Let ${\mathcal {A}}$ be a unital ${\mathbb {F}}$-algebra and let ${\mathcal {S}}$ be a generating set of ${\mathcal {A}}$. The length of ${\mathcal {S}}$ is the smallest number k such that ${\mathcal {A}}$ equals the ${\mathbb {F}}$-linear span of all products of length at most k of elements from ${\mathcal {S}}$. The length of ${\mathcal {A}}$, denoted by $l({\mathcal {A}})$, is defined to be the maximal length of its generating sets. We show that $l({\mathcal {A}})$ does not exceed the maximum of $\dim {\mathcal {A}} / 2$ and $m({\mathcal {A}})-1$, where $m({\mathcal {A}})$ is the largest degree of the minimal polynomial among all elements of the algebra ${\mathcal {A}}$. As an application, we show that for arbitrary odd n, the length of the group algebra of the dihedral group of order $2n$ equals n.

H. H. Chan, K. S. Chua and P. Solé [‘Quadratic iterations to $\pi $ associated to elliptic functions to the cubic and septic base’, Trans. Amer. Math. Soc. 355(4) (2002), 1505–1520] found that, for each positive integer d, there are theta series $A_d, B_d$ and $C_d$ of weight one that satisfy the Pythagoras-like relationship $A_d^2=B_d^2+C_d^2$. In this article, we show that there are two collections of theta series $A_{b,d}, B_{b,d}$ and $C_{b,d}$ of weight one that satisfy $A_{b,d}^2=B_{b,d}^2+C_{b,d}^2,$ where b and d are certain integers.

We propose a generalised version of configuration spaces defined by disallowing combinations of simultaneous collisions among the n points determined by a family of forbidden partitions. In the case where the underlying space is a finite graph, we construct a cubical complex with the same homology as this configuration space.

We investigate neighbourhood sizes in the enhanced power graph (also known as the cyclic graph) associated with a finite group. In particular, we characterise finite p-groups with the smallest maximum size for neighbourhoods of a nontrivial element in its enhanced power graph.

Consider a possibly unsaturated Fell bundle $\mathcal {A}\to G$ over a locally compact, possibly non-Hausdorff, groupoid G. We list four notions of continuity of representations of $\mathit {C_c}(G;\mathcal {A})$ on a Hilbert space and prove their equivalence. This allows us to define the full $\mathit {C}^*$-algebra of the Fell bundle in different ways.

We continue our study of exponent semigroups of rational matrices. Our main result is that the matricial dimension of a numerical semigroup is at most its multiplicity (the least generator), greatly improving upon the previous upper bound (the conductor). For many numerical semigroups, including all symmetric numerical semigroups, our upper bound is tight. Our construction uses combinatorially structured matrices and is parametrised by Kunz coordinates, which are central to enumerative problems in the study of numerical semigroups.

Determining the insdel distance of linear codes is a very challenging problem. Recently, Ji et al. [‘Strict half-Singleton bound, strict direct upper bound for linear insertion-deletion codes and optimal codes’, IEEE Trans. Inform. Theory 69(5) (2023), 2900–2910] proposed their strict half-Singleton bound for linear codes that do not include the all-one vector. A natural question asks for linear codes with both large Hamming distance and insdel distance. Almost maximum distance separable (MDS) codes are a special kind of linear code with good Hamming error-correcting capability. We present a sufficient condition for the insdel distance of almost MDS codes to be bounded by the strict half-Singleton bound. In addition, we construct a class of two-dimensional near MDS codes that achieve the strict half-Singleton bound using twisted Reed–Solomon codes. Finally, we present a construction of optimal almost MDS insdel codes from Reed–Solomon codes for large dimensions.

In 2003, Ozsváth, Szabó and Rasmussen introduced the $\tau $ invariant for knots, and in 2011, Sarkar [‘Grid diagrams and the Ozsváth–Szabó tau-invariant’, Math. Res. Lett. 18(6) (2011), 1239–1257] published a computational shortcut for the $\tau $ invariant of knots that can be represented by diagonal grid diagrams. Previously, the only knots known to have diagonal grid diagram representations were torus knots. We prove that all such knots are positive knots and we produce an example of a knot with a diagonal grid diagram representation which is not a torus knot.

Let $\mathcal {S}$ denote the class of univalent functions in the open unit disc $\mathbb {D}:=\{z\in \mathbb {C}:\, |z|<1\}$ with the form $f(z)= z+\sum _{n=2}^{\infty }a_n z^n$. The logarithmic coefficients $\gamma _{n}$ of $f\in \mathcal {S}$ are defined by $F_{f}(z):= \log (f(z)/z)=2\sum _{n=1}^{\infty }\gamma _{n}z^{n}$. The second Hankel determinant for logarithmic coefficients is defined by

$$ \begin{align*} H_{2,2}(F_f/2) = \begin{vmatrix} \gamma_2 & \gamma_3 \\ \gamma_3 & \gamma_4 \end{vmatrix} =\gamma_2\gamma_4 -\gamma_3^2. \end{align*} $$

We obtain sharp upper bounds of the second Hankel determinant of logarithmic coefficients for starlike and convex functions.

We consider the pseudorelativistic Hartree equation

$$ \begin{align*} i\partial_t\psi=(\sqrt{-c^2\Delta +m^2c^4}-mc^2)\psi-(|x|^{-1}*|\psi|^2)\psi\quad \text{with } (t,x)\in\mathbb{R}\times\mathbb{R}^3, \end{align*} $$

which describes the dynamics of pseudorelativistic boson stars in the mean-field limit. We study the travelling waves of the form $\psi (t,x)=e^{it\mu }\varphi _{c}(x-vt)$, where $v\in \mathbb {R}^3$ denotes the travelling velocity. We prove that $\varphi _{c}$ converges strongly to the minimiser $\varphi _{\infty }$ of the limit energy $E_{\infty }(N)$ in $H^1(\mathbb {R}^3)$ as the light speed $c\to \infty $, where $E_{\infty }(N)$ is the corresponding energy for the limit equation

$$ \begin{align*} -\frac{1}{2m}\Delta\varphi_{\infty}+i(v\cdot\nabla)\varphi_{\infty}-({|x|^{-1}}*|\varphi_{\infty}|^2)\varphi_{\infty}=-\lambda\varphi_{\infty}. \end{align*} $$

Since the operator $-\Delta $ is the classical kinetic operator, we call this the nonrelativistic limit. We prove the existence of the minimiser for the limit energy $E_{\infty }(N)$ by using concentration-compactness arguments.

If $\mu $ is a smooth measure supported on a real-analytic submanifold of ${\mathbb {R}}^{2n}$ which is not contained in any affine hyperplane, then the Weyl transform of $\mu $ is a compact operator.

The concept of stability has proved very useful in the field of Banach space geometry. In this note, we introduce and study a corresponding concept in the setting of Banach algebras, which we call multiplicative stability. As we shall prove, various interesting examples of Banach algebras are multiplicatively unstable, and hence unstable in the model-theoretic sense. The examples include Fourier algebras over noncompact amenable groups, $C^*$-algebras and the measure algebra of an infinite compact group.