Approaches Infinity Research Articles

Singularity formation for the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime

AbstractWe study the formation of singularities of smooth solutions to the relativistic Euler equations of Chaplygin gases in Schwarzschild spacetime. The system is in the spherically symmetric form, and its coefficients and nonhomogeneous terms contain a parameter reflecting the mass of the black hole, which makes it highly nonlinear and complicated. To overcome the influence of the mass parameter of black hole, we introduce a pair of suitable auxiliary variables related to it and derive their characteristic decompositions to establish the estimates of the smooth solution. We show that, for a kind of initial data, the smooth solution develops singularity in finite time and the mass‐energy density itself approaches infinity at the blowup point.

Deflection of light by wormholes and its shadow due to dark matter within modified symmetric teleparallel gravity formalism

Abstract We explore the possibility of traversable wormhole formation in the dark matter halos in the context of $f(Q)$ gravity. We obtain the exact wormhole solutions with anisotropic matter source based on the Bose-Einstein condensate, Navarro-Frenk-White, and pseudo-isothermal matter density profiles. Notably, we present a novel wormhole solution supported by these dark matters using the expressions for the density profile and rotational velocity along with the modified field equations to calculate the redshift and shape functions of the wormholes. With a particular set of parameters, we demonstrate that our proposed wormhole solutions fulfill the flare-out condition against an asymptotic background. Additionally, we examine the energy conditions, focusing on the null energy conditions at the wormhole's throat, providing a graphical representation of the feasible and negative regions. Our study also examines the wormhole's shadow in the presence of various dark matter models, revealing that higher central densities result in a shadow closer to the throat, whereas lower values have the opposite effect. Moreover, we explore the deflection of light when it encounters these wormholes, particularly noting that light deflection approaches infinity at the throat, where the gravitational field is extremely strong.

Derivation of weakly interacting lumps for the (2+1)-dimensional Yu–Toda–Sasa–Fukuyama equation via degeneracy of lump chains

This paper introduces two distinct pathways for degenerating normally interacting lump chains into weakly interacting lump waves for Yu–Toda–Sasa–Fukuyama equation, which will enrich the correlation between lump chains and weakly interacting lump waves. The first pathway involves letting the periods of M similarly-velocity lump chains approach infinity directly. The second pathway first transforms M normally interacting lump chains into weakly interacting lump chains with similar dynamic behaviors. Then, by allowing their periods to approach infinity, MM+12 weakly interacting lump waves are produced. Distance between weakly interacting lump chains in this equation is proportional to ln|t|, while between weakly interacting lump waves is proportional to |t|3. These findings will contribute valuable theoretical insights to the study of wave theory, ocean science and related disciplines.

Piezo-viscous micropolar fluid flow between two parallel annular disks

A theoretical analysis of the piezo-viscous dependence of micropolar fluid flow between parallel annular disks has been conducted. It is assumed that viscosity changes exponentially with pressure. Pressure is analytically approximated using a small perturbation method with the viscosity coefficient as the perturbation parameter. Squeeze film time, pressure, and load carrying capacity are calculated and graphically demonstrated across various viscosity–pressure characteristics. The study examines how an increase in cross-viscosity correlates with notable growth in pressure, response time, and bearing load. According to the results, micropolar fluid offers an increase in the load-bearing capacity and therefore lengthens the response time to prevent the contact of parallel annular disks. These observations are useful for bearings, seals, and lubrication systems, where controlling viscosity and pressure is crucial for smooth operation and productivity. As the limiting case of micro-polar fluid when the coupling number tends to zero or the couple stress parameter approaches infinity, without the annular ring, the present derivation agrees well with the Newtonian viscous case of parallel planar squeezing disks described by Bird et al. [Transport Phenomena (John Wiley & Sons, 2006)].

A robust iterative learning control for linear system with variable initial state and trail length

AbstractTo address the variable initial state and trail length this paper first presents a robust PD‐type open‐closed‐loop iterative learning control (ILC) law for a multiple‐input‐multiple‐output (MIMO) linear discrete‐time system. It is demonstrated that the convergence condition is dependent on the PD‐type feed‐forward learning gains, while an appropriate feedback learning gain can improve the ILC convergence performance. As a special case of PD‐type open‐closed‐loop ILC law, P‐type and D‐type open‐closed‐loop ILC laws are deduced. The three developed ILC laws ensure that as the number of iterations approaches infinity, the expectation of ILC tracking error will be constrained within a limited range, where the boundary is proportional to the initial state variation. Through a numerical simulation, the effectiveness of the proposed ILC laws is illustrated.

Synchronization of Bidirectionally Coupled Fractional-Order Chaotic Systems with Unknown Time-Varying Parameter Disturbance in Different Dimensions

In this article, the synchronization of bidirectionally coupled fractional-order chaotic systems with unknown time-varying parameter disturbance in different dimensions is investigated. The scale matrices are designed to address the problem of the synchronization for fractional-order chaotic systems across two different dimensions. Congelation of variables is used to deal with the unknown time-varying parameter disturbance. Based on Lyapunov’s stability theorem, the synchronization controllers in different dimensions are obtained. At the same time, adaptive laws of the unknown disturbance can be designed. Benefiting from the proposed methods, we verify all the synchronization errors can converge to zero as time approaches infinity, regardless of whether in n-D or m-D synchronization, simultaneously ensuring that both control and estimation signals are bounded. Finally, simulation studies based on fractional-order financial systems are carried out to validate the effectiveness of the proposed synchronization method.

Evaluation of the Komlos Conjecture Using Multi-Objective Optimization

The Komlos conjecture, which explores the existence of a constant upper bound in the realm of n-dimensional vectors, specifically addresses the function K(n). This function, intricately defined as encapsulates the maximal discrepancy within a set of n -dimensional vectors. This paper endeavors to unravel the mysteries of K(n), by meticulously evaluating its behavior for lower dimensions . Our findings revealed through systematic exploration, showcase intriguing values such as , , , and , shedding light on the intricate relationships within n-dimensional spaces. Venturing into higher dimensions, we introduce the function as a potentially robust lower bound for K(n). This innovative approach aims to provide a deeper understanding of the limiting behavior of K(n) as the dimensionality expands. As a culmination of our comprehensive analysis, we arrive at a significant revelation the Komlos conjecture stands refuted. This conclusion stems from the suspected divergence of K(n), as n approaches infinity, as evidenced by . This seminal result challenges established notions and added a valuable dimension to the ongoing discourse in optimization and discrepancy theory.

From the Janis–Newman–Winicour Naked Singularities to the Einstein–Maxwell Phantom Wormholes

The Janis–Newman–Winicour spacetime corresponds to a static spherically symmetric solution of Einstein equations with the energy momentum tensor of a massless quintessence field. It is understood that the spacetime describes a naked singularity. The solution has two parameters, b and s. To our knowledge, the exact physical meaning of the two parameters is still unclear. In this paper, starting from the Janis–Newman–Winicour naked singularity solution, we first obtain a wormhole solution by a complex transformation. Then, letting the parameter s approach infinity, we obtain the well-known exponential wormhole solution. After that, we embed both the Janis–Newman–Winicour naked singularity and its wormhole counterpart in the background of a de Sitter or anti-de Sitter universe with the energy momentum tensor of massive quintessence and massive phantom fields, respectively. To our surprise, the resulting quintessence potential is actually the dilaton potential found by one of us. It indicates that, by modulating the parameters in the charged dilaton black hole solutions, we can obtain the Janis–Newman–Winicour solution. Furthermore, a charged wormhole solution is obtained by performing a complex transformation on the charged dilaton black hole solutions in the background of a de Sitter or anti-de Sitter universe. We eventually find that s is actually related to the coupling constant of the dilaton field to the Maxwell field and b is related to a negative mass for the dilaton black holes. A negative black hole mass is physically forbidden. Therefore, we conclude that the Janis–Newman–Winicour naked singularity solution is not physically allowed.

Limiting out-of-sample performance of optimal unconstrained portfolios

This paper studies the out-of-sample Sharpe ratio of an unconstrained portfolio that combines the global minimum-variance with a hedge portfolio. Furthermore, we investigate how this ratio behaves as the number of risky assets and observations approaches infinity while maintaining a constant ratio. Under these conditions, it becomes possible to simultaneously account for estimation risk and achieve analytical tractability when optimizing the out-of-sample Sharpe ratio. This analysis also provides valuable insights to enhance out-of-sample performance in the finite case by introducing additional deterministic factors to the portfolio components.

The Wasserstein Metric between a Discrete Probability Measure and a Continuous One

This paper examines the Wasserstein metric between the empirical probability measure of n discrete random variables and a continuous uniform measure in the d-dimensional ball, providing an asymptotic estimation of their expectations as n approaches infinity. Furthermore, we investigate this problem within a mixed process framework, where n discrete random variables are generated by the Poisson process.

Final nanoparticle size distribution under unusual parameter regimes.

We explore the large-scale behavior of a stochastic model for nanoparticle growth in an unusual parameter regime. This model encompasses two types of reactions: nucleation, where n monomers aggregate to form a nanoparticle, and growth, where a nanoparticle increases its size by consuming a monomer. Reverse reactions are disregarded. We delve into a previously unexplored parameter regime. Specifically, we consider a scenario where the growth rate of the first newly formed particle is of the same order of magnitude as the nucleation rate, in contrast to the classical scenario where, in the initial stage, nucleation dominates over growth. In this regime, we investigate the final size distribution as the initial number of monomers tends to infinity through extensive simulation studies utilizing state-of-the-art stochastic simulation methods with an efficient implementation and supported by high-performance computing infrastructure. We observe the emergence of a deterministic limit for the particle's final size density. To scale up the initial number of monomers to approximate the magnitudes encountered in real experiments, we introduce a novel approximation process aimed at faster simulation. Remarkably, this approximating process yields a final size distribution that becomes very close to that of the original process when the available monomers approach infinity. Simulations of the approximating process further support the conjecture of the emergence of a deterministic limit.

Dynamical behavior of fluid–structure interaction in ducts with rigid and flexible interfaces: Modeling and analysis

This study investigates the propagation of fluid–structure coupled waves in a duct featuring both rigid and flexible interfaces. Two cases are considered: in the first case, with a rigid interface, the truncated form of solution preserves the conservation law, while in the second setting, the power balance is not preserved initially, but is rectified as the number of modes, denoted as N, approaches infinity. The discrepancy is explained by specific equations and the presence of delta functions to enforce edge conditions on the membrane at the interface. The Galerkin approach is adopted to address this issue, expressing the membrane’s displacement using a generalized Fourier series. The investigation establishes a framework for modeling higher-order boundary conditions within a duct, employing the mode-matching (MM) method. The study also delves into the scattering components of power, which vary significantly for the two cases. These findings have implications for noise control devices, particularly in active noise control measures utilizing flexible components to dampen both noise and structural vibrations.

Robust well-balanced method with flow resistance terms for accurate wetting and drying modeling in shallow water simulations

Hydrodynamic simulations in shallow water environments require careful consideration of the Wetting and Drying (WD) processes, which poses challenges to accurately modeling moving boundaries. This study introduces a novel method called the flow resistance method (FRM), which builds upon the foundation of the Negative-Depth Method (NDM) to tackle the intricacies of the moving boundary problem. Inspired by the Navier-Stokes/Brinkman (NSB) model from porous media theory, FRM incorporates a continuous function related to additional flow resistance that is proportional to the flow velocity. This approach facilitates a seamless transition between the exposed bed and fluid area wherein the additional flow resistance becomes 0 within the fluid area and approaches infinity in the exposed bed. Consequently, FRM adeptly and implicitly manages the moving boundary problem, causing a rapid decay of flow velocity to 0 in the exposed bed. In order to test the performance of FRM, four typical numerical experiments were conducted, along with an examination of a real-life case. Accuracy, robustness, and computational efficiency were assessed as key performance indicators. The simulations demonstrate that FRM adeptly tracks the moving water front, yielding precise results. Furthermore, when compared to established methods such as the Element Removal Method (ERM) and NDM, FRM exhibits broader applicability and achieves significant enhancements in the key performance indicators. These findings underscore the promising potential and broad applications of FRM in the field.

Quadratic base change and resonance sums for holomorphic cusp forms on Γ0(N)

Let D,k be integers with D square free and k even. Let N be a positive integer so that (N,D)=1 when D has residue one modulo four and (N,4D)=1 when D has residue two or three modulo four. In this paper the asymptotic behavior of a resonance sum SX(α,β;π) attached to the quadratic base change lift of a holomorphic cusp form f of level N and weight k over the quadratic extension generated by D is computed. First a Voronoi summation formula is derived that expresses SX(α,β;π) in terms of the Meier-G function. Then, using the known asymptotics of the Meier-G function the asymptotic behavior of SX(α,β;π) as X approaches infinity is determined. It is then shown that using only finitely many Fourier coefficients of the form, one can recover the weight k and the level N, which is a special case of the multiplicity one theorem.

Stochastic viability in an island model with partial dispersal: Approximation by a diffusion process in the limit of a large number of islands

In this paper, we investigate a finite population undergoing evolution through an island model with partial dispersal and without mutation, where generations are discrete and non-overlapping. The population is structured into D demes, each containing N individuals of two possible types, A and B, whose viability coefficients, sA and sB, respectively, vary randomly from one generation to the next. We assume that the means, variances and covariance of the viability coefficients are inversely proportional to the number of demes D, while higher-order moments are negligible in comparison to 1/D. We use a discrete-time Markov chain with two timescales to model the evolutionary process, and we demonstrate that as the number of demes D approaches infinity, the accelerated Markov chain converges to a diffusion process for any deme size N≥2. This diffusion process allows us to evaluate the fixation probability of type A following its introduction as a single mutant in a population that was fixed for type B. We explore the impact of increasing the variability in the viability coefficients on this fixation probability. At least when N is large enough, it is shown that increasing this variability for type B or decreasing it for type A leads to an increase in the fixation probability of a single A. The effect of the population-scaled variances, σA2 and σB2, can even cancel the effects of the population-scaled means, μA and μB. We also show that the fixation probability of a single A increases as the deme-scaled migration rate increases. Moreover, this probability is higher for type A than for type B if the population-scaled geometric mean viability coefficient is higher for type A than for type B, which means that μA−σA2/2>μB−σB2/2.

Evaluating the motion of a charged solid body having a globular cavity

The dynamical motion of a charged symmetrical solid body (SB) about its main symmetric axis is addressed in this work. The body is supposed to have a chamber with a globular shape, in which it contains a mobile mass and filled with a viscous liquid. This mass is assumed to be coupled via a double elastic band at a point that is situated along the dynamical symmetry axis and produces viscous friction. The body’s motion is influenced by a gyrostatic moment (GM), whose first two components on the body’s main axes are chosen to be zero. Additionally, the body is subject to the effects of an electromagnetic field, owing to a located point charge on the body’s dynamic symmetry axis. On the indicated motion, the combined impact of the viscous liquid and the mobile mass is investigated. The achieved outcomes show that, in the presence of internal dissipation, the system's motion tends towards a constant rotation around its axis at the point of highest inertia as time approaches infinity. To demonstrate how the body’s parameters affect the motion, the obtained outcomes are graphed. The method of fourth-order Runge-Kutta (4RKM) is employed to generate the governing system's numerical results, which are then displayed in additional diagrams. This work's relevance is concentrated on its considerable potential for usage in the submarine and gyroscope industries. The novelty of this work lies in the intricate interplay of rotational dynamics due to the GM, viscous liquid and electromagnetic effects to achieve new results, besides the gained numerical outcomes. Therefore, these results are considered novel and have significant theoretical and practical relevance. Moreover, it can be used to study the motion of celestial bodies, such as planets or moons, which might have internal cavities filled with fluids or other materials. The charged nature could be relevant for bodies with significant magnetic fields.

Maximum weighted correntropy filters for nonlinear non‐Gaussian systems

AbstractIn this paper, we focus on the filtering issue for nonlinear non‐Gaussian systems. Considering the limitations of the Gaussian function with a single kernel bandwidth, we design a novel extended version called weighted Gaussian function, which consists of a weighting coefficient and two distinct kernel bandwidths. The additional coefficient can directly balance two kernel bandwidths, which improves the flexibility and performance of the correntropy. We develop a cost function utilizing the suggested weighted Gaussian function and statistical linearization method, followed by deriving a maximum weighted correntropy filter based on the maximum correntropy criterion. The proposed algorithm is demonstrated to converge to a Gaussian filter as the kernel bandwidths approach infinity. In addition, we derive the corresponding information filter, which takes the form of information matrix and information vector. It is computationally efficient and easier to generalize to multisensor systems. The performance of the proposed algorithms is compared with other filters based on the third‐order spherical cubature rule and the fixed point iteration technique in a target tracking system. Simulation results confirm the effectiveness of the new approaches.

Proposed theoretical value for TSP constant

The traveling salesman problem (TSP) involves determining the shortest length (optimal tour) for a complete circuit through an arbitrary number of points. In the special case that the points to be visited are randomly distributed in the plane within a unit square, and travel costs correspond to the Euclidean distance between points, it is known that the length of the optimal tour divided by the square root of the number of points ([Formula: see text]) asymptotically approaches a constant ([Formula: see text] as [Formula: see text] becomes large. If individual points are randomly drawn from a uniform distribution, the optimal lengths for a sufficiently large number of TSP instances of the same size ([Formula: see text] comprise a normal distribution whose mean approaches [Formula: see text] as [Formula: see text] increases. Moreover, as [Formula: see text] approaches infinity, this distribution gradually narrows such that its standard deviation asymptotically approaches zero. Although the precise value of [Formula: see text] is unknown, published studies indicate its magnitude is approximately 0.712. In this paper, possibly for the first time, precise formulae are proposed for the parameters of the underlying normal distribution and [Formula: see text], based on intuition, mathematical reasoning, and empirical fits to both published and experimentally derived data.

Quantum nonlocality phenomena in scalar and Dirac fields near a Schwarzschild black hole

In this work, we explore the impact of the Hawking effect on quantum nonlocality, specifically the maximum violation of the Bell-CHSH inequality, for both scalar and Dirac fields within Schwarzschild spacetime. We reveal the nuanced behavior of the Bell-CHSH inequality as the Hawking temperature (HT) varies. For scalar fields, we observe distinct degradation patterns in the Bell-CHSH inequality, influenced by the state parameter α and its normalized counterpart 1−α2 . In contrast, Dirac fields exhibit two distinct degradation trajectories, each converging to different nonzero minimum values as HT approaches infinity T→∞ , however, this does not apply to the maximally entangled state. Remarkably, the maximum violation of Bell-CHSH inequality becomes zero for scalar fields at infinite HT, while it stabilizes at a finite nonzero value for Dirac fields. Additionally, the quantum nonlocality robustness of the Dirac fields is stronger than that of the scalar fields. Therefore, this study enhances our understanding of the dynamic characteristics of nonlocality within scalar and Dirac fields near a Schwarzschild black hole.

Phylogenetic inference of inter-population transmission rates for infectious diseases.

Estimating transmission rates is a challenging yet essential aspect of comprehending and controlling the spread of infectious diseases. Various methods exist for estimating transmission rates, each with distinct assumptions, data needs, and constraints. This study introduces a novel phylogenetic approach called transRate, which integrates genetic information with traditional epidemiological approaches to estimate inter-population transmission rates. The phylogenetic method is statistically consistent as the sample size (i.e. the number of pathogen genomes) approaches infinity under the multi-population susceptible-infected-recovered model. Simulation analyses indicate that transRate can accurately estimate the transmission rate with a sample size of 200 ~ 400 pathogen genomes. Using transRate, we analyzed 40,028 high-quality sequences of SARS-CoV-2 in human hosts during the early pandemic. Our analysis uncovered significant transmission between populations even before widespread travel restrictions were implemented. The development of transRate provides valuable insights for scientists and public health officials to enhance their understanding of the pandemic's progression and aiding in preparedness for future viral outbreaks. As public databases for genomic sequences continue to expand, transRate is increasingly vital for tracking and mitigating the spread of infectious diseases.