In the first miniature we recall a puzzle proposed by Martin Gardner in 1957 and refreshed by Blasco in 2014. We prove some properties of the matrices involved. We show that Gardner magical matrices are tropically singular. In the second miniature we construct a counterexample to the following conjecture by R. Flores: for a square matrix A = (aij ) of size n, if for each permutation σ there exists a 2 × 2 minor of A such that aσ(k)k aσ(l)l - aσ(k)l aσ(l)k vanishes, then determinant of A vanishes.
This paper examines the weak Rationality Conjecture for (r −1)-connected CW-complexes X admitting a unique spherical cohomology class u ∈ Hr (X, Z) for some r ≥ 2. We affirm this conjecture for such complexes of dimension kr where uk ̸= 0. We provide a non-exhaustive list of finite CW-complexes satisfying the conjecture. Furthermore, we investigate some high-torsion Lens spaces Lm2n+1, which are defined as orbit spaces S2n+1 /Zm . The generating functions of these lens spaces confirm both the Weak and Strong Rationality Conjectures under certain conditions.
We study Lipschitz algebras of holomorphic functions of the order k, 0 ≤ k ≤ ∞, and the exponent α, α ∈ (0, 1]. The Gel’fand theory and maximal ideal spaces of these algebras are discussed. Further, we solve the corona problem (1962) and Gleason’s problem (1964) for these algebras on certain bounded pseudoconvex/poly domains G in Cn (e.g., the ball and polydisc). As a welcome bonus, we affirmatively solve Fornæss and Øvrelid’s problem (1983) for holomorphic Hölder and Lipschitz spaces. In fact, we establish an equivalency between the two problems for these algebras. As an application, we establish the I.J. Schark’s theorem for Lipschitz algebras on these G’s. Indeed, we extend our recent work on Gleason’s problem, based on the functional analytic approach, as well as extend recent results of Clos for these algebras, and apply the usual Banach algebra method.
Let C be the class of all finite solvable groups and n ≥ 2 be an integer. In this paper, we present constructions of free pro-C products of n pro-C groups with amalgamated subgroups, and of free pro-C products of n pro-C groups with commuting subgroups. We also provide conditions under which a given free pro-C product of three pro-C groups with amalgamated subgroups can be written as a free pro-C product with amalgamated subgroup of two free pro-C products with amalgamated subgroups. Furthermore, we characterize –using cohomological conditions– when a pro-C group is the free pro-C product of three of its subgroups with amalgamated subgroups. Finally, we obtain a similar characterization for free pro-C products of two pro-C groups with commuting subgroups.
This paper could be considered the third in the series The Hitchhiker Guide to Categorical Banach Space Theory [24, 25]. We explore (quasi) Banach space formulations and applications for advanced categorical topics, such as relative homology (with either respect to size or relative to an operator ideal), Buchsbaum’s satellites (and homological derivation in quasi-Banach spaces), coherent functors and sheaves, Quillen’s adjunction theorem, Auslander’s formula, the Heart of the categories of Banach and quasi-Banach spaces, the vanishing of some Ext2 functors, Martsinkovsky-Russell stabilizations, nets of Banach spaces or 3-space problems.
