expectation of .

Classical Analysis

2019 ◽  
pp. 51-69
Author(s):  
John Stillwell
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Uniform Continuity ◽  
Cantor Set ◽  
Binary Trees ◽  
Real Numbers ◽  
Limit Concept ◽  
Closed Intervals ◽  
Basic Concepts

This chapter explores the basic concepts that arise when real numbers and continuous functions are studied, particularly the limit concept and its use in proving properties of continuous functions. It gives proofs of the Bolzano–Weierstrass and Heine–Borel theorems, and the intermediate and extreme value theorems for continuous functions. Also, the chapter uses the Heine–Borel theorem to prove uniform continuity of continuous functions on closed intervals, and its consequence that any continuous function is Riemann integrable on closed intervals. In several of these proofs there is a construction by “infinite bisection,” which can be recast as an argument about binary trees. Here, the chapter uses the role of trees to construct an object—the so-called Cantor set.

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