# Search Results

###### The Derivations of Temporal Logic Formulas

. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics , 1( 3 ):529-536, 1990. [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics , 1( 1 ):55-65, 1990. [7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics , 1( 1 ):153-164, 1990. [8] Czesław Bylinski. Partial functions. Formalized Mathematics , 1( 2 ):357-367, 1990. [9] Mariusz Giero. The axiomatization of propositional linear time temporal logic. Formalized Mathematics

###### The Axiomatization of Propositional Linear Time Temporal Logic

. Temporal Logic and State Systems . Springer-Verlag, 2008. [10] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics , 1( 1 ):115-122, 1990. [11] Andrzej Trybulec. Defining by structural induction in the positive propositional language. Formalized Mathematics , 8( 1 ):133-137, 1999. [12] Zinaida Trybulec. Properties of subsets. Formalized Mathematics , 1( 1 ):67-71, 1990. [13] Edmund Woronowicz. Many-argument relations. Formalized

###### The Properties of Sets of Temporal Logic Subformulas

. [10] Czesław Bylinski. Partial functions. Formalized Mathematics , 1( 2 ):357-367, 1990. [11] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics , 1( 1 ):47-53, 1990. [12] Agata Darmochwał. Finite sets. Formalized Mathematics , 1( 1 ):165-167, 1990. [13] Mariusz Giero. The axiomatization of propositional linear time temporal logic. FormalizedMathematics , 19( 2 ):113-119, 2011, doi: 10.2478/v10037-011-0018-1. [14] Mariusz Giero. The derivations of temporal logic formulas

###### Weak Completeness Theorem for Propositional Linear Time Temporal Logic

. Formalized Mathematics , 1( 1 ):165-167, 1990. [16] Mariusz Giero. The axiomatization of propositional linear time temporal logic. Formalized Mathematics , 19( 2 ):113-119, 2011, doi: 10.2478/v10037-011-0018-1. [17] Mariusz Giero. The derivations of temporal logic formulas. Formalized Mathematics , 20( 3 ):215-219, 2012, doi: 10.2478/v10037-012-0025-x. [18] Mariusz Giero. The properties of sets of temporal logic subformulas. Formalized Mathematics , 20( 3 ):221-226, 2012, doi: 10.2478/v10037-012-0026-9. [19

###### Propositional Linear Temporal Logic with Initial Validity Semantics

1 This work was supported by the University of Bialystok grants: BST447 Formalization of temporal logics in a proof-assistant. Application to System Verification , and BST225 Database of mathematical texts checked by computer . R eferences [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics , 1( 1 ):41–46, 1990. [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics , 1( 1 ):91–96, 1990. [3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite

###### A BGK model for charge transport in graphene

## Abstract

The classical Boltzmann equation describes well temporal behaviour of a rarefied perfect gas. Modified kinetic equations have been proposed for studying the dynamics of different type of gases. An important example is the transport equation, which describes the charged particles flow, in the semi-classical regime, in electronic devices. In order to reduce the difficulties in solving the Boltzmann equation, simple expressions of a collision operator have been proposed to replace the standard Boltzmann integral term. These new equations are called kinetic models. The most popular and widely used kinetic model is the Bhatnagar-Gross-Krook (BGK) model. In this work we propose and analyse a BGK model for charge transport in graphene.

###### Model Checking. Part II

## Model Checking. Part II

This article provides the definition of linear temporal logic (LTL) and its properties relevant to model checking based on [9]. Mizar formalization of LTL language and satisfiability is based on [2, 3].

###### An inversion method based on random sampling for real-time MEG neuroimaging

## Abstract

The MagnetoEncephaloGraphy (MEG) has gained great interest in neurorehabilitation training due to its high temporal resolution. The challenge is to localize the active regions of the brain in a fast and accurate way. In this paper we use an inversion method based on random spatial sampling to solve the real-time MEG inverse problem. Several numerical tests on synthetic but realistic data show that the method takes just a few hundredths of a second on a laptop to produce an accurate map of the electric activity inside the brain. Moreover, it requires very little memory storage. For these reasons the random sampling method is particularly attractive in real-time MEG applications.

###### Model Checking. Part III

## Model Checking. Part III

This text includes verification of the basic algorithm in Simple On-the-fly Automatic Verification of Linear Temporal Logic (LTL). LTL formula can be transformed to Buchi automaton, and this transforming algorithm is mainly used at Simple On-the-fly Automatic Verification. In this article, we verified the transforming algorithm itself. At first, we prepared some definitions and operations for transforming. And then, we defined the Buchi automaton and verified the transforming algorithm.

MML identifier: MODELC 3, version: 7.9.03 4.108.1028

###### On an optimal control strategy in a kinetic social dynamics model

## Abstract

Kinetic models have so far been used to model wealth distribution in a society. In particular, within the framework of the kinetic theory for active particles, some important models have been developed and proposed. They involve nonlinear interactions among individuals that are modeled according to game theoretical tools by introducing parameters governing the temporal dynamics of the system. In this present paper we propose an approach based on optimal control tools that aims to optimize this evolving dynamics from a social point of view. Namely, we look for time dependent control variables concerning the distribution of wealth that can be managed, for instance, by the government, in order to obtain a given social profile.