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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]...

Maximal almost disjoint families of functions

Dilip Raghavan (2009)

Fundamenta Mathematicae

We study maximal almost disjoint (MAD) families of functions in ω ω that satisfy certain strong combinatorial properties. In particular, we study the notions of strongly and very MAD families of functions. We introduce and study a hierarchy of combinatorial properties lying between strong MADness and very MADness. Proving a conjecture of Brendle, we show that if c o v ( ℳ ) < , then there no very MAD families. We answer a question of Kastermans by constructing a strongly MAD family from = . Next, we study the...

Maximal free sequences in a Boolean algebra

J. D. Monk (2011)

Commentationes Mathematicae Universitatis Carolinae

We study free sequences and related notions on Boolean algebras. A free sequence on a BA A is a sequence 〈 a ξ : ξ < α 〉 of elements of A , with α an ordinal, such that for all F , G ∈ [ α ] < ω with F < G we have ∏ ξ ∈ F a ξ · ∏ ξ ∈ G - a ξ ≠ 0 . A free sequence of length α exists iff the Stone space Ult ( A ) has a free sequence of length α in the topological sense. A free sequence is maximal iff it cannot be extended at the end to a longer free sequence. The main notions studied here are the spectrum function 𝔣 sp ( A ) = { | α | : A has an infinite maximal free sequence of length α } and the associated min-max function 𝔣 ( A ) = min ( 𝔣 sp ( A ) ) . Among the results...

Measurable cardinals and category bases

Andrzej Szymański (1991)

Commentationes Mathematicae Universitatis Carolinae

We show that the existence of a non-trivial category base on a set of regular cardinality with each subset being Baire is equiconsistent to the existence of a measurable cardinal.

Measurable cardinals and fundamental groups of compact spaces

Adam Przeździecki (2006)

Fundamenta Mathematicae

We prove that all groups can be realized as fundamental groups of compact spaces if and only if no measurable cardinals exist. If the cardinality of a group G is nonmeasurable then the compact space K such that G = π₁K may be chosen so that it is path connected.

Measurable cardinals and the cofinality of the symmetric group

Sy-David Friedman, Lyubomyr Zdomskyy (2010)

Fundamenta Mathematicae

Assuming the existence of a P₂κ-hypermeasurable cardinal, we construct a model of Set Theory with a measurable cardinal κ such that 2 κ = κ ⁺ ⁺ and the group Sym(κ) of all permutations of κ cannot be written as the union of a chain of proper subgroups of length < κ⁺⁺. The proof involves iteration of a suitably defined uncountable version of the Miller forcing poset as well as the “tuning fork” argument introduced by the first author and K. Thompson [J. Symbolic Logic 73 (2008)].

Measurable envelopes, Hausdorff measures and Sierpiński sets

Márton Elekes (2003)

Colloquium Mathematicae

We show that the existence of measurable envelopes of all subsets of ℝⁿ with respect to the d-dimensional Hausdorff measure (0 < d < n) is independent of ZFC. We also investigate the consistency of the existence of ℋ d -measurable Sierpiński sets.

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