Infinitesimal Time Research Articles

Celestial sector in CFT: Conformally soft symmetries

We show that time intervals of width \Delta \tauΔτ in 3-dimensional conformal field theories (CFT_33) on the Lorentzian cylinder admit an infinite dimensional symmetry enhancement in the limit \Delta \tau → 0Δτ→0. The associated vector fields are approximate solutions to the conformal Killing equations in the strip labelled by a function and a conformal Killing vector on the sphere. An Inonu-Wigner contraction yields a set of symmetry generators obeying the extended BMS_44 algebra. We analyze the shadow stress tensor Ward identities in CFT_dd on the Lorentzian cylinder with all operator insertions in infinitesimal time intervals separated by \piπ. We demonstrate that both the leading and subleading conformally soft graviton theorems in (d-1)(d−1)-dimensional celestial CFT (CCFT_{d-1}d−1) can be recovered from the transverse traceless components of these Ward identities in the limit \Delta \tau → 0Δτ→0. A similar construction allows for the leading conformally soft gluon theorem in CCFT_{d-1}d−1 to be recovered from shadow current Ward identities in CFT_dd.

Genomic inference of a severe human bottleneck during the Early to Middle Pleistocene transition.

Population size history is essential for studying human evolution. However, ancient population size history during the Pleistocene is notoriously difficult to unravel. In this study, we developed a fast infinitesimal time coalescent process (FitCoal) to circumvent this difficulty and calculated the composite likelihood for present-day human genomic sequences of 3154 individuals. Results showed that human ancestors went through a severe population bottleneck with about 1280 breeding individuals between around 930,000 and 813,000 years ago. The bottleneck lasted for about 117,000 years and brought human ancestors close to extinction. This bottleneck is congruent with a substantial chronological gap in the available African and Eurasian fossil record. Our results provide new insights into our ancestry and suggest a coincident speciation event.

A modified Newmark block method for determining the seismic displacement of a slope reinforced by prestressed anchors

The seismic permanent displacement is an essential aspect to evaluate the seismic stability of slope. The conventional Newmark block method commonly adopts a constant yield acceleration to determine the seismic permanent displacement of slope. In order to reflect a more realistic behavior of seismic displacement, the dynamic kinetic failure mechanism and the instant displacement model of slope reinforced by prestressed anchors in infinitesimal time are established. The time-dependent formula for dynamic yield acceleration is deduced based on upper bound method. A modified method is proposed to determine the seismic permanent displacement within the framework of Newmark block theory. The results are compared with the existing work with regard to the yield acceleration and seismic permanent displacement. Three typical seismic ground motions are performed to study the responses of the anchor force, the dynamic yield acceleration and the seismic permanent displacement. Results show that the anchor force, the dynamic yield acceleration and the seismic displacement all increase continuously during the excitation of seismic ground motion. The anchor causes the yield acceleration to increase, and the seismic permanent displacement to decrease significantly. The conventional method adopting a constant yield acceleration greatly overestimates the seismic permanent displacement.

How time-discretization can break the asymptotics of inverse scattering

Inverse scattering by the Marchenko or Gelfand–Levitan equations consists of two steps. The first step consists of solving an integral equation to retrieve the wavefield in the interior of a scattering medium from waves recorded outside of the scattering region. The second step consists of estimating the scattering potential from this wavefield. The estimation of the potential can be justified either by taking a high-frequency limit, or by evaluating the wavefield an infinitesimal time before or after the direct wave. Waveforms obtained in experiments are discretized in time with a sampling interval d t . We show an example that this time discretization precludes the extraction of the potential. The reason is that the frequencies of discretized waveforms are below the Nyquist frequency. This limitation precludes taking the high-frequency limit that is used for estimating the potential from the waveforms in the interior of the scattering region. We present an alternative method to estimate the potential from wavefields in the interior of the scattering region. This method can be used even when the waves are in the strong scattering regime.

Periodic event-based robust output regulation for a class of uncertain linear systems with inter-event analysis

This paper investigates the robust output regulation problem of uncertain linear systems by using a periodic event-triggered controller. A new type of event triggers based on local sampled signals is proposed to monitor events. An event detector only uses local information, avoiding infinitesimal inter-event times as in some continuous event-triggered schemes. A periodic event-triggered system naturally excludes Zeno behavior as the system operates at discrete-time instants. Convergence of tracking error holds for a class of decreasing triggering positive functions. When exponential functions bound the triggering function, analytical characterization of the relationship among sampling, event triggering, and inter-event behavior is established.

γ rays run on time, and propagate tailgating gravitational waves

Significant absorption of radiation is usually accompanied by refraction.This is not the case for γ rays travelling cosmic distances. We show that the realand imaginary parts of the refraction index are indeed commensurable, as they are relatedby dispersion relations, but when turning to physical observables, the (finite) optical depthis way larger than the (infinitesimal) time delay of the gamma rays relative to gravitational radiation.The numerically large factor solving the apparent contradiction is E γ /H 0arising from basic wave properties (Bouguer-Beer-Lambert law)and the standard cosmological model, respectively.In consequence, no delay of the γ-ray propagation affects multimessenger astronomy. We particularlypredict no such delay between gravitational waves and γ photons from binary mergers such as GW170817,save for that induced at the source, nor from more energetic events at cosmic distances.

Two-dimensional Darcy–Bénard convection evolving in Fourier space

Nonlinear transient convection in a porous rectangle heated from below is studied by an analytically based method. Fourier series for the temperature and streamfunction are applied, where each Fourier coefficient evolves in time according to a coupled set of ordinary differential equations. The mathematical method can be considered as a recursive nonlinear mapping in Fourier space from a given state at a time t to an updated state at time t + dt, with an infinitesimal time increment. This nonlinear evolution in Fourier space requires normal-mode compatible boundary conditions along the entire boundary. Each Fourier coefficient gets three contributions during its updating in time: (i) one decay term due to thermal diffusion, governed by linear theory; (ii) one growth term due to buoyancy, governed by linear theory; and (iii) quadratic nonlinearities from the convection term in the heat equation, involving all pairwise interactions between the Fourier modes. We present numerical computations with a standard Runge–Kutta method, with Rayleigh numbers up to ten times the well-known critical value 4π2. Our plots are produced with Fourier series truncated to include 15 normal modes in both the horizontal and vertical directions. For validation of our method, we present tables for the Nusselt number of steady convection, with a higher number (up to 20) of normal modes included in the truncated system of equations. Our computations of transient nonlinear convection lead to steady states. A final steady state is not unique for a given geometry, but depends on the initial state and the Rayleigh number. This ambiguity of steady states is depicted by a hysteresis loop. The Malkus hypothesis of maximal heat transfer is put into perspective. This hypothesis does not pick a preferred cell width, but it nevertheless constrains the hysteresis loop.

Overdamped and underdamped Langevin equations in the interpretation of experiments and simulations

The Brownian motion (BM) is not only a natural phenomenon but also a fundamental concept in several scientific fields. The mathematical description of the BM for students of various disciplines is most often based on Langevin’s equation with the Stokes friction force and the random force modeling Brownian particle (BP) collisions with surrounding molecules. For many phenomena, such a description is insufficient, as it assumes an infinitesimal correlation time of random force. This shortcoming is overcome by the generalized Langevin equation (GLE), which is now one of the most widely used equations in physics. In the present work, we offer a simple way of solving this equation, consisting of its transformation into an integro-differential equation for the mean square displacement of the BP, which is then effectively solved using the Laplace transform (LT). We demonstrate the use of this method to solve both the standard Langevin equation and the GLE for the BP in an external harmonic field. We analyze the cases of overdamped (when frictional forces prevail over inertial forces and the BP mass is considered zero in the equation) and underdamped (inertial effects are not neglected) equations. We show under what conditions an overdamped solution can be used instead of complicated solutions of the underdamped equation. We also demonstrate the effectiveness of the use of the LT on a microscopic approach to the derivation of the GLE. Graduate students are offered several problems in which the internal shortcomings of the overdamped Langevin equations manifest themselves.

Entropy variation of rotating BTZ black hole under Hawking radiation

An axially symmetric rotating Banados-Teitelboim-Zanelli (BTZ) black hole is considered for comprehending its interior information. The largest space-like hyper-surface is chosen to estimate its maximal interior volume as a time-dependent quantity. Similarly, the quantum mode entropy of the scalar field associated with this volume is found to increase with Eddington time. An evolution relation between the variation of quantum mode entropy and Bekenstein-Hawking entropy is obtained for an infinitesimal time interval. On comparing to higher-dimensional black holes, the characteristic feature of this evolution relation is its divergent character.

Real-time motion of open quantum systems: Structure of entanglement, renormalization group, and trajectories

In this paper, we provide a complete description of the life cycle of entanglement during the real-time motion of open quantum systems. The quantum environment can have arbitrary (e.g., structured) spectral density. The entanglement can be seen constructively as a Lego: its bricks are the modes of the environment. These bricks are connected to each other via operator transforms. The central result is that during each infinitesimal time interval one new (incoming) mode of the environment gets coupled (entangled) to the open system, and one new (outgoing) mode gets irreversibly decoupled (disentangled from the future). Moreover, each moment of time, only a few relevant modes (three to four in the considered cases) are non-negligibly coupled to the future quantum motion. These relevant modes change (flow or renormalize) with time. As a result, the temporal entanglement has the structure of a matrix-product operator. This allows us to pose a number of questions and to answer them in this paper: What is the intrinsic quantum complexity of a real time motion? Does this complexity saturate with time or grow without bounds? How does one do the real-time renormalization group in a justified way? How do the classical Brownian stochastic trajectories emerge from the quantum evolution? How does one construct the few-mode representations of non-Markovian environments? We provide illustrative simulations of the spin-boson model for various spectral densities of the environment: semicircle, subohmic, ohmic, and superohmic.

Theoretical and experimental study of motion and sinking time of Saxon bowls

This work focuses on investigating the time of sinking of a Saxon bowl proposed by ‘International Young Physicists’ Tournament in 2020. A quasi-static model is built to simulate the motion path of the bowl and predict the sinking time subsequently. The model assumes an open axisymmetric bowl with a hole in its base. The hole is modelled as a pipe for which the flow profile is governed by a modified Bernoulli’s equation which has a coefficient of discharge (C d) added to account for energy losses. The motion of the entire bowl is assumed to be in quasi-static equilibrium for an infinitesimal time interval to calculate the volumetric flow rate through the hole. The model is used to predict the sinking times of various bowls against independent variables—hole radius, bowl dimensions, mass of bowl, mass distribution of bowl, and coefficient of discharge—and predict the motion path of bowls of different, axisymmetric geometries. Characterisation of C d was done by draining a bowl filled with water and measuring the time taken to do so. Experimental verification was completed through measuring sinking times of 3D printed hemispherical bowls of the different variables in water. Motion tracking of bowls with different geometries was done using computational pixel tracking software to verify the model’s predictive power. Data from experiments for sinking time against the variables corroborate with the model to a great degree. The motion path tracked, matched the modelled motion path to a high degree for bowls of different shapes, namely a hemisphere, cylinder, frustum, and a free-form axisymmetric shape. The work is poised for an undergraduate level of readership.

Dynamic calibration of differential equations using machine learning, with application to turbulence models

We present a methodology for calibration of parametric ordinary and partial differential equation models, using off-the-shelf software for back-propagation in Neural Networks (NN). As a prototypical example, we consider calibration of a Reynolds-averaged Navier-Stokes (RANS) turbulence closure model, against ground truth data from direct numerical simulations (DNS) of two different turbulent flows. Numerical time integration is represented as a custom NN, where only the RANS model parameters are trainable. A loss function is defined to quantify the mismatch between the NN prediction and the ground truth over a predefined, finite time integration window. This loss function is then minimized using a gradient descent method utilizing the back-propagation algorithm. This dynamic approach to training is to be contrasted with a static approach, wherein a least square regression estimate for parameters is obtained in the limit of an infinitesimal time integration window. In a first test of static and dynamic approaches against ground truth data generated by the model, the former proves to be significantly faster and more accurate than the latter at recovering the parameters. When both calibration approaches are tested against DNS data, for which it is known that the model cannot achieve a perfect fit, the static approach yields a good prediction only for short times, while the dynamic approach results in physical and stable predictions over the entire integration window. After optimization of the dynamic approach for time step, spatial resolution, stability, and physics-based constraints, we obtain a 50% improvement of outcomes over those obtained from the existing, manually calibrated set of parameters, demonstrating the merits of this systematic and automated procedure.

Vortex Filament Equation for a Regular Polygon in the Hyperbolic Plane

The aim of this paper is twofold. First, we show the evolution of the vortex filament equation (VFE) for a regular planar polygon in the hyperbolic space. Unlike in the Euclidean space, the planar polygon is open and both of its ends grow up exponentially, which makes the problem more challenging from a numerical point of view. However, using a finite difference scheme in space combined with a fourth-order Runge–Kutta method in time and fixed boundary conditions, we show that the numerical solution is in complete agreement with the one obtained by means of algebraic techniques. Second, as in the Euclidean case, we claim that, at infinitesimal times, the evolution of VFE for a planar polygon as the initial datum can be described as a superposition of several one-corner initial data. As a consequence, not only can we compute the speed of the center of mass of the planar polygon, but the relationship also allows us to compare the time evolution of any of its corners with the evolution in the Euclidean case.

Evolution of coherent states as quantum counterpart of classical dynamics

Quantum dynamics of coherent states is studied within quantum field theory using two complementary methods: by organizing the evolution as a Taylor series in elapsed time and by perturbative expansion in coupling within the interaction-picture formalism. One of the important aspects of our analysis consists in utilizing the operators and the vacuum of interacting theory in constructing the states, without invoking asymptotic particles. Focusing on a coherent state describing a spatially homogeneous field configuration, it is demonstrated that both adopted methods successfully account for nonlinear classical dynamics, giving distinguishable quantum effects. In particular, according to the time-expansion analysis the initial field acceleration, with which the field departs from its initial expectation value, is governed by the tree-level potential with renormalized mass and bare coupling constant. The interaction-picture computation, instead, can be manipulated to give the nonlinear dynamics, determined in terms of renormalized coupling and mass. However, it results in a logarithmic initial-time singularity in the field acceleration, reminiscent of the similar behavior encountered within semiclassical formalism, for certain choices of the initial state for fluctuations. Within our coherent-state analysis, the above-mentioned peculiarities are artifacts of an expansion: in the first case over infinitesimal time, while in the second case in the coupling constant. Despite this, we show that the evolution obtained within the interaction-picture analysis is valid for an extended period of time. Moreover, on top of the desired classical dynamics, it serves us with interesting quantum corrections, previously proposed by Dvali-Gomez-Zell.

The Differential Transform

One of the methods of mathematical analysis in many cases makes it possible to reduce the study of differential operators, pseudo-differential operators and certain types of integral operators and the solution of equations containing them, to an examination of simpler algebraic problems. The development and systematic use of operational calculus began with the work of O. Heaviside (1892), who proposed formal rules for dealing with the differentiation operator d/dt and solved a number of applied problems. However, he did not give operational calculus a mathematical basis; this was done with the aid of the Laplace transform; J. Mikusinski (1953) put operational calculus into algebraic form, using the concept of a function ring [1]. Thereupon I’m suggesting here the use of the integration operator dt to make an extension for the systematic use of operational calculus. Throughout designing and analyzing a control system, we need to treat the functionality of the system with respect to time. The reaction of the system instructs us how to stable it by amplifiers and feedbacks [2], thereafter the Differential Transform is a good tool for doing this, and it’s a technique to frustrate difficulties we may bump into, also it has the methods to find the immediate reaction of the system with respect to infinitesimal (tiny) time which spares us from the hard work in finding the time dependent function, this could be done by producing finite series.

Euler-Milstein and relevant approaches for high-precision stochastic simulation of quantum trajectories

For quantum diffusive measurements, the system’s dynamics can be described by the Itô stochastic master equation, which works well for an infinitesimal time resolution. However, in practical quantum experiments, one cannot make a time step to be infinitesimal, as it can introduce correlation of noise in time, making the Markovian assumption invalid. On the other hand, increasing a time step can cause errors from non-commuting operations describing the system’s dynamics. We therefore consider implementing the Euler-Milstein and relevant approaches, namely the Itô map, the high-order completely positive map, and the quantum Bayesian, to simulate quajntum trajectory for a quantum system under diffusive continuous measurements. In particular, we numerically simulate trajectories for a qubit measurement in z basis. We show the comparison of individual trajectories and their averaged trajectories among these approaches. We find that the high-order completely positive map approach yields the most accurate averaged quantum trajectory. Furthermore, we also investigate the trace distance from true stochastic quantum trajectories, comparing the four approaches using the numerical simulation. We show that, for a realistic time resolution (as in a superconducting qubit, experiment), the high-order map does give the most accurate estimate of the qubit trajectories.

Singular dynamics for morphing aircraft switching on the velocity boundary

It has singularity problem for morphing aircraft to switch its structure in an infinitesimal short time. Many physical parameters such as aerodynamic coefficients and moment of inertia will be altered, and the thrust of aero engine may be different for different states of morphing aircraft. For a typical case, the sweep angle of wing changes to a relative large angle when the speed of morphing aircraft is higher than a critical speed, while it switches to a small angle for the speed of such an aircraft below the critical speed. It could have possibility that the speed of such a morphing aircraft chatters around the critical speed. If it is in unmanned operation, the aircraft will switch its structure in a high frequency for the aforementioned condition, which may be a high risk for the control and dynamic of the morphing aircraft. Therefore, it is necessary to study whether the morphing aircraft should switch its structure when the speed arrives the critical speed. To simplify the problem, the switchability of such a morphing aircraft will be studied based on the dynamical system in the longitudinal direction. The analytical passable and non-passable conditions for flow switching on the velocity boundary for such a morphing aircraft will be developed. For non-passable flows, the analytical conditions for onset and vanishing for flow sliding on the velocity boundary will also be given. Mapping structure will be used to describe the periodic motions of dynamical system for such an aircraft. Numerical simulations will be carried out to show the complexity for such a switchability problem. It will be found that with the development of switchability conditions, the morphing aircraft could obviously decrease the chatter phenomenon compared with the direct switch case.

A nonlinear impulsive Cauchy–Poisson problem. Part 1. Eulerian description

Abstract

Method for obtaining a differential equation of the asphalt concrete mix behavior under railway transport load

The use of asphalt concrete in the railway track construction allows you to ensure high strength and stability of the railway. The supporting asphalt layer is subjected to force from the railway transport. The study of material behavior under load is of great practical importance. Rheological modeling is a popular tool for studying the behavior of a material under load. The accepted rheological model is described by the corresponding differential equation. Establishing the initial conditions of a differential equation is difficult if its order is 2 or higher. The article describes an establishing initial conditions method for solving the rheological equation of the model. The model’s behavior analytical description over an infinitesimal time period allows you to set initial conditions. The proposed method for establishing the model equation initial conditions simplifies the process of material under load analytical description. This approach to the asphalt concrete layer under load study allows a more conscious approach to the choice of its parameters when designing the railway track elements.

Testing the weak cosmic censorship conjecture in torus-like black hole under charged scalar field

We investigate weak cosmic censorship conjecture in charged torus-like black hole by the complex scalar field scattering. Using the relation between the conserved quantities of a black hole and the scalar field, we can calculate the change of the energy and charge within the infinitesimal time. The change of the enthalpy is connected to the change of energy, then we use those results to test whether the first law, the second law as well as the weak cosmic censorship conjecture are valid. In the normal phase–space, the first law of thermodynamics and the weak cosmic censorship conjecture are valid, and the second law of thermodynamics is not violated. For the specific black hole under scalar field scattering we consider, in the extended phase–space, the first law of thermodynamics and the weak cosmic censorship conjecture are valid. However, the second law of thermodynamics is violated when the black hole’s initial charge reaches a certain value.