Arbitrary Values Research Articles

Attractors for Hopfield lattice model in weighted spaces

The investigation into the dynamic behavior of infinite lattice systems holds paramount significance in the realm of physical phenomena, particularly in mechanics. This intricate domain has captivated the attention of both mathematicians and physicists.In acknowledgment of the inherent noise prevalent in real-world environments, our study embraces this aspect by introducing a random term into our model. This deliberate inclusion of stochasticity engenders a novel perspective, giving rise to a stochastic lattice differential equation. This model proves to be a versatile tool for accurately characterizing spatial structures characterized by discrete components and the associated uncertainties that pervade them. This research elucidates the intricate interplay between lattice dynamics and environmental noise, shedding light on the complex behavior of such systems in a realistic context. Our result generalizes many results in three directions: extending the connections between the terms to non-linear, extending the connection neighborhood from 3 (as in most cases) to arbitrary value n, and extending the results that are in ℓ2 to ℓρ2.

Generalised conditions for rapid-turn inflation

Rapid-turn slow-roll inflationary trajectories have been shown to be an attractor in two-field models, provided the turn rate is near constant and larger than the slow-roll parameters. These trajectories can produce primordial spectra consistent with current observations on CMB scales. We present the generalized consistency condition for sustained rapid-turn inflationary trajectory with two fields, arbitrary field-space metric and potential valid for any value of the turn rate. This has to be supplemented by a second condition to ensure slow roll evolution. Both conditions together constitute a tool to identify inflationary trajectories with arbitrary values of the turning rate without having to solve the equations of motion. We present a Python package for the numerical identification of regions in field-space and parameter space that allow for rapid-turn trajectories.

Comparative Analysis of the Antibacterial Properties of Various Periodontal Bone Graft Materials In Vitro.

Periodontal bone grafting is a common procedure in dentistry to restore bone loss resulting from periodontal disease. Various graft materials are available, each with different antibacterial properties. Understanding these properties is crucial for successful grafting outcomes. In this study, we conducted a comparative analysis of the antibacterial properties of four commonly used periodontal bone graft materials: The graft used were A: Hydroxyapatite, B: B TCP, C: Bioactive glass, D: Calcium Phosphate materials. Each material was subjected to in vitro testing using a standardized bacterial culture assay. Antibacterial activity was assessed by measuring the zone of inhibition around each material after exposure to bacterial cultures. Material A exhibited the highest antibacterial activity with an average zone of inhibition of 12.3 mm, followed by Material C with 9.8 mm, Material D with 8.5 mm, and Material B with 6.2 mm. These arbitrary values reflect the relative efficacy of each material in inhibiting bacterial growth. Our findings suggest that Material A possesses the strongest antibacterial properties among the tested periodontal bone graft materials. This may contribute to improved clinical outcomes in periodontal bone grafting procedures by reducing the risk of postoperative infections. Further research is warranted to elucidate the mechanisms underlying the antibacterial effects of these materials and to validate our in vitro findings in clinical settings.

Arbitrary relaxation rate under non-Hermitian matrix iterations

We study the exponential relaxation of observables propagated with a non-Hermitian transfer matrix, an example being out-of-time-ordered correlations (OTOC) in brick-wall (BW) random quantum circuits. Until a time that scales as the system size, the exponential decay of observables is not usually determined by the second largest eigenvalue of the transfer matrix, as one can naively expect, but it is, in general, slower—this slower decay rate was dubbed “phantom eigenvalue.” Generally, this slower decay is given by the largest value in the pseudospectrum of the transfer matrix; however, we show that the decay rate can be an arbitrary value between the second largest eigenvalue and the largest value in the pseudospectrum. This arbitrary decay can be observed, for example, in the propagation of OTOC in periodic boundary conditions BW circuits. To explore this phenomenon, we study a matrix iteration made from a simple tridiagonal Toeplitz matrix. This setting can be used to propagate OTOC in random circuits with open boundary conditions and to describe a one-dimensional biased random walk with dissipation at the edges. Published by the American Physical Society 2024

Nonuniqueness in defining the polarization: Nonlocal surface charges and the electrostatic, energetic, and transport perspectives

Ionic crystals, ranging from dielectrics to solid electrolytes to complex oxides, play a central role in the development of modern technologies for energy storage, sensing, actuation, and other functional applications. Mesoscale descriptions of these crystals are based on the continuum polarization density field to represent the effective physics of charge distribution at the scale of the atomic lattice. However, a long-standing difficulty is that the classical electrostatic definition of the macroscopic polarization — as the dipole or first moment of the charge density in a unit cell — is not unique; rather, it is sensitive to translations of the unit cell in an infinite periodic system. This unphysical non-uniqueness has been shown to arise from starting directly with an infinite system — wherein the boundaries are ill-defined — rather than starting with a finite body and taking appropriate limits. This limit process shows that the electrostatic description requires not only the bulk polarization density, but also the surface charge density, as the effective macroscopic descriptors; that is, a nonlocal effective description. Other approaches to resolve this difficulty include the popular modern theory of polarization that completely sets aside the polarization as a fundamental quantity in favor of the change in polarization from an arbitrary reference value and then relates the change in polarization to the transport of charge (the “transport” definition); or, in the spirit of classical continuum mechanics, to define the polarization as the energy-conjugate to the electric field (the “energetic” definition).This work examines the relation between the classical electrostatic definition of polarization, and the transport and energy-conjugate definitions of polarization. We show the following: (1) The transport of charge does not correspond to the change in polarization in general; instead, one requires additional simplifying assumptions on the electrostatic definition of polarization for these approaches to give rise to the same macroscopic electric fields. Thus, the electrostatic definition encompasses the transport definition as a special case. (2) The energy-conjugate definition has both bulk and surface contributions; while traditional approaches neglect the surface contribution, we find that accounting for the nonlocal surface contributions is essential to be consistent with the classical definition and obtain the correct macroscopic electric fields.

Impact of network degradation on load frequency control of large interconnected power system

Due to the extensive geographical coverage of the power system, the load frequency control (LFC) performance is impacted by communication delays and packet loss that occur during the exchange of control and measurement signals. In the literature on LFC of large interconnected power systems, communication delays are often assumed to be arbitrary or time-varying values based on probability distributions or estimation models. Most studies neglect realistic delays based on the actual communication technology present in the system despite its practical significance. Furthermore, the impact of packet loss in LFC studies has been largely overlooked in the existing literature. This research work presents a comprehensive investigation into the impact of real-time communication delay and packet loss on LFC performance in large interconnected power systems. The simulation results demonstrate the significant impact of communication network impairments on LFC performance. The classical simulation results are validated using Real-Time Hardware-in-the-Loop (RT-HIL) simulation.

Evaluation of the Sealing Ability of Various Root Canal Obturation Techniques in Permanent Molars: An In vitro Study.

Achieving a reliable seal in root canal obturation is essential for the success of endodontic treatment. Despite the availability of multiple techniques, the comparative sealing abilities of these methods remain a topic of investigation, particularly in permanent molars. This in vitro study involved 60 extracted permanent molars, divided randomly into three groups: Group A (lateral compaction), Group B (thermoplasticized gutta-percha), and Group C (single cone). After instrumentation and obturation, the sealing ability was assessed using a dye penetration method. Arbitrary values were assigned to quantify the extent of dye penetration. The mean dye penetration values were as follows: Group B (thermoplasticized gutta-percha) had the lowest penetration (2.1 ± 0.3 mm), followed by Group A (lateral compaction) (3.5 ± 0.4 mm), and Group C (single cone) exhibited the highest penetration (5.2 ± 0.6 mm). In this study, thermoplasticized gutta-percha demonstrated superior sealing ability compared to lateral compaction and single cone techniques in permanent molars. These findings underscore the importance of selecting an effective obturation technique to ensure optimal seal integrity in root canal therapy.

Magnetoelectric fractals, Magnetoelectric parametric resonance and Hopf bifurcation

In the present work, we study the dynamics of the magnetic nanodisc coupled through a magnetoelectric coupling to a ferroelectric crystal. The model of our interest is nonlinear, and we explore the problem under different limits of weak and strong non-linearity. By applying two electric fields with different frequencies, we control the form of the confinement potential of a ferroelectric subsystem and realize different types of dynamics. We proved that the system is more sensitive to magnetoelectric coupling in the case of double-well potential. In particular, in the case of strong non-linearity, arbitrary small values of magnetoelectric coupling lead to chaotic dynamics. In essence, magnetoelectric coupling plays a role akin to the small perturbations destroying invariant tori according to the KAM theorem. We showed that bifurcations in the system are of Hopf’s type. We observed the formation of magnetoelectric fractals in the system. In the limit of weak non-linearity, we studied a problem of parametric nonlinear resonance and enhancement of magnetic oscillations through magnetoelectric coupling.

Switching periodic event‐triggered global prescribed performance control of uncertain strict‐feedback systems with sensor faults

AbstractThis study investigates the global adaptive prescribed performance control (PPC) of a class of uncertain strict‐feedback nonlinear systems with sensor faults based on the switching periodic event‐triggering mechanism (SPETM). Due to the existence of sensor faults, the controlled system is first remodeled by utilizing the available variables. To reduce the communication frequency, a novel SPETM is proposed by combining the advantages of the static and the dynamic event‐triggering mechanisms. This mechanism can not only avoid continuously monitoring the event‐triggering condition and avoid the Zeno phenomenon in mechanism, but also reduce the trigger frequency while ensuring the system performance by adjusting the event‐triggering threshold dynamically. Meanwhile, a time‐varying scaling function, whose reciprocal is considered as a prescribed performance function, is designed to achieve the global PPC by combining the nonlinear transformation technique. The adaptive control algorithm design is completed by employing the backstepping methodology, which can guarantee that all closed‐loop signals are bounded and the actual system output signal evolves within the prescribed performance boundary for arbitrary initial values. The effectiveness and the advantages of the proposed control algorithm are illustrated through an application example of the network‐based robotic manipulator system.

Tunable broadband polarization retarders.

We theoretically propose a type of tunable polarization retarder, which is composed of sequences of half-wave and quarter-wave polarization retarders, allowing operation at broad spectral bandwidth. The constituent retarders are composed of stacked standard half-wave retarders and quarter-wave retarders rotated at designated angles relative to their fast polarization axes. The proposed composite retarder (CR) can be tuned to an arbitrary value of the retardance by varying the middle retarder alone while maintaining its broadband spectral bandwidth intact.

Multi-component ion equilibria and transport in ion-exchange membranes

At the interface between an ion-exchange membrane and a multi-electrolyte solution, charged species redistribute themselves to minimize the free energy of the system. In this paper, we explore the Donnan equilibrium of membranes with quaternary electrolyte (Na+/Mg2+/K+/Ca2+/Cl−) solutions, experimentally. The data was used to calculate the ion activity coefficients for six commercial cation-exchange membranes (CEMs). After setting one of the activity coefficients to an arbitrary value, we used the remaining (N-1) activity coefficients as fitting parameters to describe the equilibrium concentrations of (N) ionic species with a mean relative error of 3 %. At increasing solution ionic strengths, the equivalent ion fractions of monovalent counter-ions inside the membrane increased at the expense of the multivalent ones in alignment with the Donnan equilibrium theory. The fitted activity coefficients were employed in a transport model that simulated a Donnan dialysis experiment involving all four cations simultaneously. The arbitrary value assigned to one activity coefficient affects the calculated Donnan potential at the membrane interface. Nevertheless, this arbitrary value does not affect the prediction of the ion concentrations inside the membrane and consequently does not affect the modeled ion fluxes.

Renormalization of asymmetric staple-shaped Wilson-line operators in lattice and continuum perturbation theory

In this work, we study the renormalization of nonlocal quark bilinear operators containing an asymmetric staple-shaped Wilson line at the one-loop level in both lattice and continuum perturbation theory. These operators enter the first-principle calculation of transverse momentum-dependent parton distribution functions (TMDPDFs) in lattice QCD using the formulation of large momentum effective theory. We provide appropriate RI′-type conditions that address the power and logarithmic divergences, as well as the mixing among staple operators of different Dirac structures, using a number of different possible projectors. A variant of RI′, including calculations of rectangular Wilson loops, which cancel the pinch-pole singularities of the staple operators at infinite length and reduce residual power divergences, is also employed. We calculate at one-loop order the conversion matrix, which relates the quasi-TMDPDFs in the RI′-type schemes to the reference scheme MS¯ for arbitrary values of the renormalization momentum scale and of the dimensions of the staple. Published by the American Physical Society 2024

Flow of an Oldroyd-B fluid in a slowly varying contraction: theoretical results for arbitrary values of Deborah number in the ultra-dilute limit

Pressure-driven flows of viscoelastic fluids in narrow non-uniform geometries are common in physiological flows and various industrial applications. For such flows, one of the main interests is understanding the relationship between the flow rate $q$ and the pressure drop $\Delta p$ , which, to date, is studied primarily using numerical simulations. We analyse the flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, contracting channels and present a theoretical framework for calculating the $q-\Delta p$ relation. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity profile remains parabolic and Newtonian, resulting in a one-way coupling between the velocity and polymer conformation tensor. This one-way coupling enables us to derive closed-form expressions for the conformation tensor and the flow rate–pressure drop relation for arbitrary values of the Deborah number ( $De$ ). Furthermore, we provide analytical expressions for the conformation tensor and the $q-\Delta p$ relation in the high-Deborah-number limit, complementing our previous low-Deborah-number lubrication analysis. We reveal that the pressure drop in the contraction monotonically decreases with $De$ , having linear scaling at high Deborah numbers, and identify the physical mechanisms governing the pressure drop reduction. We further elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following the contraction and show that the downstream distance required for such relaxation scales linearly with $De$ .

Complete description of effects on reconstruction of dynamic objects from time-averaged digital holography to high-speed digital holography

Time-averaged holography (TAH) is a specialized technique for studying objects subjected to sinusoidal vibration, characterized by presenting a Bessel J0 envelope in the object’s reconstruction, a condition that occurs when the vibration period is much shorter than the hologram exposure time. In this work, we present an analytical expression that describes the reconstruction effects when both the exposure time and the period can take arbitrary values, allowing the application of the TAH technique for exposure times as fractions of the period. We observe that the presented function contains higher-order Bessel functions. Additionally, we found that the envelope not only depends on the relationship between the exposure time and the vibration period but is also directly related to the vibration amplitude. The expression we introduce applies to conditions where exposure times are very short, possible with pulsed lasers, called high-speed holography (HSH), where the object reconstructs as if it were static. This mathematical expression serves as a bridge that continuously connects the techniques of HSH and TAH, enabling a smooth transition between both techniques.

Making Predictions Using Poorly Identified Mathematical Models

Many commonly used mathematical models in the field of mathematical biology involve challenges of parameter non-identifiability. Practical non-identifiability, where the quality and quantity of data does not provide sufficiently precise parameter estimates is often encountered, even with relatively simple models. In particular, the situation where some parameters are identifiable and others are not is often encountered. In this work we apply a recent likelihood-based workflow, called Profile-Wise Analysis (PWA), to non-identifiable models for the first time. The PWA workflow addresses identifiability, parameter estimation, and prediction in a unified framework that is simple to implement and interpret. Previous implementations of the workflow have dealt with idealised identifiable problems only. In this study we illustrate how the PWA workflow can be applied to both structurally non-identifiable and practically non-identifiable models in the context of simple population growth models. Dealing with simple mathematical models allows us to present the PWA workflow in a didactic, self-contained document that can be studied together with relatively straightforward Julia code provided on GitHub. Working with simple mathematical models allows the PWA workflow prediction intervals to be compared with gold standard full likelihood prediction intervals. Together, our examples illustrate how the PWA workflow provides us with a systematic way of dealing with non-identifiability, especially compared to other approaches, such as seeking ad hoc parameter combinations, or simply setting parameter values to some arbitrary default value. Importantly, we show that the PWA workflow provides insight into the commonly-encountered situation where some parameters are identifiable and others are not, allowing us to explore how uncertainty in some parameters, and combinations of parameters, regardless of their identifiability status, influences model predictions in a way that is insightful and interpretable.

Coupled bulk and interfacial transport of surfactants governs the settling of a drop towards a wall

Surfactant-like impurities are omnipresent in multiphase emulsions and may substantially affect the motion of small droplets by altering their interfacial properties. Usually these surfactants are soluble in the bulk and undergo adsorption–desorption onto the interface which modifies their surface concentration and hence their overall influence on droplet motion. Yet, the impact of the bulk solubility and transport of surfactants on droplet dynamics, especially in the presence of bounding walls, remains poorly understood. As such, in this article, we assess the impact of bulk soluble surfactants on the settling of a spherical drop towards a plane wall. We consider coupled bulk and interfacial transport of surfactants, mediated by adsorption–desorption processes and construct a semi-analytical framework for arbitrary values of ‘bulk interaction parameter’, which dictates the strength of adsorption–desorption kinetics compared with bulk diffusion. Our results indicate that while mass exchange between the bulk and the interface can remobilize the drop, a finite bulk diffusion rate restricts this process and therefore slows down the drop. This also results in bulk concentration depletion near the south pole and accumulation near the north pole, the extent of which becomes strongly asymmetric with an enhanced intensity of depletion, as the drop approaches the wall. Presence of the wall and bulk solubility are found to aid each other towards remobilizing the drop by aptly modifying the interfacial concentration. Our results may provide fundamental insights into the kinetics of surfactant-laden drops, with potential applications in food and pharmaceutical industries, separation processes, etc.

Deterministic preparation of optical qubits with coherent feedback control

We propose a class of preparation schemes for orbital angular momentum and polarization qubits carried by single photons or classical states of light based on coherent feedback control by an ancillary degree of freedom of light. The preparation methods use linear optics and include the transcription of an arbitrary polarization state onto a two-level OAM system (swap) for arbitrary OAM values ±ℓ within a light beam, i.e., without a spatial interferometer. The preparations can be carried out with unit efficiency independent from the potentially unknown initial state of the system. The swap scheme also allows us to implement arbitrary unitary gates on OAM qubits (±ℓ) by reducing them to polarization gates. In addition, we show how to translate measurement-based qubit control channels into coherent feedback schemes for optical implementation.

Testing the scalar weak gravity conjecture in no-scale supergravity

We explore possible extensions of the Weak Gravity Conjecture (WGC) to scalar field theories. To avoid charged black hole remnants, the WGC requires the existence of a particle with a mass m ≤ gqMP, with charge q and U(1) gauge coupling g, allowing the decay to shed the black hole charge. Although there is no obvious problem that arises in the absence of a U(1) charge, it has been postulated that gravity must remain the weakest force even when extended to scalar interactions. Quantifying this conjecture may be done by comparing scalar and gravitational amplitudes, or as we advocate here by comparing scattering cross sections. In theories with non-trivial field space geometries, by working out examples with perturbation theory around arbitrary field values and performing tadpole resummations, we argue that the conjecture must be applied only at extrema of the scalar potential (when expressed in locally canonical coordinates). We consider several toy models in the context of no-scale supergravity and also consider examples of inflationary models.

Soliton structures for the (3 + 1)-dimensional Painlevé integrable equation in fluid mediums

The (3 + 1)-dimensional Painlevé integrable equation are a class of nonlinear differential equations with special properties, which play an important role in nonlinear science and are of great significance in solving various practical problems, such as many important models in fields such as quantum mechanics, statistical physics, nonlinear optics, and celestial mechanics. In this work, we utilize the Hirota bilinear form and Mathematica software to formally obtain the interaction solution among lump wave, solitary wave and periodic wave, which has not yet appeared in other literature. Additionally, using the (G′/G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(G'/G)$$\\end{document}-expansion method, we provide a rich set of exact solutions for the (3 + 1)-dimensional Painlevé integrable equation, which includes two functions with arbitrary values. This method is the first to be applied to the (3 + 1)-dimensional Painlevé integrable equation. By giving some 3D graphics and density maps, the dynamic properties are analyzed and demonstrated, which is beneficial for promoting understanding and application of the (3 + 1)-dimensional Painlevé integrable equation.

An extended graphical solution for undrained cylindrical cavity expansion in K0‐consolidated Mohr–Coulomb soil

AbstractThis paper develops a general and complete solution for the undrained cylindrical cavity expansion problem in nonassociated Mohr‐Coulomb soil under nonhydrostatic initial stress field (i.e., arbitrary values of the earth pressure coefficient), by expanding a unique and efficient graphical solution procedure recently proposed by Chen and Wang in 2022 for the special in situ stress case with . It is interesting to find that the cavity expansion deviatoric stress path is always composed of a series of piecewise straight lines, for all different case scenarios of K0 being involved. When the cavity is sufficiently expanded, the stress path will eventually end, exclusively, in a major sextant with Lode angle θ in between and or on the specific line of . The salient advantage/feature of the present general graphical approach lies in that it can deduce the cavity expansion responses in full closed form, nevertheless being free of the limitation of the intermediacy assumption for the vertical stress and of the difficulty existing in the traditional zoning method that involves cumbersome, sequential determination of distinct Mohr–Coulomb plastic regions. Some typical results for the desired cavity expansion curves and the limit cavity pressure are presented, to investigate the impacts of soil plasticity parameters and the earth pressure coefficient on the cavity responses. The proposed graphical method/solution will be of great value for the interpretation of pressuremeter tests in cohesive‐frictional soils.