# Search Results

###### Arithmetic Operations on Short Finite Sequences

Publishing, 2015. ISBN 978-3-319-20614-1. doi:10.1007/978-3-319-20615-8_17. [3] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning , 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6. [4] Alexander Elizarov, Alexander Kirillovich, Evgeny Lipachev, and Olga Nevzorova. Digital ecosystem OntoMath: Mathematical knowledge analytics and management. In Leonid Kalinichenko, Sergei O

###### Tarski Geometry Axioms. Part III

. Archive of Formal Proofs, October 2012. Formal proof development. [9] Timothy James McKenzie Makarios. A further simplification of Tarski’s axioms of geometry. Note di Matematica, 33(2):123-132, 2014. [10] Julien Narboux. Mechanical theorem proving in Tarski’s geometry. In F. Botana and T. Recio, editors, Automated Deduction in Geometry, volume 4869 of Lecture Notes in Computer Science, pages 139-156. Springer, 2007. [11] William Richter, Adam Grabowski, and Jesse Alama. Tarski geometry axioms. Formalized Mathematics, 22

###### Fundamental Properties of Fuzzy Implications

References [1] Michał Baczyński and Balasubramaniam Jayaram. Fuzzy Implications . Springer Publishing Company, Incorporated, 2008. doi:10.1007/978-3-540-69082-5. [2] Grzegorz Bancerek, Czesław Byliński, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pąk. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning , 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6. [3] Didier Dubois and Henri Prade. Fuzzy Sets and Systems: Theory and Applications

###### Extension of tumor fingers: A comparison between an individual-cell based model and a measure theoretic approach

, and N. J. Poplawski, Magnetization to morphogenesis: a brief history of the Glazier-Graner-Hogeweg model, in Single-Cell-Based Models in Biology and Medicine (A. R. A. Anderson, M. A. J. Chaplain, and K. A. Rejniak, eds.), Mathematics and Biosciences in Interactions, pp. 79–106, Birkaüser, 2007. 11. F. Graner and J. A. Glazier, Simulation of biological cell sorting using a two dimensional extended Potts model, Phys. Rev. Lett. , vol. 69, pp. 2013–2017, 1992. 12. M. Scianna and L. Preziosi, Multiscale developments of the cellular Potts model

###### Ordered Rings and Fields

## Summary

We introduce ordered rings and fields following Artin-Schreier’s approach using positive cones. We show that such orderings coincide with total order relations and give examples of ordered (and non ordered) rings and fields. In particular we show that polynomial rings can be ordered in (at least) two different ways [8, 5, 4, 9]. This is the continuation of the development of algebraic hierarchy in Mizar [2, 3].

###### Formal Introduction to Fuzzy Implications

## Summary

In the article we present in the Mizar system the catalogue of nine basic fuzzy implications, used especially in the theory of fuzzy sets. This work is a continuation of the development of fuzzy sets in Mizar; it could be used to give a variety of more general operations, and also it could be a good starting point towards the formalization of fuzzy logic (together with t-norms and t-conorms, formalized previously).

###### Klein-Beltrami Model. Part II

## Summary

Tim Makarios (with Isabelle/HOL^{1}) and John Harrison (with HOL-Light^{2}) have shown that “the Klein-Beltrami model of the hyperbolic plane satisfy all of Tarski’s axioms except his Euclidean axiom” [2, 3, 15, 4].

With the Mizar system [1], [10] we use some ideas are taken from Tim Makarios’ MSc thesis [12] for formalized some definitions (like the tangent) and lemmas necessary for the verification of the independence of the parallel postulate. This work can be also treated as a further development of Tarski’s geometry in the formal setting [9].

###### Prime Factorization of Sums and Differences of Two Like Powers

## Abstract

Representation of a non zero integer as a signed product of primes is unique similarly to its representations in various types of positional notations [4], [3]. The study focuses on counting the prime factors of integers in the form of sums or differences of two equal powers (thus being represented by 1 and a series of zeroes in respective digital bases).

Although the introduced theorems are not particularly important, they provide a couple of shortcuts useful for integer factorization, which could serve in further development of Mizar projects [2]. This could be regarded as one of the important benefits of proof formalization [9].

###### Lagrange–Galerkin methods for the incompressible Navier-Stokes equations: a review

## Abstract

We review in this paper the development of Lagrange-Galerkin (LG) methods to integrate the incompressible Navier-Stokes equations (NSEs) for engineering applications. These methods were introduced in the computational fluid dynamics community in the early eighties of the past century, and at that time they were considered good methods for both their theoretical stability properties and the way of dealing with the nonlinear terms of the equations; however, the numerical experience gained with the application of LG methods to different problems has identified drawbacks of them, such as the calculation of specific integrals that arise in their formulation and the calculation of the ow trajectories, which somehow have hampered the applicability of LG methods. In this paper, we focus on these issues and summarize the convergence results of LG methods; furthermore, we shall briefly introduce a new stabilized LG method suitable for high Reynolds numbers.