Motivated by a classical theorem of Schoenberg, we prove that an n + 1 point finite metric space has strict 2-negative type if and only if it can be isometrically embedded in the Euclidean space $\mathbb{R}^{n}$ of dimension n but it cannot be isometrically embedded in any Euclidean space $\mathbb{R}^{r}$ of dimension r < n. We use this result as a technical tool to study ‘roundness’ properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space (X,d): (1) X is ultrametric, (2) X has infinite roundness, (3) X has infinite generalized roundness, (4) X has strict p-negative type for all p ≥ 0 and (5) X admits no p-polygonal equality for any p ≥ 0. As all ultrametric spaces have strict 2-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class $\mathcal{M}$ of all finite metric spaces that may be isometrically embedded into ℓ2 as an affinely independent set. The results of this paper show that Shkarin's class $\mathcal{M}$ consists of all finite metric spaces of strict 2-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly ℘ for each ℘ ∈ [1, ∞].

In this paper, we follow and extend a group-theoretic method introduced by Greenleaf–Iosevich–Liu–Palsson (GILP) to study finite points configurations spanned by Borel sets in $\mathbb{R}^n,n\geq 2,n\in\mathbb{N}.$ We remove a technical continuity condition in a GILP’s theorem in [Revista Mat. Iberoamer 31 (2015), 799–810]. This allows us to extend the Wolff–Erdogan dimension bound for distance sets to finite points configurations with k points for $k\in\{2,\dots,n+1\}$ forming a $(k-1)$ -simplex.

In the usual definition of an inductive limit of locally convex spaces, one is given a linear space E, a family (Eα) of locally convex spaces and a set (iα) of linear maps from Eα into E. Garling in [2] studies an extension of this, looking at absolutely convex subsets Sα of Eα and restrictions jα of iα to such sets. If, in the definition of Garling [2, p. 3], each Sα is instead a balanced semiconvex set, then the finest linear (not necessarily locally convex) topology on E for which the maps ja are continuous, will be referred to as the generalized *-inductive limit topology of the semiconvex sets. This topology is our object of study in the present paper; we find applications in the closed graph theorem.

In this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$ .

Given a commutative ring A with identity, let W–1(A) denote the Witt group of skew-symmetric bilinear forms over A (cf. [1] or [7] for the definition of W–1 (A)).

We show that if $\mathcal S$ is a compact Riemann surface of genus $g=p+1$, where $p$ is prime, with a group of automorphisms $G$ such that $|G|\geq\lambda(g-1)$ for some real number $\lambda>6$, then for all sufficiently large $p$ (depending on $\lambda$), $\mathcal S$ and $G$ lie in one of six infinite sequences of examples. In particular, if $\lambda=8$ then this holds for all $p\geq 17$.

Let $e$ and $q$ be fixed co-prime integers satisfying $1\lt e\lt q$. Let $\mathscr {C}$ be a certain family of deformations of the curve $y^e=x^q$. That family is called the $(e,q)$-curve and is one of the types of curves called plane telescopic curves. Let $\varDelta$ be the discriminant of $\mathscr {C}$. Following pioneering work by Buchstaber and Leykin (BL), we determine the canonical basis $\{ L_j \}$ of the space of derivations tangent to the variety $\varDelta =0$ and describe their specific properties. Such a set $\{ L_j \}$ gives rise to a system of linear partial differential equations (heat equations) satisfied by the function $\sigma (u)$ associated with $\mathscr {C}$, and eventually gives its explicit power series expansion. This is a natural generalisation of Weierstrass’ result on his sigma function. We attempt to give an accessible description of various aspects of the BL theory. Especially, the text contains detailed proofs for several useful formulae and known facts since we know of no works which include their proofs.

Let E be a Banach space such that its dual E* is separable. We show that there exists a hypercyclic bounded operator T on E such that its adjoint T* is also hypercyclic on E*. We also exhibit a new kind of dual hypercyclic operator. Thus answers affirmatively two of the questions raised by Henrik Petersson in a recent paper.

In the thirty years since it was proved that 0, 1 and 144 were the only perfect squares in the Fibonacci sequence [1, 9], several generalisations have been proved, but many problems remain. Thus it has been shown that 0, 1 and 8 are the only Fibonacci cubes [6] but there seems to be no method available to prove the conjecture that 0, 1, 8 and 144 are the only perfect powers.

In this paper we study prime and maximal ideals in a polynomial ring R[X], where R is a ring with identity element. It is well-known that to study many questions we may assume Ris prime and consider just R-disjoint ideals. We give a characterizaton for an R-disjoint ideal to be prime. We study conditions under which there exists an R-disjoint ideal which is a maximal ideal and when this is the case how to determine all such maximal ideals. Finally, we prove a theorem giving several equivalent conditions for a maximal ideal to be generated by polynomials of minimal degree.