We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$-algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$. We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$.

The local analysis of convergence for Newton’s method has been extensively studied by numerous researchers under a plethora of sufficient conditions. However, the complexity of extending the convergence domain requires very general conditions such as the ones depending on the majorant principle in order to include as large classes of operators as possible. In the present article, such an analysis is developed under the weak majorant condition. The new results extend earlier ones using similar information. Finally, the numerical examples complement the theory.

We show that under mild conditions, the connected sum $M\# N$ of simply connected, closed, orientable n-dimensional Poincaré Duality complexes M and N is hyperbolic and has no homotopy exponent at all but finitely many primes, verifying a weak version of Moore’s conjecture. This is derived from an elementary framework involving $CW$-complexes satisfying certain conditions.

We study some analytic properties of the Asai lifts associated with cuspidal Hilbert modular forms, and prove sharp bounds for the second moment of their central L-values.

We characterize the ideals of the semicrossed product $C_0(X)\times _\phi {\mathbb Z}_+$, associated with suitable sequences of closed subsets of X, with left (resp. right) approximate unit. As a consequence, we obtain a complete characterization of ideals with left (resp. right) approximate unit under the assumptions that X is metrizable and the dynamical system $(X,\phi )$ contains no periodic points.

In this paper, we consider the closed spacelike solution to a class of Hessian quotient equations in de Sitter space. Under mild assumptions, we obtain an existence result using standard degree theory based on a priori estimates.

The Thue–Morse sequence $\{t(n)\}_{n\geqslant 0}$ is the indicator function of the parity of the number of ones in the binary expansion of nonnegative integers n, where $t(n)=1$ (resp. $=0$) if the binary expansion of n has an odd (resp. even) number of ones. In this paper, we generalize a recent result of E. Miyanohara by showing that, for a fixed Pisot or Salem number $\beta>\sqrt {\varphi }=1.272019\ldots $, the set of the numbers

$$\begin{align*}1,\quad \sum_{n\geqslant1}\frac{t(n)}{\beta^{n}},\quad \sum_{n\geqslant1}\frac{t(n^2)}{\beta^{n}},\quad \dots, \quad \sum_{n\geqslant1}\frac{t(n^k)}{\beta^{n}},\quad \dots \end{align*}$$

$\mathbb {Q}(\beta )$

$\varphi :=(1+\sqrt {5})/2$

$k\geqslant 1$

$a_1,a_2,\ldots ,a_k\in \mathbb {Q}(\beta )$

$a_1t(n)+a_2t(n^2)+\cdots +a_kt(n^k)\}_{n\geqslant 1}$

Let $S=K[x_1,\ldots ,x_n]$ be the polynomial ring over a field K, and let A be a finitely generated standard graded S-algebra. We show that if the defining ideal of A has a quadratic initial ideal, then all the graded components of A are componentwise linear. Applying this result to the Rees ring $\mathcal {R}(I)$ of a graded ideal I gives a criterion on I to have componentwise linear powers. Moreover, for any given graph G, a construction on G is presented which produces graphs whose cover ideals $I_G$ have componentwise linear powers. This, in particular, implies that for any Cohen–Macaulay Cameron–Walker graph G all powers of $I_G$ have linear resolutions. Moreover, forming a cone on special graphs like unmixed chordal graphs, path graphs, and Cohen–Macaulay bipartite graphs produces cover ideals with componentwise linear powers.

By using fixed point argument, we give a proof for the existence of singular rotationally symmetric steady and expanding gradient Ricci solitons in higher dimensions with metric $g=\frac {da^2}{h(a^2)}+a^2g_{S^n}$ for some function h where $g_{S^n}$ is the standard metric on the unit sphere $S^n$ in $\mathbb {R}^n$ for any $n\ge 2$. More precisely, for any $\lambda \ge 0$ and $c_0>0$, we prove that there exist infinitely many solutions ${h\in C^2((0,\infty );\mathbb {R}^+)}$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-1)+rh_r(r)(rh_r(r)-\lambda r-(n-1))$, $h(r)>0$, in $(0,\infty )$ satisfying $\underset {\substack {r\to 0}}{\lim }\,r^{\sqrt {n}-1}h(r)=c_0$ and prove the higher-order asymptotic behavior of the global singular solutions near the origin. We also find conditions for the existence of unique global singular solution of such equation in terms of its asymptotic behavior near the origin.

In this note, we study the Li coefficients $\lambda _{n,a}$ for the quadrilateral zeta function. Furthermore, we give an arithmetic and asymptotic formula for these coefficients. Especially, we show that for any fixed $n \in {\mathbb {N}}$, there exists $a>0$ such that $\lambda _{2n-1,a}> 0$ and $\lambda _{2n,a} < 0$.

This paper concerns the $L^4$ norm of Littlewood polynomials on the unit circle which are given by

$$ \begin{align*}q_n(z)=\sum_{k=0}^{n-1}\pm z^k;\end{align*} $$

$\{-1,1\}$

$$ \begin{align*}||q_n||_4^4=\frac{1}{2\pi}\int_0^{2\pi}|q_n(e^{i\theta})|^4 d\theta.\end{align*} $$

$||q_n||_4/\sqrt {n}\rightarrow \sqrt [4]{2}$

$n\to \infty $