We describe the full group of isometries of each left-invariant Riemannian metric on the simply connected unimodular nilpotent or solvable $(R)$-type Lie groups of dimension four. As an application of these results, we answer the following question: Does every nontrivial compact subgroup of the automorphism group arise as the isotropy subgroup of some left-invariant Riemannian metric?

We obtain conditions sufficient for the mutual absolute continuity of two standard or pluriharmonic Riesz products defined on the unit sphere of the complex Euclidean space.

We consider the question of determining whether two binary cubic forms over an arbitrary field $K$ whose characteristic is not $2$ or $3$ are equivalent under the actions of either $\mathrm{GL}(2,K)$ or $\mathrm{SL}(2,K)$, deriving two necessary and sufficient criteria for such equivalence in each case. One of these involves an algebraic invariant of binary cubic forms, which we call the Cardano invariant, as it is closely connected to classical formulas; it also appears in the work of Bhargava et al. The second criterion is in terms of the base field itself and also gives explicit matrices in $\mathrm{SL}(2,K)$ or $\mathrm{GL}(2,K)$ transforming one cubic into the other, if any exist, in terms of the coefficients of bilinear factors of a bicovariant of the two cubics. We also consider automorphisms of a single binary cubic form, show how to use our results to test equivalence of binary cubic forms over an integral domain such as $\mathbb Z$, and briefly recall some connections between binary cubic forms and the arithmetic of elliptic curves. The methods used are elementary and similar to those used in our work with Fisher concerning equivalences between binary quartic forms.

We give a simple construction of Markov traces for Iwahori–Hecke algebras associated with the crystallographic Coxeter groups of types $A_n, B_n$ and $D_n$. In types $B$ and $D$, it is new and generalises a known construction in type $A$ employing symmetric polynomials in multiplicative Jucys–Murphy elements.

Given two closed curves on a surface consider their algebraic intersection divided by their length product. In this article, we compute the maximum of this ratio. This ratio, called the interaction strength, is difficult to compute for translation surfaces with several singularities as geodesics can change direction at singularities. The main contribution of this paper is to deal with this type of surfaces. Namely, we study the interaction strength of the regular $n$-gons for $n \equiv 2 \pmod 4$ and the Bouw–Möller surfaces $S_{m,n}$ with $1 \lt \gcd (m,n) \lt n$. This answers a conjecture of the author, and it completes the study of the algebraic interaction strength KVol on the regular polygon Veech surfaces. Our results on Bouw–Möller surfaces extends results of the author with Pasquinelli. This is also the first exact computation of KVol on translation surfaces with several singularities, and the pairs of curves that achieve the best ratio are singular geodesics made of two saddle connections with different directions.

We analyze the average behavior of various arithmetic functions at the values of degree $d$ binary forms ordered by height, with probability $1$. This approach yields averaged versions of the Chowla conjecture and the Bateman–Horn conjecture for random binary forms. Furthermore, we show that the rational Hasse principle holds for almost all Châtelet varieties defined by a fixed norm form of degree $e$ and by varying binary forms of fixed degree $d$, provided $e$ divides $d$. This proves an average version of a conjecture of Colliot-Thélène.

We introduce the concept of a restriction semigroupoid $\mathcal{S}$, which unifies the notion of restriction semigroups and restriction categories within a single structure. We prove a representation theorem, showing that every restriction semigroupoid can be embedded into a certain category of partial maps. Furthermore, we construct the Szendrei expansion $Sz(\mathcal{S})$ of $\mathcal{S}$ and establish that each premorphism between two restriction semigroupoids $\mathcal{S}$ and $\mathcal{T}$ is uniquely factorized by a morphism between the Szendrei expansion $Sz(\mathcal{S})$ and $\mathcal{T}$.

We show that the quasilinear Schrödinger equation $-\epsilon ^{2}\Delta u + Q(x)u -\Delta (u^{2})u= h(u)$ in $\mathbb{R}^{N}$ has $\wp$ solutions related to the $\wp$ global minima of the potential $Q$. We work with the Nehari manifold associated with the energy functional. We evaluate the energy of the solutions to distinguish them from each other.

We compute the $RO({\mathcal{K}})$-graded coefficients of the equivariant Eilenberg–Mac Lane spectrum associated with various Hill–Hopkins–Ravenel norms of the constant-$\mathbb{F}_2$ Mackey functor, where $\mathcal{K}$ is the Klein-four group. Further, we analyse the multiplicative structure of these ${\textrm {RO}}({\mathcal{K}})$-graded Tambara functors.

In this paper, we construct an implementable algorithm that solves the conjugacy problem in twisted right-angled Artin groups (T-RAAGs). In certain cases, the complexity is known to be linear, by reducing the problem to the twisted conjugacy problem in RAAGs. We also show that T-RAAGs are biautomatic, providing an alternative solution to the conjugacy problem.

In this paper, we show that for each sufficiently large integer $H$, there is a real cubic number $\alpha$ and a rational number $r$, both of height smaller than $H$, such that $0\lt \alpha -r\lt 272/H^4$. The exponent $4$ of $H$ in this inequality is best possible. The numbers $\alpha$ and $r$ are both constructed explicitly. We also show that a necessary and sufficient condition on $(u,v)$ under which there is a positive constant $c_1=c_1(d,u,v)$ such that the inequality $|\alpha -\beta | \geq c_1 H(\alpha )^{-u} H(\beta )^{-v}$ holds for all real algebraic numbers $\alpha$ of degree $d \geq 2$ and all real algebraic numbers $\beta$ of degree $d-1$ is $u \geq d-1$ and $v \geq d$.

The goal of this article is to find some geometric characterizations of Kenmotsu and almost Kenmotsu manifolds that admit a non-gradient $m$-quasi-Einstein structure. First, we prove that a Kenmotsu manifold admits a closed $m$-quasi-Einstein structure is an Einstein manifold. Next, we prove that if a three-dimensional Kenmotsu manifold admits a non-trivial $m$-quasi-Einstein structure $(g, V,m,\lambda )$ with $V$ as a conformal vector field, then it is of constant sectional curvature $-1$. Finally, we prove that if a non-Kenmotsu almost Kenmotsu $(k,\mu )^{\prime}$-manifold admits a closed $m$-quasi-Einstein structure $(g, V,m,\lambda )$ with $m\ne 1$, then it is locally isometric to the product space $\mathbb{H}^{n+1}(-4)\times \mathbb{R}^{n}$.

In this article we introduce a gluing operation on dimer models. This allows us to construct dimer quivers on arbitrary surfaces. We study how the associated dimer and boundary algebras behave under the gluing and how to determine them from the gluing components. We also use this operation to construct homogeneous dimer quivers on annuli.

We study the Ambrozio–Carlotto–Sharp (ACS) criterion on minimal isoparametric hypersurfaces $N^{n+1}\subset S^{n+2}$ with positive Ricci curvature, motivated by the Schoen–Marques–Neves conjecture on Morse index. For $g=4$ distinct principal curvatures with multiplicities $m_1,m_2$, we prove that the pointwise ACS inequality holds if and only if $\min \{m_1,m_2\}\ge 4$. Sufficiency is obtained via a moment-relaxation technique yielding the sharp bound $4a^2$ on the quadratic part of the integrand; necessity follows from an explicit extremal configuration in the top eigenspace of the shape operator. We also verify the ACS condition for $g=3$ with $m_1=m_2\in \{4,8\}$. As a consequence, for any closed embedded minimal hypersurface $M^n$ in such an ambient manifold, $\operatorname {index}(M)\ge \tfrac {2}{d(d-1)}\, b_1(M)$ with $d=n+3$.

Using the CR structure of the Heisenberg group, we construct several different Kähler structures in the Siegel domain. However, all these are PCR-Kähler equivalent, that is, essentially the same when restricted to the CR structure.

We find upper and lower bounds on the number of rational points with bounded denominators that are contained in a rectangular neighbourhood of some $n$-dimensional $p$-adic integer. To find the upper bound, we use lattice point counting techniques on $p$-adic approximation lattices, and for the corresponding lower-bound statement, a classical pigeonhole principle-style argument is used. We apply this counting result to prove a statement in the setting of weighted simultaneous $p$-adic Diophantine approximation on coordinate hyperplanes. For the lower-bound Hausdorff dimension result, we construct a local ubiquitous system of rectangles and then apply the recent Mass Transference Principle result of Wang and Wu (Math. Ann., 2021).

A known condition for the integrability of the one-dimensional Fourier transform, in which the derivative is assumed to belong to the real Hardy space, is extended to the $n$-dimensional case, in terms of the $n$-th Riesz derivative. We obtain results for the Riesz derivatives of other orders. The function spaces involved in this study are compared and analyzed.

We show that, for a finite spectrum $X$, Spanier–Whitehead duality induces an isomorphism between the cohomological and homological Atiyah–Hirzebruch spectral sequences. As an application, it follows that Poincaré duality for a Poincaré duality complex that is oriented over a ring spectrum $\mathcal{R}$ induces an isomorphism between the two spectral sequences.