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.

In this paper we study the (co)homology of Tanabe algebras, which are a family of subalgebras of the partition algebras exhibiting a Schur–Weyl duality with certain complex reflection groups. The homology of the partition algebras has been shown to be related to the homology of the symmetric groups by Boyd–Hepworth–Patzt and the results they obtain depend on a parameter. In all known results, the homology of a diagram algebra is dependent on one of two things: the invertibility of a parameter in the ground ring or the parity of the positive integer indexing the number of pairs of vertices. We show that the (co)homology of Tanabe algebras is isomorphic to the (co)homology of the symmetric groups and that this is independent of both the parameter and the parity of the index. To the best of our knowledge, this is the first example of a result of this sort. Along the way we will also study the (co)homology of uniform block permutation algebras and totally propagating partition algebras as well collecting cohomological analogues of known results for the homology of partition algebras and Jones annular algebras.

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 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.

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 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.

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.

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.

The action of the subgroup G2 of SO(7) (resp. Spin(7) of SO(8)) on the Grassmannian space M = SO(7)/(SO(5)×SO(2)) (resp. M = SO(8)/(SO(5)×SO(3)) ) is still transitive. We prove that the spectrum (i.e. the collection of eigenvalues of its Laplace-Beltrami operator) of a symmetric metric g0 on M coincides with the spectrum of a G2-invariant (resp. Spin(7)-invariant) metric g on M only if g0 and g are isometric. As a consequence, each non-flat compact irreducible symmetric space of non-group type is spectrally unique among the family of all currently known homogeneous metrics on its underlying differentiable manifold.

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.