Abstract 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.
Let $p$ be an odd prime, and let $E_1$ and $E_2$ be two elliptic curves defined over a number field $K$ , with good ordinary reduction at $p$ . We compare the $\Lambda$ -ranks and (generalized) Iwasawa invariants of the Pontryagin duals of the Selmer groups of $E_1$ and $E_2$ over ${\mathbb{Z}}_p^d$ -extensions $\mathbb{L}_\infty$ of $K$ for general $d \ge 1$ under the hypothesis that $E_1[p^i] \cong E_2[p^i]$ as Galois modules for a sufficiently large $i$ . This generalizes and complements previous work over ${\mathbb{Z}}_p$ -extensions. The comparison of generalized Iwasawa invariants is related via an up-down approach to the comparison of the variation of classical Iwasawa invariants over the ${\mathbb{Z}}_p$ -extensions of $K$ which are contained in $\mathbb{L}_\infty$ .
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)'-manifold admits a closed m-quasi-Einstein structure (g, V, m, lambda) with m not equal 1, then it is locally isometric to the product space Hn+1( - 4) & times; R-n.
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 < alpha - r < 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| >= c(1)H(alpha)H-u (beta)(-v) holds for all real algebraic numbers alpha of degree d >= 2 and all real algebraic numbers beta of degree d-1 is u >= d-1 and v >= d.
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, $\operatorname {index}(M)\ge frac {2}{d(d-1)}\, b_1(M)$ with $d=n+3$ .
