Apollonius Surfaces, Circumscribed Spheres of Tetrahedra, Menelaus’s and Ceva’s Theorems in S2 × R and H2 × R Geometries

In the present paper we study $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries, which are homogeneous Thurston 3-geometries. We define and determine the generalized Apollonius surfaces and with them define the ‘surface of a geodesic triangle’. Using the above Apollonius surfaces we develop a procedure to determine the centre and the radius of the circumscribed geodesic sphere of an arbitrary $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ tetrahedron. Moreover, we generalize the famous Menelaus’s and Ceva’s theorems for geodesic triangles in both spaces. In our work we will use the projective model of $\mathbf{S}^2\!\times\!\mathbf{R}$ and $\mathbf{H}^2\!\times\!\mathbf{R}$ geometries described by E. Molnár in [6].

Tamagawa Products of Elliptic Curves Over ℚ

We explicitly construct the Dirichlet series $$\begin{equation*}L_{\mathrm{Tam}}(s):=\sum_{m=1}^{\infty}\frac{P_{\mathrm{Tam}}(m)}{m^s},\end{equation*}$$ where $P_{\mathrm{Tam}}(m)$ is the proportion of elliptic curves $E/\mathbb{Q}$ in short Weierstrass form with Tamagawa product m. Although there are no $E/\mathbb{Q}$ with everywhere good reduction, we prove that the proportion with trivial Tamagawa product is $P_{\mathrm{Tam}}(1)={0.5053\dots}$. As a corollary, we find that $L_{\mathrm{Tam}}(-1)={1.8193\dots}$ is the average Tamagawa product for elliptic curves over $\mathbb{Q}$. We give an application of these results to canonical and Weil heights.

Low Pseudomoments of Euler Products

In this paper, we determine the order of magnitude of the 2 q-th pseudomoment of powers of the Riemann zeta function $\zeta(s)^{\alpha}$ for $0 \lt q\le 1/2$ and $0 \lt \alpha \lt 1$, completing the results of Bondarenko, Heap and Seip, and Gerspach. Our results also apply to more general Euler products satisfying certain conditions.

Algebras associated with invariant means on the subnormal subgroups of an amenable group

Let G be an amenable group. We define and study an algebra ${\mathcal{A}}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of G. For a just infinite amenable group G, we show that ${\mathcal{A}}_{sn}(G)$ is nilpotent if and only if G is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $\textrm{rad}\, \ell^1(G)^{**}$ for an amenable branch group G and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely generated counterexamples to a question of Dales and Lau [4], first resolved by the author in [10], which asks whether we always have $(\textrm{rad}\, \ell^1(G)^{**})^{\Box 2} = \{0 \}$. We further study this question by showing that $(\textrm{rad}\, \ell^1(G)^{**})^{\Box 2} = \{0 \}$ imposes certain structural constraints on the group G.

Fourier Optimization and Quadratic Forms

We prove several results about integers represented by positive definite quadratic forms, using a Fourier analysis approach. In particular, for an integer $\ell\ge 1$, we improve the error term in the partial sums of the number of representations of integers that are a multiple of $\ell$. This allows us to obtain unconditional Brun–Titchmarsh-type results in short intervals and a conditional Cramér-type result on the maximum gap between primes represented by a given positive definite quadratic form.

The Image Milnor Number and Excellent Unfoldings

We show three basic properties of the image Milnor number µI(f) of a germ $f\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with isolated instability. First, we show the conservation of the image Milnor number, from which one can deduce the upper semi-continuity and the topological invariance for families. Second, we prove the weak Mond’s conjecture, which states that µI(f) = 0 if and only if f is stable. Finally, we show a conjecture by Houston that any family $f_t\colon(\mathbb{C}^{n},S)\rightarrow(\mathbb{C}^{n+1},0)$ with $\mu_I(\,f_t)$ constant is excellent in Gaffney’s sense. For technical reasons, in the last two properties, we consider only the corank 1 case.

Presentations for Temperley–Lieb Algebras

We give a new and conceptually straightforward proof of the well-known presentation for the Temperley–Lieb algebra, via an alternative new presentation. Our method involves twisted semigroup algebras, and we make use of two apparently new submonoids of the Temperley–Lieb monoid.

A Unified Approach to the Arens Regularity and Related Problems for a Class of Banach Algebras Associated With Locally Compact Groups

Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity.