The sleep-wake distribution contributes to the peripheral rhythms in PERIOD-2

In the mouse, Period-2 (Per2) expression in tissues peripheral to the suprachiasmatic nuclei (SCN) increases during sleep deprivation and at times of the day when animals are predominantly awake spontaneously, suggesting that the circadian sleep-wake distribution directly contributes to the daily rhythms in Per2. We found support for this hypothesis by recording sleep-wake state alongside PER2 bioluminescence in freely behaving mice, demonstrating that PER2 bioluminescence increases during spontaneous waking and decreases during sleep. The temporary reinstatement of PER2-bioluminescence rhythmicity in behaviorally arrhythmic SCN-lesioned mice submitted to daily recurring sleep deprivations substantiates our hypothesis. Mathematical modelling revealed that PER2 dynamics can be described by a damped harmonic oscillator driven by two forces: a sleep-wake-dependent force and a SCN-independent circadian force. Our work underscores the notion that in peripheral tissues the clock gene circuitry integrates sleep-wake information and could thereby contribute to behavioral adaptability to respond to homeostatic requirements.

Adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator

In this work we present a simple and elegant approach to study the adiabatic and nonadiabatic evolution of a generalized damped harmonic oscillator which is described by the generalized Caldirola–Kanai Hamiltonian, in both classical and quantum contexts. Based on time-dependent dynamical invariants, we find that the geometric phase acquired when the damped oscillator evolves adiabatically in time provides a direct connection between the classical Hannay’s angle and the quantum Berry’s phase. In addition, we solve the time-dependent Schrödinger equation for this system and calculate various quantum properties of the damped generalized harmonic one, such as coherent states, expectation values of the position and momentum operators, their quantum fluctuations and the associated uncertainty product.

London superconductor and time-varying mesoscopic LC circuits

In this work we study the classical and quantum dynamics of a London superconductor and of a time-dependent mesoscopic or nanoscale LC circuit by assuming that the inductance and capacitance vary exponentially with time at constant rate. Surprisingly, we find that the behavior of these two systems are equivalent, both classically and quantum mechanically, and can be mapped into a standard damped harmonic oscillator which is described by the Caldirola-Kanai Hamiltonian. With the aid of the dynamical invariant method and Fock states, we solve the time-dependent Schr\"odinger equation associated with this Hamiltonian and calculate some important physical properties of these systems such as expectation values of the charge and magnetic flux, their variances and the respective uncertainty principle.

A Fixed Point Technique for Set-Valued Contractions with Supportive Applications

In this manuscript, exciting fixed point results for a pair of multivalued mappings justifying rational Gupta-Saxena type Ω -contractions in the setting of extended b -metric-like spaces are established. The theoretical results have also been strengthened by some nontrivial examples. Finally, the theoretical results are used to study the existence of the solution of Fredholm integral equation which arises from the damped harmonic oscillator, to study initial value problem which arises from Newton’s law of cooling and to study infinite systems of fractional ordinary differential equations (ODEs).

The Damped Harmonic Oscillator at the Classical Limit of the Snyder-de Sitter Space

Valtancoli in his paper entitled (P. Valtancoli, Canonical transformations and minimal length, J. Math. Phys. 56, 122107 2015) has shown how the deformation of the canonical transformations can be made compatible with the deformed Poisson brackets. Based on this work and through an appropriate canonical transformation, we solve the problem of one dimensional (1D) damped harmonic oscillator at the classical limit of the Snyder-de Sitter (SdS) space. We show that the equations of the motion can be described by trigonometric functions with frequency and period depending on the deformed and the damped parameters. We eventually discuss the influences of these parameters on the motion of the system.