Approaches Infinity Research Articles

Bound states in the continuum in clusters of acoustic and elastic resonators

This study investigates the localization of waves within highly symmetric clusters of scatterers, focusing on the resonances that emerge when resonators are strategically positioned along the perimeter of a circumference. Notably, the quality factor of these resonances exhibits a noteworthy enhancement with an increasing number of resonators. We demonstrate that, specifically for flexural waves, as the number of scatterers approaches infinity, a defining condition emerges for the formation of bound states in the continuum (BICs). Additionally, we present an analytical expression for the design of these BICs. The findings extend beyond flexural waves, being applicable to various types of classical waves. To validate our theoretical framework, we conduct an experimental study in acoustics, showcasing the realization of an acoustic BIC open resonator. This resonator is constructed by coupling polygonally arranged holes within a two-dimensional waveguide. Our proposed design enables the scanning of the acoustic field, facilitating the retrieval of mode shapes and the estimation of quality factors through the analysis of signal decay rates.

Structurally Invariant Higher-Order Ince-Gaussian Beams and Their Expansions into Hermite-Gaussian or Laguerre-Gaussian Beams

Paraxial beam modes, which propagate in space and focus without changing their transverse intensity pattern, are of great value for multiplexing transmitted data in optical communications, both in waveguides and in free space. The best-known paraxial modes are the Hermite-Gaussian and Laguerre-Gaussian beams. Here, we derive explicit analytical expressions for Ince-Gaussian (IG) beams for several first values of the indices p = 3, 4, 5, and 6. In total, we obtain expressions for the amplitudes of 24 IG beams. These formulae are written as superpositions of the Laguerre-Gaussian (LG) or Hermite-Gaussian (HG) beams, with the superposition coefficients explicitly depending on the ellipticity parameter. Due to simultaneous representation of the IG modes via the LG and HG modes, it is easy to obtain the IG modes in the limiting cases wherein the ellipticity parameter is zero or approaches infinity. The explicit dependence of the obtained expressions for the IG modes on the ellipticity parameter makes it possible to change the intensity pattern at the beam cross-section by continuously varying the parameter values. For the first time, the intensity distributions of the IG beams are obtained for negative values of the ellipticity parameter. The obtained expressions could facilitate a theoretical analysis of properties of the IG modes and could find practical applications in the numerical simulation or generation of such beams with a liquid-crystal spatial light modulator.

Insight on the surface fluctuating pressures considering distorted turbulence around a rectangular bluff body

The distortion of turbulence approaching a bluff body plays a dominant role in determining the unsteadiness of the fluctuating surface pressures. A three-dimensional (3D) spectral approach is proposed to decouple the effects of the distorted turbulence into the blocking effect, distortion effect, and 3D effect. Wind tunnel tests are conducted to investigate the unsteady behavior of the windward fluctuating surface pressures on a rectangular bluff body, considering the effect of the ratio of the turbulence scale to the structural feature size λ = Lu/D. The results show that the fluctuating pressure on the surface of a rectangular cylinder is mainly affected by the low-frequency blocking effect, the high-frequency turbulent distortion effect, and the full-frequency turbulence 3D effect. The low-frequency blocking effect is related to the surface spatial position, but is less affected by λ; the high-frequency distortion effect does not depend on the spatial position of the pressure but is affected by λ; the 3D effect is affected by both the spatial position of the pressure and λ. When λ approaches infinity, the distortion and 3D effects can be ignored, and the quasi-steady theory is valid for the fluctuating pressure. Finally, one-wavenumber and 2D fluctuating pressure generalized spectrum models considering the unsteady effects and the spatial position of the surface pressure are proposed, and their accuracy is effectively verified by the test results.

Thermodynamic analysis of anomalous region, critical point, and transition from subcritical to supercritical states: Application to van der Waals and five real fluids

All fluids exhibit large property-variations near the critical point in a region identified as the anomalous state. The anomaly starts in the liquid and extends well into the supercritical state, which can be identified thermodynamically using the Gibbs free energy (g). The specific heat, isobaric expansion, and isothermal compressibility parameters governing the transitions are: (cp/T), (vβ), and (vκ), rather cp, β, and κ. They are essentially the second-order derivatives of g and have two extrema (minimum, maximum); only maxima reported ever. When applied to the van der Waals fluid, these extrema exhibit closed loops on the phase-diagram to satisfy d3g = 0 and map the anomalous region. The predicted liquid-like to gas-like transitions are related to the ridges reported earlier, and the Widom delta falls between these loops. Evidently, in the anomalous region, both the liquid and the supercritical fluid need to be treated differently. Beyond the anomalous states, the supercritical fluids show monotonic, gradual changes in their properties. The analysis for argon, methane, nitrogen, carbon dioxide, and water validates the thermodynamic model, supports the stated observations, and identifies their delimiting pressures and temperatures for the anomalous states. It also demonstrates the applicability of the law of corresponding states. Notably, the critical point is a state where d3g = 0, the anomaly in the fluid's properties/behavior is maximal, and the governing parameters approach infinity. Also the following are presented: (a) the trajectory of the liquid–vapor line toward the melt-solid boundary and (b) a modified phase diagram (for water) exhibiting the anomalous region.

Handshake mechanism selection in MANETs with directional antennas: A new evaluation view

AbstractHandshake mechanism plays a significant role in neighbour discovery (ND) algorithms in mobile ad hoc networks (MANETs) with directional antennas. In this letter, aiming at 1) the MANET scenario of uniformly‐distributed nodes and 2) the random beam‐scanning manner, the upper and lower bounds of ND time of the network are theoretically derived for both 1‐way and 2‐way. This analysis indicates that the ND time is approximately linear growth with NDT‐Mutual, where the NDT‐Mutual is defined as the expected time for a node to complete mutual discovery with its neighbours. Then the theoretical expression for NDT‐mutual is derived, which is a function of the number of nodes. The asymptotic analysis reveals that when the number of nodes is very small, the NDT‐mutual of 2‐way is lower than 1‐way, while 1‐way performs better when the number of nodes approaches infinity. The number of nodes when 1‐way and 2‐way perform as well is further presented, which can be utilized to guide the selection of handshake mechanism in MANETs with directional antennas. Simulation results demonstrate the accuracy of our modelling for ND and the effectiveness of using NDT‐mutual to approximate who performs better with 1‐way and 2‐way.

A general analytical solution for fluid flow and heat convection through arbitrary-shaped triangular ducts: A variational analysis

This paper presents an analytical solution for fluid flow and heat transfer inside arbitrarily-shaped triangular ducts for the first time. The former analytical solutions are limited to the special case of isosceles triangular ducts. The literature has no report about the analytical solution for the general case of arbitrarily-shaped triangular ducts. Due to the significant role of fluid flow through non-circular channels in industry and the large number of triangular shapes, a method for solving the heat transfer problem for all triangular shapes is needed. The heat transfer of a fluid flow through a channel with an arbitrary triangular cross-section for the case of constant heat flux at the walls is solved in this work for the first time, considering viscous dissipation. Here, the functionals of flow and heat transfer equations are derived, and the resulting Euler–Lagrange equations are solved using the Ritz method. The effect of the duct geometry on the velocity profile and friction coefficient is studied in detail. The effect of the Brinkman number on the temperature distribution and Nusselt number is investigated for both cooling and heating cases. The results reveal that the critical Brinkman Number distinguishes between the cooling and heating cases and represents the critical point at which the Nusselt number approaches infinity. The value of the Nusselt number decreases with the increase of the Brinkman number in both the wall cooling and heating modes. It is also found that the equilateral triangle exhibits the minimum friction coefficient and the maximum value of the Poiseuille number.

Optimized bipartite formation control for multiagent systems with obstacle and collision avoidance

This paper considers an optimal bipartite formation control problem for high-order nonlinear multiagent systems (MASs) with the consideration of obstacle/collision avoidance. Contrary to existing literature, the control strategy designed in this paper can render the agents in a bipartite formation to complete the obstacle/collision avoidance tasks without requiring the obstacles to be located on the coordinate axis. To save resources, an identifier-critic-actor-based optimized formation control scheme is designed for MASs. An obstacle/collision avoidance approach based on dynamical systems (DS) is designed for high-order nonlinear MASs. Different from the existing related results, the proposed DS-based obstacle/collision avoidance approach can avoid the phenomenon that the control quantities approach infinity near obstacles or other agents, and eliminate the dependence on global information. Finally, a simulation example is given to validate the proposed control method.

Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition

We generalize the one-dimensional population model of Anguige & Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller & Šverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. TE check: Please check the reference citation in abstract.

Frequency-Domain Identification of Discrete-Time Systems using Sum-of-Rational Optimization

This paper proposes a new computationally tractable method to fit coefficients of a fixed-order discrete-time transfer function to a measured frequency response, with stability guaranteed. The problem is formulated as a non-convex global sum-of-rational optimization problem whose objective function is the sum of weighted squared residuals at each observed frequency datapoint. Stability is enforced using a polynomial matrix inequality constraint. The problem is solved by a moment-sum-of-squares hierarchy of semidefinite programs through a framework for sum-of-rational-functions optimization. Convergence of the moment-sum-of-squares program is guaranteed as the bound on the degree of the sum-of-squares polynomials approaches infinity. The performance of the proposed method is demonstrated using numerical simulation examples.

Pressure-Tuned Classical–Quantum Crossover in Magnetic Field-Induced Quantum Phase Transitions of a Triangular-Lattice Antiferromagnet

he correspondence principle states that as quantum numbers approach infinity, the nature of a system described by quantum mechanics should match that described by classical mechanics. Quantum phenomena, such as quantum superposition and quantum correlation, generally become unobservable when a system approaches this regime. Conversely, as quantum numbers decrease, classical descriptions give way to observable quantum effects. The external approach to classical–quantum crossover has attracted research interest. This study aims to demonstrate a method for achieving such control in materials.

Quantum Fisher information in moving reference frame

In the field of quantum metrology, an important application is quantum parameter estimation. As the fundamental theory of quantum parameter estimation, quantum Cramér-Rao inequality shows that the variance of parameter estimation is determined by the inverse of quantum Fisher information. Higher quantum Fisher information corresponds to a lower variance, thereby improving the precision of parameter estimation. Quantum Fisher information has been extensively investigated in many aspects of non-relativistic quantum mechanics, including entanglement structure detection, quantum teleportation, quantum phase transition, quantum chaos, and quantum computation. However, there are few researches considering the influence of relativistic effect on quantum Fisher information, and therefore, we attempt to investigate this topic in this work. The relativistic transformation of particle states is employed, and the quantum Fisher information about amplitude parameter <inline-formula><tex-math id="M3">\begin{document}$ \theta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M3.png"/></alternatives></inline-formula> and phase parameter <inline-formula><tex-math id="M4">\begin{document}$\varphi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M4.png"/></alternatives></inline-formula> are investigated in moving reference frame. In this work, the parameters to be estimated are encoded into the spin degree of freedom, and the pure single-qubit state and the pure two-qubit state are both considered. The quantum Fisher information about <inline-formula><tex-math id="M5">\begin{document}$ \theta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M5.png"/></alternatives></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$\varphi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M6.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M6.png"/></alternatives></inline-formula> of single-qubit state and two-qubit state in moving reference frame are numerically calculated, respectively. It can be observed that the quantum Fisher information is associated with rapidity, amplitude parameter, and the ratio of the width to the particle mass <inline-formula><tex-math id="M7">\begin{document}${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M7.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M7.png"/></alternatives></inline-formula>. The quantum Fisher information of the estimated parameters decreases with rapidity increasing for both single-qubit state and two-qubit state. As rapidity approaches infinity, i.e. increases to the speed of light, the quantum Fisher information reaches to a constant which decreases as the ratio <inline-formula><tex-math id="M8">\begin{document}${{{\sigma _r}} \mathord{\left/ {\vphantom {{{\sigma _r}} m}} \right. } m}$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M8.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M8.png"/></alternatives></inline-formula> increases. More importantly, for the phase parameter <inline-formula><tex-math id="M9">\begin{document}$ \varphi $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M9.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M9.png"/></alternatives></inline-formula>, it is observed that the quantum Fisher information of two-qubit state reduces more significantly than that of single-qubit state. While, for the amplitude parameter <inline-formula><tex-math id="M10">\begin{document}$\theta $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M10.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="5-20231394_M10.png"/></alternatives></inline-formula>, the quantum Fisher information of two-qubit state is greater than that of single-qubit state. These results are useful and valuable for improving the precision of parameter estimation under the influence of relativistic effect.

Sensitivity enhancement at higher order exceptional point based on coupled whispering gallery modes in microstructured optical fibers

Sensitivity enhancement at higher order exceptional point (EP) was demonstrated based on coupled whispering gallery modes in microstructured optical fibers. Nonlinear wavelength shift response near the third-order EP (EP(3)) could be obtained at four typical refractive indices (RIs) by exploiting the adjustable RI feature of functional materials. The maximum sensitivity near EP reaches about 1.5 × 104 nm/RIU, and the sensitivity at EP approaches infinity. Besides, the sensing characteristics of a second-order EP system and a diabolic point system are compared with the EP(3) system for the same RI measurement range around 1.45 and the calculation results indicate that EP(3) system experiences an increase in RI sensitivity around EP by 6 and 10 times, respectively. In addition, tuning the real part of the medium RI of gain & loss microcavities helps to achieve tuning of the sample material RI with high sensitivity. Finally, another orthogonal state of the eigenvectors has been found by analyzing the eigenvector evolution from Hermitian regime to EP state for the EP(3) system and a lower limit of detection could be obtained as well. These finds would be of great significance for noise suppression in sensing applications based on interrogation of eigenfrequency splitting near higher order EPs.

Asymptotic approximations of expectations of power means

In this paper we study how the expectations of power means behave asymptotically as some relevant parameter approaches infinity and how to approximate them accurately for general nonnegative continuous probability distributions. We derive approximation formulae for such expectations as distribution mean increases, and apply them to some commonly used distributions in statistics and financial mathematics. By numerical computations we demonstrate the accuracy of the proposed formulae which behave well even for smaller sample sizes. Furthermore, analysis of behavior depending on sample size contributes to interesting connections with the power mean of probability distribution.

Core of the Non-Universe in Jaina Cosmology as a Cube of Eight Space-Points

According to Jaina cosmology, the non-universe is a hollow sphere beyond and exists all round the universe, whose middle region is the horizontal universe resting on a circular disc. The Bhagavatī Sūtra and the Sthānāṅga Sūtra each mention that there is the core of the horizontal universe where a cube of eight space-points is and from those eight space-points ten directions originate. This paper presents a conjectural explanation for understanding that cube. The core of the non-universe is the core of a core-sphere when the core-sphere is formed using n core-circles such that they are stacked up and down in the same order from the nth core-circle to the first core-circle. The upper four points of the cube of eight points form the core of the horizontal universe and the cube of eight points is the core of the non-universe. The nth core-circle becomes the horizontal universe when n is innumerable and the nth core-sphere forms the non-universe when n approaches infinity. This paper also shows that the intermediate and cardinal directions are linear and planar respectively while the zenith and nadir ones are three-dimensional.

Simple versus nonsimple loops on random regular graphs

AbstractIn this note, we solve the “birthday problem” for loops on random regular graphs. Namely, for fixed , we prove that on a random ‐regular graph with vertices, as approaches infinity, with high probability: (i) almost all primitive nonbacktracking loops of length are simple, that is, do not self‐intersect, and (ii) almost all primitive nonbacktracking loops of length self‐intersect.

Asymptotic properties of hierarchical clustering in high-dimensional settings

In this study, three asymptotic behaviors of hierarchical clustering are defined and studied with strict conditions under several asymptotic settings, from large samples to high dimensionality, when having two independent populations. We proceed with the current comprehension of the asymptotic properties of hierarchical clustering in high-dimensional, low-sample-size (HDLSS) settings. For high-dimensional data, the asymptotic properties of hierarchical clustering are demonstrated under mild and practical settings, and we present simulation studies and hierarchical clustering performance discussions. Furthermore, hierarchical clustering was theoretically investigated when both the dimension and sample size approach infinity, and we generalized a latent number of populations considering hierarchical clustering in multiclass HDLSS settings.

Numerical instability and its treatment for steady state heat conduction problems in exchanger tubes using the dual boundary element method

In this paper, we employed the boundary element method (BEM) to determine the heat conduction shape factor (CSF) for the steady-state heat conduction problem in the heat exchanger tube. Various doubly-connected geometric shapes were considered. Numerical results of the conduction shape factor matched well with the theoretical data. Numerical instability was found for a special size. Degenerate scale occurred for all cases of regular N-gons in determining the conduction shape factor and matched the data as analytically predicted. Numerical results showed that a degenerate scale is only dependent on the outer boundary instead of the inner boundary. Three regularization techniques, the dual BEM, the CLEEF(UT), and the CLEEF(LM), were provided to avoid numerical instability. It was found that the larger number of sides for the N-gon shape is given, the lower the CSF is obtained. As N approaches infinity, it converges to the minimum value of the solution for an annulus.

Evolutionary rescue on genotypic fitness landscapes.

Populations facing adverse environments, novel pathogens or invasive competitors may be destined to extinction if they are unable to adapt rapidly. Quantitative predictions of the probability of survival through adaptation, evolutionary rescue, have been previously developed for one of the most natural and well-studied mappings from an organism's traits to its fitness, Fisher's geometric model (FGM). While FGM assumes that all possible trait values are accessible via mutation, in many applications only a finite set of rescue mutations will be available, such as mutations conferring resistance to a parasite, predator or toxin. We predict the probability of evolutionary rescue, via de novo mutation, when this underlying genetic structure is included. We find that rescue probability is always reduced when its genetic basis is taken into account. Unlike other known features of the genotypic FGM, however, the probability of rescue increases monotonically with the number of available mutations and approaches the behaviour of the classical FGM as the number of available mutations approaches infinity.

Performance metric and analytical gain optimality for set-based robust fault detection

This paper proposes a new performance metric for set-based robust fault detection on linear time-invariant systems, which quantitatively characterizes all faults that are unguaranteed to be detected by a specific set-based fault detection method at next time instant. Based on this metric, a novel online optimal design method is proposed for the set-theoretic unknown input observer, where the optimal parameters of the observer are obtained by analytically solving an optimization problem with bilinear matrix inequalities. Under a mild assumption, it is demonstrated that the observer is internally stable under the proposed optimal design. Moreover, the proposed online optimal design method is directly related to an algebra Riccati equation as time approaches infinity, and hence an offline optimal design method is further proposed to reduce the computational complexity. Under the proposed optimal designs, it is demonstrated that the observer can detect faults more timely and with much less computational complexity. At the end of this paper, an electric-circuit example and a high-dimensional numerical example are used to illustrate the effectiveness of the proposed methods.

Predefined-Time Adaptive Fault-Tolerant Control for Switched Odd-Rational-Power Multi-Agent Systems

This paper studies the distributed adaptive neural network (NN) predefined-time (PT) fault-tolerant consensus tracking control for unknown non-strict feedback nonlinear multi-agent systems (MASs) with time-varying full state constraints (TVFSCs), arbitrary switching behavior functions (possibly asynchronous), actuator faults and high powers. Compared with existing results, the convergence time estimation bound in this paper is less conservative due to the upper bound of convergence time presents as a tuning parameter. This paper avoids the control gain of each virtual controller to appear in the next virtual controller by using separable function technique, so it can reduce the control action complexity. Besides, this paper does not use any filters or piecewise continuous function in the controller design process, which can effectively avoid the potential singularity problem. An adaptive NN fault-tolerant consensus tracking controller is designed via backstepping technique and tangent barrier Lyapunov functions (BLF-Tans), which can ensure that all the closed-loop signals are bounded within a PT and the TVFSC bounds are not violated. Finally, numerical simulation and practical example further verify the effectiveness of the presented control algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —Many practical (underactuated, weakly coupled and unstable mechanical systems) control systems can be modeled as uncertain high-order MAS with high powers. Due to the influence of environment and other factors, the values of some variables in the practical systems are limited and cannot be arbitrarily selected. In addition, the actual systems may fail due to human factors and in most of the relevant literatures, it is often assumed that the powers are equal to 1 and only achieve the controlled systems asymptotic stability as time approaches infinity. Therefore, in this paper, an adaptive PT fault-tolerant control scheme is developed for MASs with high powers, time-varying full state constraints and actuator faults. The effectiveness of the control scheme is illustrated based on simulation study.