Infinite System Research Articles

Index Concepts for Linear Differential-Algebraic Equations in Infinite Dimensions

Different index concepts for regular linear differential-algebraic equations are defined and compared in the general Banach space setting. For regular finite dimensional linear differential-algebraic equations, all these indices exist and are equivalent. For infinite dimensional systems, the situation is more complex. It is proven that although some indices imply others, in general they are not equivalent. The situation is illustrated with a number of examples.

Size-dependency and lattice-discreetness effect on fracture toughness in 2D crystals under antiplanar loading

AbstractFracture toughness is the material property characterizing resistance to failure. Predicting its value from the solid structure at the atomistic scale remains elusive, even in the simplest situations of brittle fracture. We report here numerical simulations of crack propagation in two-dimensional fuse networks of different periodic geometries, which are electrical analogs of bidimensional brittle crystals under antiplanar loading. Fracture energy is determined from Griffith’s analysis of energy balance during crack propagation, and fracture toughness is determined from fits of the displacement fields with Williams’ asymptotic solutions. Significant size dependencies are evidenced in small lattices, with fracture energy and fracture toughness both converging algebraically with system size toward well-defined material-constant values in the limit of infinite system size. The convergence speed depends on the loading conditions and is faster when the symmetry of the considered lattice increases. The material constants at infinity obey Irwin’s relation and properly define the material resistance to failure. Their values are approached up to $$\sim 15\%$$ ∼ 15 % using the recent analytical method proposed in Nguyen and Bonamy (Phys Rev Lett 123:205503, 2019). Nevertheless, the deviation remains finite and does not vanish when the system size goes to infinity. We finally show that this deviation is a consequence of the lattice discreetness and decreases when the super-singular terms of Williams’ solutions (absent in a continuum medium but present here due to lattice discreetness) are taken into account.

First order Büchi automata and their application to verification of LTL specifications

Büchi automata have applications in formal verification, e.g., in deciding whether a system satisfies given properties. We provide a definition of Büchi automata based on first order logics for representing infinite state systems, and investigate rules for proving emptiness and non-emptiness of such automata. We then apply these rules to solve the problem of verifying correctness of concurrent transition systems, leading to a relatively complete approach for proving and disproving LTL (Linear Temporal Logic) specifications. This approach overcomes weaknesses of existing work based on well-founded sets in the sense that the relative completeness does not depend on additional specification for ensuring progress of non-stuttering transitions. On the practical aspect, we provide a set of examples with an experimental verification condition generation tool to demonstrate the potential applicability of the approach for the verification of concurrent systems.

System size effects on the free energy landscapes from molecular dynamics of phase-separating bilayers.

The "lipid raft" hypothesis proposes that cell membranes contain distinct domains of varying lipid compositions, where "rafts" of ordered lipids and cholesterol coexist with disordered lipid regions. Experimental and theoretical phase diagrams of model membranes have revealed multiple coexisting phases. Molecular dynamics (MD) simulations can also capture spontaneous phase separation of bilayers. However, these methods merely determine the signof the free energy change upon phase separation-whether or not it is favorable-but not the amplitude. Recently, we developed a workflow to compute the free energy of phase separation from MD simulations using the weighted ensemble method. However, while theoretical treatments generally focus on infinite systems and experimental measurements on mesoscopic to macroscopic systems, MD simulations are comparatively small. Therefore, if we are to put the results of these calculations into the appropriate context, we need to understand the effects the finite size of the simulation has on the computed free energy landscapes. In this study, we investigate this phenomenon by computing free energy profiles for a model phase-separating system as a function of system size, ranging from 324 to 10 110 lipids. The results suggest that, within the limits of statistical uncertainty, bulk-like behavior emerges once the systems contain roughly 4000 lipids.

A Dynamic Principal-Agent Problem with One-Sided Commitment

In this paper, we consider a principal-agent problem where the agent is allowed to quit by incurring a cost. When the current agent quits the job, the principal will hire a new one, possibly with a different type. We characterize the principal’s dynamic value function, which could be discontinuous at the boundary, as the (unique) minimal solution of an infinite dimensional system of Hamilton-Jacobi-Bellman equations, parameterized by the agent’s type. This dynamic problem is time consistent in a certain sense. Some interesting findings are worth mentioning. First, self-enforcing contracts are typically suboptimal. The principal would rather let the agent quit and hire a new one. Next, the standard contract for a committed agent may also be suboptimal because of the presence of different types of agents in our model. The principal may prefer no commitment from the agent; then, the principal can hire a cheaper one from the market at a later time by designing the contract to induce the current agent to quit. Moreover, because of the cost incurred to the agent, the principal will see only finitely many quittings. Funding: J. Zhang is supported in part by the NSF [Grants DMS-1908665 and DMS-2205972]. Z. Zhu is supported by The Hong Kong University of Science and Technology (Guangzhou) Start-Up Fund [Grant G0101000240] and the Guangzhou-HKUST(GZ) Joint Funding Program [Grant 2024A03J0630].

Transition of dimuonium through foil

This paper presents a study of the passage of dimuonium through the foil of ordinary matter. First, we provide an overview of how dimuonium is planned to be produced for such a type of experiment and how it is expected to interact with the ordinary atoms—predominantly electromagnetically via the screened Coulomb potential of the atomic nuclei. Then, we describe the transport equations that represent the evolution of dimuonium states during the passage and their solution methods. Finally, for three different foils (beryllium, aluminium, and lead), we present the results of this study. To estimate the impact of uncertainties in the potential of a target atom, we study 15 different approximations of the atomic potential and show that the corresponding atomic-potential-model-dependent error in the yields of the low-lying states of dimuonium is quite small within the framework of the applied Born approximation. The convergence of the results after truncation of the infinite system of transport equations to the finite number of quantum states of dimuonium is also studied, and good convergence for the yields of low-lying states is demonstrated. Published by the American Physical Society 2024

Towards enhanced metamaterials via shape memory alloy-embedded nonlinear resonators

Standard acoustic metamaterials create an attenuation around specific frequency bands. Finite structures, with distinct boundary conditions and a limited number of unit cells, present less attenuation and narrower bands than those predicted by idealized infinite systems. A critical engineering goal is enhancing performance to motivate practical applications. To that aim, this work incorporates shape memory alloys into a quasi-zero stiffness resonator design, allowing the tuning of each cell's internal resonance. Such a feature enables the system reconfiguration to achieve different purposes, from following a disturbance frequency fluctuation to realizing graded nonlinear metamaterials. While the former can work under an adaptive framework, the latter can result in time-invariant locally resonant metamaterials with wideband vibration attenuation. The proposed design combines a frame that axially compresses an SMA beam in the vicinity of its buckling. The critical buckling force changes by thermally activating the SMA, affecting the residual stiffness, hence the unit cell resonant frequency. The unit cell concept is developed analytically and via finite elements, then built and validated experimentally. The promising results indicate the viability of the proposed concept. Preliminary numerical simulations and experiments show the advantages of the proposed smart resonator over a linear time-invariant counterpart.

Some Results on Pursuit Games for an Infinite System of Ternary Differential Equations

This paper aims to study a one-pursuer, one evader pursuit differential game for a higher level of infinite system that is an infinite system of first order ternary differential equations, and prove completion of pursuit in the game. Both integral constraints and geometric constraints are subjected on the players' control functions, thus two separate cases of pursuit games are examined. In the game, the pursuer wants to take the state of the system into the origin of l2 space at some finite time interval, whereas evader avoids this from happening. For every case, we solve the control problem by establishing the admissible control function. In order to achieve the pursuer's objective, we then construct an admissible strategy for the pursuer and develop an equation for the guaranteed pursuit time of the game.

Tunable low-frequency broadband band gap in nonlinear locally resonant metamaterial inspired by sarcomere structure

Since traditional locally resonant metamaterials (LRMs) cannot achieve a broadband band gap in the low-frequency band, the suppression and isolation of low-frequency broadband vibration has always been a difficult problem in the field of vibration control. Therefore, in this paper, inspired by the structure of the sarcomere, a novel nonlinear LRM is proposed, aiming to open a wider band gap in the low-frequency band. Firstly, a statics model of the nonlinear local resonator (LR) is established to explore the mechanical nonlinear characteristics of the structure with different geometric parameters. And the inflection point fitting method is used to approximate the nonlinear statics model of nonlinear LR. Then a dynamics model was established, the nonlinear dispersion relationship of the infinite periodic system was obtained by Bloch's theorem and l-P perturbation method. And an analytical investigation of the theoretically calculated band gap is conducted through the band gap ratio and relative band gap width. In addition, according to the influence of different structural parameters on the real and imaginary parts of the wave vector, the regulation rules and mechanisms for nonlinear LRM to open low-frequency broadband band gaps are revealed. Finally, the Runge-Kutta method was used to numerically calculate the vibration transmissibility curve of the nonlinear LRM composed of a limited number of nonlinear LRs. And the correctness of the theoretically calculated band gap was verified. The research results provide theoretical support for the application of nonlinear LRMs in the suppression and isolation of low-frequency broadband vibrations.

Gelation in input-driven aggregation.

We investigate irreversible aggregation processes driven by a source of small mass clusters. In the spatially homogeneous situation, a well-mixed system consists of clusters of various masses whose concentrations evolve according to an infinite system of nonlinear ordinary differential equations. We focus on the cluster mass distribution in the long-time limit. An input-driven aggregation with rates proportional to the product of merging partners undergoes a percolation transition. We examine this process analytically and numerically. There are two theoretical schemes and two natural ways of numerical integration on the level of a truncated system with a finite number of equations. After the percolation transition, the behavior depends on the adopted approach: The giant component quickly engulfs the entire system (Flory approach), or a nontrivial stationary mass distribution emerges (Stockmayer approach). We also outline ageneralization to ternary aggregation.

An Enhanced Symmetric Sand Cat Swarm Optimization with Multiple Strategies for Adaptive Infinite Impulse Response System Identification

An infinite impulse response (IIR) system might comprise a multimodal error surface and accurately discovering the appropriate filter parameters for system modeling remains complicated. The swarm intelligence algorithms facilitate the IIR filter’s parameters by exploring parameter domains and exploiting acceptable filter sets. This paper presents an enhanced symmetric sand cat swarm optimization with multiple strategies (MSSCSO) to achieve adaptive IIR system identification. The principal objective is to recognize the most appropriate regulating coefficients and to minimize the mean square error (MSE) between an unidentified system’s input and the IIR filter’s output. The MSSCSO with symmetric cooperative swarms integrates the ranking-based mutation operator, elite opposition-based learning strategy, and simplex method to capture supplementary advantages, disrupt regional extreme solutions, and identify the finest potential solutions. The MSSCSO not only receives extensive exploration and exploitation to refrain from precocious convergence and foster computational efficiency; it also endures robustness and reliability to facilitate demographic variability and elevate estimation precision. The experimental results manifest that the practicality and feasibility of the MSSCSO are superior to those of other methods in terms of convergence speed, calculation precision, detection efficiency, regulating coefficients, and MSE fitness value.

An Optimized Power-Angle and Excitation Dual Loop Virtual Power System Stabilizer for Enhanced MMC-VSG Control and Low-Frequency Oscillation Suppression

Modular Multilevel Converter Virtual Synchronous Generator (MMC-VSG) technology is gaining widespread attention for its ability to enhance the inertia and frequency stability of the power grid integrated with converter-interfaced renewable energy sources. However, the excitation voltage regulation in the MMC-VSG can generate equivalent negative damping torque and cause low-frequency oscillation problems similar to those in synchronous machines. This article aims to improve the system’s damping torque and minimize low-frequency oscillations by introducing a Virtual Power System Stabilizer (VPSS) into the power control loop. Building on the study of dynamic interactions between various control links of the MMC, this research establishes a reduced-order model (ROM) and a Phillips–Heffron state equation for the MMC-VSG single machine infinite bus system, using a hybrid modeling approach and a zero-pole truncation method. It also analyzes the mechanism of low-frequency oscillations in the MMC-VSG system through the damping torque method. The analysis reveals that the negative damping torque produced during the excitation voltage regulation process causes changes in the virtual power angle, which in turn increases the risk of low-frequency oscillation in the MMC-VSG. To address this issue, the article proposes an optimized control method for the MMC-VSG dual power loop architecture (power-angle/excitation) VPSS. This strategy compensates for the inadequate damping torque of a single loop VPSS and effectively suppresses low-frequency oscillations in the system.

Parametric oscillations of the sessile drop

Waves are formed on the surface of a sessile drop driven through substrate vibrations oriented at a slanting angle from the normal. A mathematical model is derived, which leads to an infinite system of coupled Mathieu equations governing the wave dynamics that are solved using Floquet theory. The spatial structure of the waves is described by the mode number pair $[\ell,m]$ , where $\ell$ and $m$ are the polar and azimuthal mode numbers, respectively. Limiting cases corresponding to horizontal and vertical vibrations are discussed with predictions agreeing well with prior literature. We focus our results on three drop motions – (1) harmonic $[1,1]$ rocking mode, (2) harmonic $[2,0]$ pumping mode, and (3) subharmonic rocking $[1,1]$ mode – as they depend upon the slanting angle, static contact angle, and contact-line conditions, which we assume to be either pinned or freely moving with fixed contact angle. New theoretical predictions are tested through experiments over a range of parameters, showing good agreement.

A crack in a confocal elliptical inhomogeneity embedded in an infinite matrix subjected to uniform heat flux

We derive a series-form analytical solution to the thermoelastic problem of an insulated and traction-free crack in a confocal elliptical isotropic elastic inhomogeneity embedded in an infinite isotropic elastic matrix subjected to uniform remote heat flux. Using complex variable techniques such as conformal mapping, analytic continuation and Laurent series expansions, the original thermoelastic problem is reduced to an infinite system of linear algebraic equations, the solution of which will yield the mode I and mode II stress intensity factors at the crack tip. An exact closed-form solution is derived when the inhomogeneity and the matrix have identical shear moduli. Detailed numerical results are presented to demonstrate the series-form and closed-form solutions with an emphasis on the particular case of a vanishingly thin inhomogeneity.

Optimizing the geometry of tube-in-tube Thermosyphon heat transport device from instability Perspective: A computational study

Since the Fukushima disaster, there has been a significant surge in the adoption of infinite time passive decay heat removal systems. In this context, the thermosyphon heat transport device (THTD) has distinct advantages over natural circulation loops (NCLs). Previous studies proved that the THTD experiences flow instability characterized by recirculating loops under specific geometric and operational conditions. However, identifying the critical value for these parameters remains unresolved. Therefore, the present study deals with the geometric optimization of THTD through computational (CFD) simulations.Initially, the CFD model is validated with earlier experimental data, which resulted in a minimal deviation of −2 % to 5 %. The parameters for geometric optimization of THTD are height of the system (H), the center-line elevation difference between the source and the sink (Δz), heater length (Lh), cooler length (Lc), hot leg length (Lhl), and area ratio (AR) between hot leg and cold leg, under the heater power range of 100 W to 1000 W. The impact of these geometric and operational variations on flow instability is evaluated by observing the presence of recirculation loops, which are identified by analyzing velocity vectors and contours. The key findings of the present study are: flow instability in the form of a single recirculating loop sweeping radially along the cross-section of the annulus, thereby leading to radial flow maldistribution; a center-line elevation greater than 0.8 m eliminates the flow instabilities; interchanging heating and cooling lengths for a given center-line elevation doesn’t alter the criteria for instability; Heater power of 100 W leads to instability irrespective of center-line elevation; the optimal cooling length can be considered as 5 % to 25 % of the total height of the THTD, whereas the optimal hot leg length can be considered as 25 % to 45 % of the total height; and the HTF flow rate increases with a rise in the area ratio until AR=2, after which a flow anomaly occurs, rendering the system unstable. In addition to these aspects, this study also suggests optimal geometric features of THTD to reduce instability and attain perfect unidirectional flow. By integrating findings from current research and past studies, the design methodology for conventional THTD is comprehensively presented to the scientific community.

Semi-infinite Simple Exclusion Process: From Current Fluctuations to Target Survival.

The symmetric simple exclusion process (SEP), where diffusive particles cannot overtake each other, is a paradigmatic model of transport in the single-file geometry. In this model, the study of currents has attracted a lot of attention, but so far most results are restricted to two geometries: (i)a finite system between two reservoirs, which does not conserve the number of particles but reaches a nonequilibrium steady state, and (ii)an infinite system which conserves the number of particles but never reaches a steady state. Here, we obtain an expression for the full cumulant generating function of the integrated current in the important intermediate situation of a semi-infinite system connected to a reservoir, which does not conserve the number of particles and never reaches a steady state. This results from the determination of the full spatial structure of the correlations, which we infer to obey the very same closed equation recently obtained in the infinite geometry and argue to be exact. Besides their intrinsic interest, these results allow us to solve two open problems: the survival probability of a fixed target in the SEP, and the statistics of the number of particles injected by a localized source.

Control approach to well-posedness and asymptotic behavior of a queueing system

In this paper, we investigate well-posedness and asymptotic behavior of an M/G/1 retrial queueing system characterized by reserved idle time and setup time. This system is intricately described by an infinite system of integro-partial differential equations. Our methodology is rooted in the feedback theory of well-posed and regular infinite-dimensional linear systems. Firstly, we convert the system into a control framework, incorporating both input and output variables. Subsequently, we establish its well-posedness by leveraging the feedback theory specific to regular infinite-dimensional linear systems. Secondly, by scrutinizing the spectral distribution of the system operator along the imaginary axis, we demonstrate that the time-evolving solution of the system converges robustly to its steady-state solution. Lastly, we also discuss the essential properties of the semigroup induced by the system operator, encompassing stability and hyperbolicity, thereby shedding light on its regularity.

Transient Synchronous Stability Modeling and Comparative Analysis of Grid-Following and Grid-Forming New Energy Power Sources

New energy power sources can be categorized into grid-following and grid-forming types based on their synchronization characteristics with the grid. Due to the different basic control operation principles of these two types of new energy power sources, they can lead to the instability of the grid-connected system through different paths. A mathematical model is established to describe the dynamic characteristics of the grid-following and the grid-forming new energy power resources, and the control block diagram of the nonlinear dynamic characteristic equation is compared and analyzed. The similarity between grid-following and grid-forming new energy power supply is revealed, and we preliminarily discuss the influencing factors of the stability of new energy sources. Using a single-machine infinite system model, the influence of key control parameters on transient stability is separately investigated for grid-following and grid-forming types. The influence of the converter synchronous loop control parameters on the system is verified, and the factors affecting the stability are discussed in combination with current limiting and frequency limiting. The research will provide reference for the design of a new energy power grid connection system.

α-, β- and γ-duals of the sequence spaces formed by a regular matrix of Tetranacci numbers

Abstract The primary objective of this paper is to create a novel infinite Toeplitz matrix by leveraging Tetranacci numbers. This matrix serves as the foundation for defining new sequence spaces denoted as c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} , where 1 ≤ p < ∞ {1\leq p<\infty} . By utilizing this newly constructed matrix, the paper also explores and examines various algebraic and topological properties inherent to the sequence spaces c 0 ⁢ ( G ) {c_{0}(G)} , c ⁢ ( G ) {c(G)} , ℓ ∞ ⁢ ( G ) {\ell_{\infty}(G)} , and ℓ p ⁢ ( G ) {\ell_{p}(G)} for values of p within the range of 1 ≤ p < ∞ {1\leq p<\infty} . At last, we also prove existence theorem with example for infinite systems of differential equations in ℓ p ⁢ ( G ) {\ell_{p}(G)} .

An Infinite Capacity Single Server Markovian Queueing System With Discouraged Arrivals Retention Of Reneged Customers and Controllable Arrival Rates With Feedback

An Infinite Capacity Single Server Markovian Queueing System With Discouraged Arrivals Retention Of Reneged Customers and Controllable Arrival Rates With Feedback