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Yuichi Futa and Yasunari Shidama

Summary

In this article, we formalize the definition of divisible ℤ-module and its properties in the Mizar system [3]. We formally prove that any non-trivial divisible ℤ-modules are not finitely-generated.We introduce a divisible ℤ-module, equivalent to a vector space of a torsion-free ℤ-module with a coefficient ring ℚ. ℤ-modules are important for lattice problems, LLL (Lenstra, Lenstra and Lovász) base reduction algorithm [15], cryptographic systems with lattices [16] and coding theory [8].

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Adam Naumowicz and Radosław Piliszek

. Formalized Mathematics , 1( 4 ):711-714, 1990. [6] Grzegorz Bancerek. Veblen hierarchy. Formalized Mathematics , 19( 2 ):83-92, 2011. doi:10.2478/v10037-011-0014-5. [7] C.C. Briggs. Simple divisibility rules for the 1st 1000 prime numbers. arXiv preprint arXiv:math/0001012 , 2000. [8] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics , 1( 1 ): 55-65, 1990. [9] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics , 1( 1 ):153-164, 1990. [10

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Rafał Ziobro

. [19] Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics , 3(2):275-278, 1992. [20] Richard Krueger, Piotr Rudnicki, and Paul Shelley. Asymptotic notation. Part II: Examples and problems. Formalized Mathematics , 9(1):143-154, 2001. [21] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics , 1(5):887-890, 1990. [22] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics , 1(5):829-832, 1990

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Dana Piciu, Christina Theresia Dan and Florentina Chirteş

, MTL-filters, divisible filters, BL-filters and regular filters in residuated lattices , Iranian Journal of Fuzzy Systems 13(1) (2016) 145-160. [6] L. Ciungu, Non-commutative Multiple - Valued Logic Algebras , Springer, 2013. [7] R. Dilworth, Non-commutative residuated lattices , Trans. Amer. Math. Soc. 46 (1939) 426-444. [8] N. Galatos, P. Jipsen, T. Kowalski, H. Ono, Residuated lattices: an algebraic glimpse at structural logic , Studies in logic and the foundation of math., Vol. 151, Elsevier Science, 2007. [9] P. Hájek, Metamathematics

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Iz-iddine EL-Fassi and Samir Kabbaj

Abstract

In this paper, we prove the hyperstability of the following mixed additive-quadratic-Jensen functional equation

2f(x+y2)+f(xy2)+f(yx2)=f(x)+f(y)
in the class of functions from an 2-divisible abelian group G into a Banach space.

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Rafał Ziobro

evens: An inquiry into students’ understanding of even and odd numbers. Educational Studies in Mathematics , 36(1):73–89, Jun 1998. doi:10.1023/A:1003149901409. [10] Rafał Ziobro. Fermat’s Little Theorem via divisibility of Newton’s binomial. Formalized Mathematics , 23( 3 ):215–229, 2015. doi:10.1515/forma-2015-0018.

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Ivan Chajda and Helmut Länger

Abstract

Since the reduct of every residuated lattice is a semiring, we can ask under what condition a semiring can be converted into a residuated lattice. It turns out that this is possible if the semiring in question is commutative, idempotent, G-simple and equipped with an antitone involution. Then the resulting residuated lattice even satisfies the double negation law. Moreover, if the mentioned semiring is finite then it can be converted into a residuated lattice or join-semilattice also without asking an antitone involution on it. To a residuated lattice L which does not satisfy the double negation law there can be assigned a so-called augmented semiring. This can be used for reconstruction of the so-called core C(L) of L. Conditions under which C(L) constitutes a subuniverse of L are provided.

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Rafał Ziobro

Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990. [6] M.I. Mostafa. A new approach to polynomial identities. The Ramanujan Journal, 8(4): 423-457, 2005. ISSN 1382-4090. doi:10.1007/s11139-005-0272-3. [7] Werner Georg Nowak. On differences of two k-th powers of integers. The Ramanujan Journal, 2(4):421-440, 1998. ISSN 1382-4090. doi:10.1023/A:1009791425210. [8] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized

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Adam Naumowicz

On the Representation of Natural Numbers in Positional Numeral Systems1

In this paper we show that every natural number can be uniquely represented as a base-b numeral. The formalization is based on the proof presented in [11]. We also prove selected divisibility criteria in the base-10 numeral system.

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Sylwia Cichacz, Dalibor Froncek, Elliot Krop and Christopher Raridan

Abstract

A distance magic labeling of a graph G = (V,E) with |V | = n is a bijection ℓ : V → {1, . . . , n} such that the weight of every vertex v, computed as the sum of the labels on the vertices in the open neighborhood of v, is a constant.

In this paper, we show that hypercubes with dimension divisible by four are not distance magic. We also provide some positive results by proving necessary and sufficient conditions for the Cartesian product of certain complete multipartite graphs and the cycle on four vertices to be distance magic.