A Gap Principle for Subvarieties with Finitely Many Periodic Points

Let $f:X\rightarrow X$ be a quasi-finite endomorphism of an algebraic variety $X$ defined over a number field $K$ and fix an initial point $a\in X$. We consider a special case of the Dynamical Mordell–Lang Conjecture, where the subvariety $V$ contains only finitely many periodic points and does not contain any positive-dimensional periodic subvariety. We show that the set $\{n\in \mathbb{Z}_{{\geqslant}0}\mid f^{n}(a)\in V\}$ satisfies a strong gap principle.

GAPS BETWEEN DIVISIBLE TERMS IN

Suppose $a^{2}(a^{2}+1)$ divides $b^{2}(b^{2}+1)$ with $b>a$. We improve a previous result and prove a gap principle, without any additional assumptions, namely $b\gg a(\log a)^{1/8}/(\log \log a)^{12}$. We also obtain $b\gg _{\unicode[STIX]{x1D716}}a^{15/14-\unicode[STIX]{x1D716}}$ under the abc conjecture.

On the size of Diophantine m-tuples in imaginary quadratic number rings

A Diophantine [Formula: see text]-tuple is a set of [Formula: see text] distinct integers such that the product of any two distinct elements plus one is a perfect square. It was recently proven that there is no Diophantine quintuple in positive integers. We study the same problem in the rings of integers of imaginary quadratic fields. By using a gap principle proven by Diophantine approximations, we show that [Formula: see text]. Our proof is relatively simple compared to the proofs of similar results in positive integers.

A Macroscopic Multi-Lane Traffic Flow Model for ACC/CACC Traffic Dynamics

An extended second-order macroscopic traffic flow model is presented that describes multi-lane traffic dynamics and also incorporates the effects of adaptive cruise control (ACC) or cooperative ACC (CACC). The extended model equations stem from a recently proposed multi-lane gas-kinetic traffic flow (GKT) model that can simulate lane changes due to vehicle interactions as well as spontaneous ones. The proposed extension that models the effects of ACC/CACC satisfies the time-gap principle of such systems on each lane and allows for consideration of mixed traffic comprising both manual and ACC/CACC vehicles. Numerical simulations are performed for a particular three-lane motorway stretch in the United Kingdom, where recurrent traffic congestion is observed during the morning peak hours, so as to compare the effects of ACC/CACC in the traffic flow conditions with those resulting from manual driving.

Common factors among pairs of consecutive integers

In this paper, we study products of consecutive integers and prove an optimal bound on their greatest common factors. In contrast, we obtain a modest upper bound for the greatest common factor in the perfect square situation. Using this and an upper bound on the size of solutions to hyperelliptic curves, we prove a gap principle when [Formula: see text] divides [Formula: see text] with some additional restrictions. We also obtain a stronger gap principle under the abc conjecture.

Influence of the electrical circuit configurations of a DVR coupling transformer with a magnetic bypass

Purpose For the proposed coupling transformer, a magnetic bypass based on the virtual air gap principle is realized by inserting auxiliary windings in a return leg added to a standard transformer. With such a setup, it is able to act as a voltage regulator as well as protect the power electronics of the dynamic voltage restorer from electrical grid fault currents. This paper focuses on the electrical design part of the coupling transformer. It aims to explain how the behavior of the auxiliary windings electrical circuit of the magnetic bypass impacts the performances of the device. Design/methodology/approach The influence of the electrical auxiliary windings circuit configurations on the operation of the coupling transformer is studied by finite element analyses with nonlinear and isotropic magnetic materials. Findings A configuration for the realization of the electrical circuit of the auxiliary windings is determined according to the finite element simulation results to achieve the design of the coupling transformer whose magnetic core was previously designed. Originality/value By studying the operation of a special coupling transformer with nonlinear saturation phenomenon by finite element analyses, a to-do list of the electrical circuit parameters is described to design this device well.