Counting Hamilton cycles in Dirac hypergraphs

A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

Exponential sums involving q-digital functions

We estimate the exponential sum [Formula: see text], where [Formula: see text] is a real number and [Formula: see text] are digital functions; in the spirit of the works of Kim and Berend–Kolesnik. A similar estimate along short intervals is also provided.

Effects of Drugs Used to Treat Infertility on Periodontium and Periodontal Disease

Infertility has been described medically as the inability of a couple to achieve conception after a year or more of regular unprotected sexual intercourse or the incapacity to carry a pregnancy to a live birth [1]. The CDC statistics show that on an average one in every eight couple suffers from infertility [2]. The World Health Organization also shows a similar estimate of about 8-10% of the world population suffering from infertility problems [3].

Effective Log-Free Zero Density Estimates for Automorphic L-Functions and the Sato–Tate Conjecture

Let $K/\mathbb{Q}$ be a number field. Let π and π′ be cuspidal automorphic representations of $\textrm{GL}_{d}(\mathbb{A}_{K})$ and $\textrm{GL}_{d^{\prime }}(\mathbb{A}_{K})$. We prove an unconditional and effective log-free zero density estimate for all automorphic L-functions L(s, π) and prove a similar estimate for Rankin–Selberg L-functions L(s, π × π′) when π or π′ satisfies the Ramanujan conjecture. As applications, we make effective Moreno’s analog of Hoheisel’s short interval prime number theorem and extend it to the context of the Sato–Tate conjecture; additionally, we bound the least prime in the Sato–Tate conjecture in analogy with Linnik’s theorem on the least prime in an arithmetic progression. We also prove effective log-free density estimates for automorphic L-functions averaged over twists by Dirichlet characters, which allows us to prove an “average Hoheisel” result for GLdL-functions.

Continuity of the Peierls barrier and robustness of laminations

We study the Peierls barrier$P_{\omega }(\xi )$for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start by deriving an estimate for the difference$\vert P_{\omega }(\xi ) - P_{q/p}(\xi ) \vert $of the Peierls barriers of rotation numbers$\omega \in {{\mathbb{R}}}$and$q/p\in {\mathbb{Q}}$. A similar estimate was obtained by Mather [Modulus of continuity for Peierls’s barrier.Proc. NATO Advanced Research Workshop on Periodic Solutions of Hamiltonian Systems and Related Topics (Il Ciocco, Italy, 13–18 October 1986) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 209).Eds. P. H. Rabinowitz, A. Ambrosetti and I. Eckeland. D. Reidel, Dordrecht, 1987, pp. 177–202] in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that$\omega \mapsto P_{\omega }(\xi )$is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers of rotation number$\omega \in {{\mathbb{R}}}\delimiter "026E30F {\mathbb{Q}}$is open in the$C^1$-topology.

An evaluation of the IDEEA™ activity monitor for estimating energy expenditure

Objective estimates of activity patterns and energy expenditure (EE) are important for the measurement of energy balance. The Intelligent Device for Energy Expenditure and Activity (IDEEA) can estimate EE from the thirty-five postures and activities it can identify and record. The present study evaluated the IDEEA system's estimation of EE using whole-body indirect calorimetry over 24 h, and in free-living subjects using doubly-labelled water (DLW) over 14 d. EE was calculated from the IDEEA data using calibration values for RMR and EE while sitting and standing, both as estimated by the IDEEA system (IDEEAest) and measured by indirect calorimetry (IDEEAmeas). Subjects were seven females and seven males, mean age 38·1 and 39·7 years, mean BMI 25·2 and 26·2 kg/m2, respectively. The IDEEAest method produced a similar estimate of EE to the calorimeter (10·8 and 10·8 MJ, NS), while the IDEEAmeas method underestimated EE (9·9 MJ, P < 0·001). After removing data from static cycling, which the IDEEA was unable to identify as an activity, both the IDEEAest and IDEEAmeas methods overestimated EE compared to the calorimeter (9·9 MJ, P < 0·001; 9·1 MJ, P < 0·05 and 8·6 MJ, respectively). Similarly, the IDEEA system overestimated EE compared to DLW over 14 d; 12·7 MJ/d (P < 0·01), 11·5 MJ/d (P < 0·01) and 9·5 MJ/d for the IDEEAest, IDEEAmeas and DLW, respectively. The IDEEA system overestimated EE both in the controlled laboratory and free-living environments. Using measured EE values for RMR, sitting and standing reduced, but did not eliminate, the error in estimated EE.

Tropical cooling and the onset of North American glaciation

We offer a test of the idea that gradual cooling in the eastern tropical Pacific led to cooling of North America and the initiation of glaciation ~3 Myr ago. Using modern climate data we estimate how warming of the eastern tropical Pacific affects North American temperature and ice-ablation. Assuming that the modern relationship holds over the past millions of years, a ~4°C warmer eastern tropical Pacific between 3–5 Ma would increase ablation in northern North America by approximately two meters per year. By comparison, a similar estimate of the ablation response to variations in Earth's obliquity gives less than half the magnitude of the tropically-induced change. Considering that variations in Earth's obliquity appear sufficient to initiate glaciations between ~1–3 Ma, we infer that the warmer eastern equatorial Pacific prior to 3 Ma suffices to preclude glaciation.

Tropical cooling and the onset of North American glaciation

We offer a test of the idea that gradual cooling in the eastern tropical Pacific led to cooling of North America and the initiation of glaciation ~3 Myr ago. Using modern climate data we estimate how warming of the eastern tropical Pacific affects North American temperature and ice-ablation. Assuming that the modern relationship holds over the past millions of years, an eastern tropical Pacific warmer by ~4° between 3–5 Ma would increase ablation in northern North America by approximately two meters per year. By comparison, a similar estimate of the ablation response to variations in Earth's obliquity gives less than half the magnitude of the tropically-induced change. Considering that variations in Earth's obliquity appear sufficient to initiate glaciations between ~1–3 Ma, we infer that the warmer eastern equatorial Pacific prior to 3 Ma suffices to preclude glaciation.

Some estimates for the Bergman Kernel and Metric in Terms of Logarithmic Capacity

For a bounded domain Ω on the plane we show the inequality cΩ(z)2 ≤ 2πKΩ(z), z ∈ Ω, where cΩ(z) is the logarithmic capacity of the complement ℂ\Ω with respect to z and KΩ is the Bergman kernel. We thus improve a constant in an estimate due to T. Ohsawa but fall short of the inequality cΩ(z)2 ≤ πKΩ(z) conjectured by N. Suita. The main tool we use is a comparison, due to B. Berndtsson, of the kernels for the weighted complex Laplacian and the Green function. We also show a similar estimate for the Bergman metric and analogous results in several variables.