An Algorithmic Approach to Limit Cycles of Nonlinear Differential Systems: the Averaging Method Revisited

This paper introduces an algorithmic approach to the analysis of bifurcation of limit cycles from the centers of nonlinear continuous differential systems via the averaging method. We develop three algorithms to implement the averaging method. The first algorithm allows to transform the considered differential systems to the normal formal of averaging. Here, we restricted the unperturbed term of the normal form of averaging to be identically zero. The second algorithm is used to derive the computational formulae of the averaged functions at any order. The third algorithm is based on the first two algorithms that determines the exact expressions of the averaged functions for the considered differential systems. The proposed approach is implemented in Maple and its effectiveness is shown by several examples. Moreover, we report some incorrect results in published papers on the averaging method.Comment: Proc. 44th ISSAC, July 15--18, 2019, Beijing, Chin

Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals

We develop techniques for the verification of the Chebyshev property of Abelian integrals. These techniques are a combination of theoretical results, analysis of asymptotic behavior of Wronskians, and rigorous computations based on interval arithmetic. We apply this approach to tackle a conjecture formulated by Dumortier and Roussarie in [Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009), 723-781], which we are able to prove for q < 2

Integrability and Transcendentality

We derive the two-loop Bethe ansatz for the sl(2) twist operator sector of N=4 gauge theory directly from the field theory. We then analyze a recently proposed perturbative asymptotic all-loop Bethe ansatz in the limit of large spacetime spin at large but finite twist, and find a novel all-loop scaling function. This function obeys the Kotikov-Lipatov transcendentality principle and does not depend on the twist. Under the assumption that one may extrapolate back to leading twist, our result yields an all-loop prediction for the large-spin anomalous dimensions of twist-two operators. The latter also appears as an undetermined function in a recent conjecture of Bern, Dixon and Smirnov for the all-loop structure of the maximally helicity violating (MHV) n-point gluon amplitudes of N=4 gauge theory. This potentially establishes a direct link between the worldsheet and the spacetime S-matrix approach. A further assumption for the validity of our prediction is that perturbative BMN (Berenstein-Maldacena-Nastase) scaling does not break down at four loops, or beyond. We also discuss how the result gets modified if BMN scaling does break down. Finally, we show that our result qualitatively agrees at strong coupling with a prediction of string theory.Comment: 45 pages LaTeX, 3 postscript figures. v2: Chapter on BMN scaling and transcendentality added. v3: version accepted for publication in JSTA

Kinematics and Robot Design II (KaRD2019) and III (KaRD2020)

This volume collects papers published in two Special Issues “Kinematics and Robot Design II, KaRD2019” (https://www.mdpi.com/journal/robotics/special_issues/KRD2019) and “Kinematics and Robot Design III, KaRD2020” (https://www.mdpi.com/journal/robotics/special_issues/KaRD2020), which are the second and third issues of the KaRD Special Issue series hosted by the open access journal robotics.The KaRD series is an open environment where researchers present their works and discuss all topics focused on the many aspects that involve kinematics in the design of robotic/automatic systems. It aims at being an established reference for researchers in the field as other serial international conferences/publications are. Even though the KaRD series publishes one Special Issue per year, all the received papers are peer-reviewed as soon as they are submitted and, if accepted, they are immediately published in MDPI Robotics. Kinematics is so intimately related to the design of robotic/automatic systems that the admitted topics of the KaRD series practically cover all the subjects normally present in well-established international conferences on “mechanisms and robotics”.KaRD2019 together with KaRD2020 received 22 papers and, after the peer-review process, accepted only 17 papers. The accepted papers cover problems related to theoretical/computational kinematics, to biomedical engineering and to other design/applicative aspects

The density of states of chaotic Andreev billiards

Quantum cavities or dots have markedly different properties depending on whether their classical counterparts are chaotic or not. Connecting a superconductor to such a cavity leads to notable proximity effects, particularly the appearance, predicted by random matrix theory, of a hard gap in the excitation spectrum of quantum chaotic systems. Andreev billiards are interesting examples of such structures built with superconductors connected to a ballistic normal metal billiard since each time an electron hits the superconducting part it is retroreflected as a hole (and vice-versa). Using a semiclassical framework for systems with chaotic dynamics, we show how this reflection, along with the interference due to subtle correlations between the classical paths of electrons and holes inside the system, are ultimately responsible for the gap formation. The treatment can be extended to include the effects of a symmetry breaking magnetic field in the normal part of the billiard or an Andreev billiard connected to two phase shifted superconductors. Therefore we are able to see how these effects can remold and eventually suppress the gap. Furthermore the semiclassical framework is able to cover the effect of a finite Ehrenfest time which also causes the gap to shrink. However for intermediate values this leads to the appearance of a second hard gap - a clear signature of the Ehrenfest time.Comment: Refereed version. 23 pages, 19 figure

Molecular Dynamics (MD) Study on the Electrochemical Properties of Electrolytes in Lithium-Ion Battery (LIB) Applications

While the high energy density and the power along with longer cycle life and less requirements of maintenance distinguish the rechargeable lithium-ion batteries (LIBs) from other energy storage devices, development of an electrolyte of LIBs with optimized properties still constitutes a challenge towards next-generation LIB systems with robust electrochemical performance. The electrolytes serve as the medium to provide ionic conduction path between the electrodes as their basic function. Conductivity of the solutions are mainly affected by their transport properties and the electrolyte electrode/separator interfacial phenomena. Although many contributions on thermodynamic properties of the electrolytes consist of alkyl carbonates mixed with salts have been previously studied, relatively little information is known regarding the correlation between interfacial properties of the electrolyte -electrode/separator with electrochemical properties of the cell. In this study, therefore, we present the impacts of salt concentration and temperature-dependent properties of LIBs on wetting behavior of various electrolytes, i.e., ethyl methyl carbonate (EMC), diethyl carbonate (DEC), and propylene carbonate (PC), in contact with the graphite anode and polyethylene (PE)/polypropylene (PP) separator using molecular dynamics (MD) computational technique. The results based on MD computations affirm the general consistent dependency of interfacial tension energies to polarity of the solvents in DEC, EMC, and PC electrolytes contained 1 M LiPF6 salt. The PC systems interestingly showed inverse trend due to the special stacking motifs of PC layers that may increase the interfacial electrostatic interactions. Temperature did not show significant effect on the interfacial energies of linear solvents whereas PC exhibited more tendency to interact with the graphite anode at T = 25 C compared to the similar solution at 0 C. Moreover, the electrolytes that incorporated same solvents had better wettability in absence of salt ions due to their lower polarity and viscosity. Accordingly, EMC: 0.752 M LiPF6 electrolyte system had the lowest interfacial energy value among the EMC solutions contained 1 M and 1.254 M salt. However, the probability of insufficient number of charge carriers in addition to the close values of interfacial energies for electrolytes with 0.752 M and 1 M LiPF6 resulted in considering EMC: 1 M LiPF6 electrolyte as a more efficient mixture. The impact of solution polarity on clustering behavior of the salt ions were investigated in DEC, EMC, and PC electrolytes with 1 M LiPF6 based on the ions coordination and their relative closest neighbors. Due to the higher dielectric constant value, PC showed higher ability of salt dissociating, which leaded that Li+ and PF6- ions were distributed more uniformly compared to the DEC and EMC electrolytes

Aspects of the PP wave/ CFT correspondence

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2005.Includes bibliographical references (leaves 187-191).In this thesis, I discuss various aspects of the PP wave/CFT duality as a concrete example of the gauge-gravity correspondence.by Umur Gürsoy.Ph.D