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Matrix of ℤ-module1

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2015)

Formalized Mathematics

In this article, we formalize a matrix of ℤ-module and its properties. Specially, we formalize a matrix of a linear transformation of ℤ-module, a bilinear form and a matrix of the bilinear form (Gramian matrix). We formally prove that for a finite-rank free ℤ-module V, determinant of its Gramian matrix is constant regardless of selection of its basis. ℤ-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm and cryptographic systems with lattices [22]...

Metric similarities in the logic of approximation.

Michael Katz (1982)

Stochastica

We describe restricted and extended versions of the logic of approximation which is meant to handle formally the problems of measurement error and of deduction under conditions of uncertainty. We apply the logic to the foundations of social and behavioral inquiry, axiomatizing in it an inexact similarity predicate which behaves like a metric approximation to identity. In the restricted version of the logic we formulate conditions for the imbeddability of similarity models in the real line, and in...

Migrativity properties of 2-uninorms over semi-t-operators

Ying Li-Jun, Qin Feng (2022)

Kybernetika

In this paper, we analyze and characterize all solutions about α -migrativity properties of the five subclasses of 2-uninorms, i. e. C k , C k 0 , C k 1 , C 1 0 , C 0 1 , over semi-t-operators. We give the sufficient and necessary conditions that make these α -migrativity equations hold for all possible combinations of 2-uninorms over semi-t-operators. The results obtained show that for G ∈ C k , the α -migrativity of G over a semi-t-operator F μ , ν is closely related to the α -section of F μ , ν or the ordinal sum representation of t-norm...

Modeling biased information seeking with second order probability distributions

Gernot D. Kleiter (2015)

Kybernetika

Updating probabilities by information from only one hypothesis and thereby ignoring alternative hypotheses, is not only biased but leads to progressively imprecise conclusions. In psychology this phenomenon was studied in experiments with the “pseudodiagnosticity task”. In probability logic the phenomenon that additional premises increase the imprecision of a conclusion is known as “degradation”. The present contribution investigates degradation in the context of second order probability distributions....

Modelling Real World Using Stochastic Processes and Filtration

Peter Jaeger (2016)

Formalized Mathematics

First we give an implementation in Mizar [2] basic important definitions of stochastic finance, i.e. filtration ([9], pp. 183 and 185), adapted stochastic process ([9], p. 185) and predictable stochastic process ([6], p. 224). Second we give some concrete formalization and verification to real world examples. In article [8] we started to define random variables for a similar presentation to the book [6]. Here we continue this study. Next we define the stochastic process. For further definitions...

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