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.
We build a tangent structure on the category of divided power algebras using a particular notion of semidirect product. We show that this tangent structure admits an adjoint tangent structure, which involves a version of Khäler differentials, and which is similar to the Zariski cotangent space for affine schemes. We study vector fields and differential bundles for these two structures, which correspond respectively to a notion of special derivation, and to the category of modules over the underlying commutative algebra of a given divided power algebra.
We reduce the polynomial cluster value problem for the algebra of bounded analytic functions, H ∞ , on the ball of Banach spaces X to the same polynomial cluster value problem for H ∞ but on the ball of those spaces which are `1 -sums of finite dimensional spaces.
This short article presents several equivalent topological characterizations for the metrizability of the strong dual of a locally convex Hausdorff space. Among our key findings, we establish that the metrizability of the strong dual is precisely equivalent to it being a q-space.
The objective of this paper is to establish initial coefficient inequalities, Upper bounds to the Hankel and Toeplitz determinants for certain normalized univalent functions defined on the open unit disk D in the complex plane related to the analytic function ϕ4L (z) = 1 + 5z/6 + z5/6 that maps the open unit disk in the complex plane onto the interior of four leaf shaped domain in the right half of the complex plane.
We investigate quasi-Einstein structures on four-dimensional non-reductive homogeneous spaces. We show that contrary to the Ricci solitons structures, quasi-Einstein structures display a strong rigidity in the sense that every such a structure is necessarily Einstein.
We derive a Binet-type formula for operator-valued sequences satisfying linear recurrence relations, extending the classical scalar case to the setting of bounded operators on Hilbert spaces. In this framework, we analyze the operator moment problem as an application, establishing new connections between recursive operator sequences and moment sequences.
Let R be a principal ideal domain, and let (T (V ), ∂) and (T (W ), δ) be two free differential graded R-algebras. Let (V, d) and (W, d0 ) denote the chain complexes of the indecomposables of (T (V ), ∂) and (T (W ), δ), respectively. Given a chain map ξ* : (V, d) → (W, d'), this paper addresses the problem of determining whether there exists a DGA-map α : (T (V ), ∂) → (T (W ), δ) such that H* (α) = H* (ξ* ).
We provide infinite-dimensional versions of analytic gap dichotomies, in the sense that a sequence of analytic hereditary families {Ip }p<ω of subsets of a countable set Ω is either countably separated or there is a tree structure inside Ω in which p-chains are sets from Ip . A topological version of this is that if K is a separable Rosenthal compact space, then either K is a continuous image of a finite-to-one preimage of a metric compactum or there is a tree structure inside K in which p-chains inside every branch form a relatively discrete family of sets.
Finding the shortest non-zero vectors in a lattice is a computationally hard problem (NP-hard in general dimensions), making results in low dimensions particularly important in lattice reduction theory. This paper focuses on the coordinates of minimal lattice vectors when expressed in a Minkowski-reduced basis. By applying Ryskov’s findings on admissible centerings and Tammela’s work characterizing Minkowski-reduced forms via a finite set of inequalities (up to dimension 6), we demonstrate sharp bounds on the absolute values of these coordinates. Specifically, we show that for dimensions n ≤ 6, the absolute values of the coordinates of any minimal vector with respect to a Minkowski-reduced basis are bounded by 1 (for n = 2, 3), 2 (for n = 4, 5), and 3 (for n = 6). This refines bounds implicitly available from Tammela’s results by combining geometric arguments from lattice theory, admissible centering theory, and reduction theory.
Let d ≥ 2. In this paper we prove that Id(ℓ, a) fills Rd face–to–face by translations. We prove that the symmetry group of Id(ℓ, a) contains the product of cyclic groups Cd × C2 as a subgroup. We compute the Lebesgue j–volume (i.e., the sum of the Lebesgue j–measures of the j–faces) of Id(ℓ, a), for 1 ≤ j < d. We compute the incidence numbers (as defined by Grünbaum) of the faces of Id(ℓ, a).
This paper answers the following two questions: What are the easiest polynomial differential systems in R3 having an invariant hyperbolic, parabolic or elliptic cylinder?, and for such polynomial differential systems what are their phase portraits on such invariant cylinders?
This article focuses on the study of Lorentzian para-Sasakian manifolds Mn . It demonstrates that a W9-semisymmetric Lorentzian para-Sasakian manifold is a W9-flat manifold. Additionally, we explore Lorentzian para-Sasakian manifolds that satisfy the ζ-W9-flat condition, revealing that they represent a special type of η-Einstein manifold. Furthermore, it is shown that a W9-flat Lorentzian para-Sasakian manifold is a flat manifold. We also investigate Lorentzian para-Sasakian manifolds that meet W9-recurrent and ϕ-W9-semisymmetric conditions, presenting several significant results from this analysis. At last, we explore η-Ricci Solitons on Lorentzian para-Sasakian manifold satisfying W9(ζ, F1 ) · S = 0.
In this paper, we generalize for an arbitrary double vector bundle, some results on linear tensor fields. Moreover we study some properties of their lifts with respect to a product preserving bundle functor.
In this article we study the n-homogeneous polynomials P that are c-continuous on bounded subsets of l1 . We show that P can be decomposed in the form R + Q, where Q and R are n-homogeneous polynomials, with R weakly star continuous and Q (x) = 0 for all x ∈ ker u for u = (1, 1, . . . , 1, . . . ). We conclude that P = Σ un−j ⊗ Rj , where R is a weakly star continuous j-homogeneous polynomial for j = 0, 1, . . . , n.
This is a survey of known results and still open problems on characteristic properties of classes of homothetic convex sets in the n-dimensional Euclidean space. These properties are formulated in terms of orthogonal projections, plane sections, homothety classes, Choquet simplices, and homothetic tilings and partitions.
