Let$u$_{$n$}be the$n$th term of a Lucas sequence or a Lehmer sequence. In this article we shall establish an estimate from below for the greatest prime factor of$u$_{$n$}which is of the form$n$exp(log$n$/104 log log$n$). In doing so, we are able to resolve a question of Schinzel from 1962 and a conjecture of Erdős from 1965. In addition we are able to give the first general improvement on results of Bang from 1886 and Carmichael from 1912.

Structures and functions of proteins play various essential roles in biological processes. The functions of newly discovered proteins can be predicted by comparing their structures with that of known functional proteins. Many approaches have been proposed for measuring the protein structure similarity, such as the template-modeling (TM)-score method, GRaphlet (GR)-Align method as well as the commonly used root-mean-square deviation (RMSD) measures. However, the alignment comparisons between the similarity of protein structure cost much time on large dataset, and the accuracy still have room to improve. In this study, we introduce a new three-dimensional (3D) Yau–Hausdorff distance between any two 3D objects. The (3D) Yau–Hausdorff distance can be used in particular to measure the similarity/dissimilarity of two proteins of any size and does not need aligning and super- imposing two structures. We apply structural similarity to study function similarity and perform phylogenetic analysis on several datasets. The results show that (3D) Yau–Hausdorff distance could serve as a more precise and effective method to discover biological relationships between proteins than other methods on structure comparison.

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We consider numerical boundary conditions for high order finite difference schemes for solving convection-diffusion equations on arbitrary geometry. The two main difficulties for numerical boundary conditions in such situations are: (1) the wide stencil of the high order finite difference operator requires special treatment for a few ghost points near the boundary; (2) the physical boundary may not coincide with grid points in a Cartesian mesh and may intersect with the mesh in an arbitrary fashion. For purely convection equations, the so-called inverse Lax-Wendroff procedure, in which we convert the normal derivatives into the time derivatives and tangential derivatives along the physical boundary by using the equations, have been quite successful. In this paper, we extend this methodology to convection-diffusion equations. It turns out that this extension is non-trivial, because totally different boundary treatments are needed for the diffusion-dominated and the convection-dominated regimes. We design a careful combination of the boundary treatments for the two regimes and obtain a stable and accurate boundary condition for general convection-diffusion equations. We provide extensive numerical tests for one- and two-dimensional problems involving both scalar equations and systems, including the compressible Navier-Stokes equations, to demonstrate the good performance of our numerical boundary conditions.

We show that given $${n \geqslant 3}$$ , $${q \geqslant 1}$$ , and a finite set $${\{y_1, \ldots, y_q \}}$$ in $${\mathbb{R}^n}$$ there exists a quasiregular mapping $${\mathbb{R}^n\to \mathbb{R}^n}$$ omitting exactly points $${y_1, \ldots, y_q}$$ .