Exact Expressions Research Articles

Study on dynamics of thin shell elements of composite materials based on absolute nodal coordinate formulation

Most solar panels use composite materials to suit the development objectives of large-scale and lightweight spaceship components, resulting in complex nonlinear dynamic characteristics. When describing the dynamic characteristics of thin shell elements in composite materials, the standard 48-DOF fully parameterized thick plate element based on absolute nodal coordinate formulation cannot accurately account for the coupling between the large range of motion and large deformation of thin shell elements, exposing issues such as shear locking, slow convergence, and low computational efficiency. In this study, the absolute nodal coordinate formulation is used to create the 36-DOF reduced-order composite thin shell element. The strain and strain energy of the thin-shell element are calculated using the Green-Lagrange strain tensor, while the bending deformation energy of the thin shell element is calculated using the exact expression of curvature, based on continuum mechanics theory and the constitutive relation of the composite material. The generalized mass matrix and stiffness matrix are developed, and the nonlinear dynamic model of the thin shell element is established using the virtual work principle, with numerical simulation used to validate the dynamic characteristics. The results reveal that the dynamic characteristics of the 36-DOF reduced-order composite thin shell element are consistent with those of the virtual prototype, confirming the validity of the model. The proposed model not only addresses the element locking issue, but it also creates a more realistic dynamic model and increases simulation accuracy. The findings could serve as a theoretical foundation for satellite solar panel on-orbit operations.

Exact Integral Formulas for False Discovery Rate and the Variance of False Discovery Proportion.

Multiple hypothesis testing is an integral component of data analysis for large-scale technologies such as proteomics, transcriptomics, or metabolomics, for which the false discovery rate (FDR) and positive FDR (pFDR) have been accepted as error estimation and control measures. The pFDR is the expectation of false discovery proportion (FDP), which refers to the ratio of the number of null hypotheses to that of all rejected hypotheses. In practice, the expectation of ratio is approximated by the ratio of expectation; however, the conditions for transforming the former into the latter have not been investigated. This work derives exact integral expressions for the expectation (pFDR) and variance of FDP. The widely used approximation (ratio of expectations) is shown to be a particular case (in the limit of a large sample size) of the integral formula for pFDR. A recurrence formula is provided to compute the pFDR for a predefined number of null hypotheses. The variance of FDP was approximated for a practical application in peptide identification using forward and reversed protein sequences. The simulations demonstrate that the integral expression exhibits better accuracy than the approximate formula in the case of a small number of hypotheses. For large sample sizes, the pFDRs obtained by the integral expression and approximation do not differ substantially. Applications to proteomics data sets are included.

Thermoelastic interactions in a micropolar poroelastic medium

The present problem investigates the two-dimensional disturbances in a homogeneous, isotropic, micropolar thermoelastic saturated porous medium under the purview of Green-Lindsay (GL) theory of thermoelasticity. The medium is assumed to be unstrained and unstressed initially and has uniform temperature. The formulation is subjected to a moving thermal shock. Normal mode technique is employed to obtain the exact expressions for the displacement components, stresses, pore pressure and temperature. Numerical computations have been carried out with the help of MATLAB software and the results are illustrated graphically. Comparisons are made to show the effects of time, velocity of the load, micropolar and porous parameters on the resulting quantities. The outcomes point out a strong impact of these parameters on the physical quantities and agree with the boundary conditions. Some particular cases of interest have been deducted from the current investigation to validate the results. The present work is useful and valuable for analysis of problems involving thermal shock, micropolar and poroelastic deformations.

A stochastic model for the early stages of highly contagious epidemics by using a state-dependent point process

The recent COVID-19 pandemic has shown that when the reproduction number is high and there are no proper measurements in place, the number of infected people can increase dramatically in a short time, producing a phenomenon that many stochastic SIR-like models cannot describe: overdispersion of the number of infected people (i.e., the variance of the number of infected people during any interval is very high compared to the average). To address this issue, in this paper we explore the possibility of modeling the total number of infections as a state-dependent self-exciting point process. In this way, infections are not independent among themselves, but any infection will increase the likelihood of a new infection while also the number of currently infected and recovered individuals are included into determining the likelihood of new infections, Since long term simulation is extremely computationally intensive, exact expressions for the moments of the processes determining the number of infected and recovered individuals are computed, while also simulation algorithms for these state-dependent processes are provided.

Entropy Production of Run-and-Tumble Particles.

We analyze the entropy production in run-and-tumble models. After presenting the general formalism in the framework of the Fokker-Planck equations in one space dimension, we derive some known exact results in simple physical situations (free run-and-tumble particles and harmonic confinement). We then extend the calculation to the case of anisotropic motion (different speeds and tumbling rates for right- and left-oriented particles), obtaining exact expressions of the entropy production rate. We conclude by discussing the general case of heterogeneous run-and-tumble motion described by space-dependent parameters and extending the analysis to the case of d-dimensional motions.

Dynamical Behaviors of Stochastic SIS Epidemic Model with Ornstein–Uhlenbeck Process

Controlling infectious diseases has become an increasingly complex issue, and vaccination has become a common preventive measure to reduce infection rates. It has been thought that vaccination protects the population. However, there is no fully effective vaccine. This is based on the fact that it has long been assumed that the immune system produces corresponding antibodies after vaccination, but usually does not achieve the level of complete protection for undergoing environmental fluctuations. In this paper, we investigate a stochastic SIS epidemic model with incomplete inoculation, which is perturbed by the Ornstein–Uhlenbeck process and Brownian motion. We determine the existence of a unique global solution for the stochastic SIS epidemic model and derive control conditions for the extinction. By constructing two suitable Lyapunov functions and using the ergodicity of the Ornstein–Uhlenbeck process, we establish sufficient conditions for the existence of stationary distribution, which means the disease will prevail. Furthermore, we obtain the exact expression of the probability density function near the pseudo-equilibrium point of the stochastic model while addressing the four-dimensional Fokker–Planck equation under the same conditions. Finally, we conduct several numerical simulations to validate the theoretical results.

Performance Prediction of Power Beacon-Aided Wireless Sensor-Powered Non-Orthogonal Multiple-Access Internet-of-Things Networks under Imperfect Channel State Information

In this paper, we investigate a novel power beacon (PB)-aided wireless sensor-powered non-orthogonal multiple-access (NOMA) Internet-of-Things (IoT) network under imperfect channel state information (CSI). Furthermore, the exact expression outage probability (OP) of two IoT users is derived to analyze the performance of the considered network. To give further insight, the expression asymptotic OP and diversity order are also expressed when the transmit power at the PB goes to infinity. Furthermore, a deep neural network (DNN) framework is proposed to concurrently forecast IoT users’ OP in relation to real-time setups for IoT users. Additionally, when compared to the traditional analysis, our created DNN shows the shortest run-time prediction, and the outcomes predicted by the DNN model almost match those of the simulation. In addition, numerical results validate our analysis, simulation, and prediction through a Monte Carlo Simulation. Furthermore, the results show the impact of the main parameter on our proposed system. Finally, these findings show that NOMA performs better than the conventional orthogonal multiple-access (OMA) techniques.

Effect of relative timescale on a system of particles sliding on a fluctuating energy landscape: Exact derivation of product measure condition.

We consider a system of hardcore particles advected by a fluctuating potential energy landscape, whose dynamics is in turn affected by the particles. Earlier studies have shown that as a result of two-way coupling between the landscape and the particles, the system shows an interesting phase diagram as the coupling parameters are varied. The phase diagram consists of various different kinds of ordered phases and a disordered phase. We introduce a relative timescale ω between the particle and landscape dynamics, and study its effect on the steady state properties. We find there exists a critical value ω=ω_{c} when all configurations of the system are equally likely in the steady state. We prove this result exactly in a discrete lattice system and obtain an exact expression for ω_{c} in terms of the coupling parameters of the system. We show that ω_{c} is finite in the disordered phase, diverges at the boundary between the ordered and disordered phase, and is undefined in the ordered phase. We also derive ω_{c} from a coarse-grained level description of the system using linear hydrodynamics. We start with the assumption that there is a specific value ω^{*} of the relative timescale when correlations in the system vanish, and mean-field theory gives exact expressions for the current Jacobian matrix A and compressibility matrix K. Our exact calculations show that Onsager-type current symmetry relation AK=KA^{T} can be satisfied if and only if ω^{*}=ω_{c}. Our coarse-grained model calculations can be easily generalized to other coupled systems.

Special Trans Function based exact expressions for the double and triple diode models of solar cells: Superior fitness, accuracy and convergence

Inspired by the fact that the Special Trans Function (STF) is more accurate than Lambert W function (LWF), this paper rigorously derives two types of STF-based exact expressions for the double-diode model (DDM) and triple-diode model (TDM) of solar cells. The former involves multi-STF (mSTF), while the latter contains single-STF (sSTF), bridging the remaining gap in describing the nonlinear I–V characteristic of solar cells using STF. The proposed mSTF and sSTF are closely linked with but different from the multi-Lambert W function (mLWF), single-LWF (sLWF) and implicit exponential function (IEF) based expressions, particularly in terms of fitness and parameter extraction. Through a fitness test involving 203 cases in the space of STF branch x∈R+, it was found that under the same parameter values, the proposed mSTF consistently outperforms in fitting the measured I–V data of various solar cell/modules, followed by the proposed sSTF, mLWF, sLWF, and finally IEF-based expression. Notably, all optimal STF branch x fall within the interval of x∈[0,6] rather than the preset x∈[0,20]. A normalized trust-region-reflective (NTRR) algorithm and a V-shaped selection strategy are developed to improve the parameter extraction of proposed mSTF and sSTF. Results from the derivative-dependent V-shaped NTRR algorithm and population-based Rcr-IJADE algorithm indicate that the proposed sSTF consistently achieves the highest accuracy and fastest convergence speed, followed by the mSTF, sLWF, mLWF and IEF-based expression. Moreover, the proposed sSTF exhibits fewer degeneracies from TDM to DDM compared to other expressions. Given these advantages, the proposed mSTF and sSTF show great promise for PV simulation and therefore deserve serious attention.

Kibble-Zurek scalings and coarsening laws in slowly quenched classical Ising chains.

We consider a one-dimensional classical ferromagnetic Ising model when it is quenched from a low temperature to zero temperature in finite time using Glauber or Kawasaki dynamics. Most of the previous work on finite-time quenches assume that the system is initially in equilibrium and focuses on the excess mean defect density at the end of the quench, which decays algebraically in quench time with Kibble-Zurek exponent. Here we are interested in understanding the conditions under which the Kibble-Zurek scalings do not hold and in elucidating the full dynamics of the mean defect density. We find that depending on the initial conditions and quench time, the dynamics of the mean defect density can be characterized by coarsening and/or the standard finite-time quench dynamics involving adiabatic evolution and Kibble-Zurek dynamics; the timescales for crossover between these dynamical phases are determined by coarsening time and stationary state relaxation time. As a consequence, the mean defect density at the end of the quench either is a constant or decays following coarsening laws or Kibble-Zurek scaling. For the Glauber chain, we formulate a low-temperature scaling theory and find exact expressions for the final mean defect density for various initial conditions. For the Kawasaki chain where the dynamic exponents for coarsening and stationary state dynamics are different, we verify the above findings numerically and examine the effect of unequal dynamic exponents.

Capacity Analysis of Hybrid Satellite-Terrestrial Systems with Selection Relaying.

A hybrid satellite-terrestrial relay network is a simple and flexible solution that can be used to improve the performance of land mobile satellite systems, where the communication links between satellite and mobile terrestrial users can be unstable due to the multipath effect, obstacles, as well as the additional atmospheric losses. Motivated by these facts, in this paper, we analyze a system where the satellite-terrestrial links undergo shadowed Rice fading, and, following this, terrestrial relay applies the selection relaying protocol and forwards the information to the final destination using the communication link subjected to Nakagami-m fading. For the considered relaying protocol, we derive the exact closed-form expressions for the outage probability, outage capacity, and ergodic capacity, presented in polynomial-exponential form for the integer-valued fading parameters. The presented numerical results illustrate the usefulness of the selection relaying for various propagation scenarios and system geometry parameters. The obtained analytical results are corroborated by an independent simulation method, based on the originally developed fading simulator.

Solution to a class of multistate Landau-Zener model beyond integrability conditions

We study a class of multistate Landau-Zener model which cannot be solved by integrability conditions or other standard techniques. By analyzing analytical constraints on its scattering matrix and performing fitting to results from numerical simulations of the Schrödinger equation, we find nearly exact analytical expressions of all its transition probabilities for specific parameter choices. We also determine the transition probabilities up to leading orders of series expansions in terms of the inverse sweep rate (namely, in the diabatic limit) for general parameter choices. We further show that this model can describe a Su-Schrieffer-Heeger chain with couplings changing linearly in time. Our work presents a new route, i.e., analytical constraint plus fitting, to analyze those multistate Landau-Zener models which are beyond the applicability of conventional solving methods.

Vibration behavior prediction of submerged nanobeams with axially traveling supports: numerical, analytical, and machine learning approaches

This work surveyed the vibration behavior and stability analysis of an underwater moving flexible nanosize beam, implementing numerical and analytical methods as well as tree-based machine learning (ML) algorithms. Dynamic modeling is performed based on the nonlocal stress-strain gradient theory (NSSGT), incorporating variable environmental conditions, surface energy, and rotational inertia effects. The natural frequencies and instability threshold are computed numerically. The exact closed-form mathematical expression for the critical towing velocity of the nanosize beam is determined analytically. Also, decision tree regression and least-squares boosting tree (LSBT) regression algorithms are exploited to predict stability boundaries and the fundamental vibration frequency, and their efficiency is surveyed accordingly. Comparative studies and parametric investigations are conducted. The impressions of geometry, added mass coefficient, scale ratio parameter, surface layer characteristics, fluid mass ratio, humidity, temperature rise, and external magnetic field intensity on the nanosystem dynamics are examined and illuminated. It is detected that the results of the analytical method and ML-based approaches are consistent with those reported by the numerical technique and literature. The results asserted that the proposed ML algorithms have acceptable performance and excellent effectiveness in computational time and cost. It is comprehended that the dynamic features of submerged traveling nanobeams drastically depend on the rotational inertia effects and thickness-dependent scale effects. The stability of the immersed movable nanoscale beams is perceived to improve by increasing the scale ratio parameter and the added mass coefficient. From the perspective of the optimal design of engineering instrumentation tools, the outcomes of the current article can be applied as an inclusive benchmark.

Towards verifications of Krylov complexity

Abstract Krylov complexity is considered to provide a measure of the growth of operators evolving under Hamiltonian dynamics. The main strategy is the analysis of the structure of Krylov subspace $\mathcal {K}_M(\mathcal {H},\eta )$ spanned by the multiple applications of the Liouville operator $\mathcal {L}$ defined by the commutator in terms of a Hamiltonian $\mathcal {H}$, $\mathcal {L}:=[\mathcal {H},\cdot ]$ acting on an operator η, $\mathcal {K}_M(\mathcal {H},\eta )=\text{span}\lbrace \eta ,\mathcal {L}\eta ,\ldots ,\mathcal {L}^{M-1}\eta \rbrace$. For a given inner product ( ·, ·) of the operators, the orthonormal basis $\lbrace \mathcal {O}_n\rbrace$ is constructed from $\mathcal {O}_0=\eta /\sqrt{(\eta ,\eta )}$ by Lanczos algorithm. The moments $\mu _m=(\mathcal {O}_0,\mathcal {L}^m\mathcal {O}_0)$ are closely related to the important data {bn} called Lanczos coefficients. I present the exact and explicit expressions of the moments {μm} for 16 quantum mechanical systems which are exactly solvable both in the Schrödinger and Heisenberg pictures. The operator η is the variable of the eigenpolynomials. Among them six systems show a clear sign of ‘non-complexity’ as vanishing higher Lanczos coefficients bm = 0, m ≥ 3.

Enhancing data rate and energy efficiency of NOMA systems using reconfigurable intelligent surfaces for millimeter-wave communications

In this article, we employ reconfigurable intelligent surface (RIS) to aid a non-orthogonal multiple access (NOMA) system utilizing millimeter-wave (mmWave) communications. Different to recent works on the RIS-supported NOMA systems where the certain carrier frequency was often normalized (fc=1 Hz), we analyze specific impacts of high carrier frequencies in the mmWave band (fc≥30 GHz) on the performance of the RIS-supported NOMA system. In particular, we derive the exact expressions of achievable data rate (ADR) and energy efficiency (EE) of the RIS-supported NOMA system using mmWave band (RIS-mmWave-NOMA system). We compare the ADR and EE of the RIS-mmWave-NOMA system with those of mmWave-NOMA system without RIS, where there are only source-user channels. Numerical illustrations clarify the benefits of utilizing RIS as well as the great impacts of high fc in mmWave band on the ADR and EE of the RIS-mmWave-NOMA system. More specifically, when fc increases from 50 to 100 GHz, the transmit power of source should be increased 8 dBm to achieve the same ADR. Meanwhile, the maximal EE with fc=100 GHz is greatly lower than the maximal EE with fc=50 GHz even though higher transmit power is used. In other words, when fc increases, higher transmit power can be used for the ADR aims but this method cannot be used for EE purposes. In these circumstances, utilizing a larger number of reflecting elements (REs) in the RIS is a good method. By using N≥200 REs, the ADR and EE of the RIS-mmWave-NOMA system are significantly higher than those of mmWave-NOMA system without RIS. Finally, we confirm the exactness of the derived formulas through Monte-Carlo simulations.

The Second-Order Features Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (2nd-FASAM-L): Mathematical Framework and Illustrative Application to an Energy System

The Second-Order Features Adjoint Sensitivity Analysis Methodology for Response-Coupled Forward/Adjoint Linear Systems (abbreviated as “2nd-FASAM-L”), presented in this work, enables the most efficient computation of exactly obtained mathematical expressions of first- and second-order sensitivities of a generic system response with respect to the functions (“features”) of model parameters. Subsequently, the first- and second-order sensitivities with respect to the model’s uncertain parameters, boundaries, and internal interfaces are obtained analytically and exactly, without needing large-scale computations. Within the 2nd-FASAM-L methodology, the number of large-scale computations is proportional to the number of model features (defined as functions of model parameters), as opposed to being proportional to the number of model parameters. This characteristic enables the 2nd-FASAM-L methodology to maximize the efficiency and accuracy of any other method for computing exact expressions of first- and second-order response sensitivities with respect to the model’s features and/or primary uncertain parameters. The application of the 2nd-FASAM-L methodology is illustrated using a simplified energy-dependent neutron transport model of fundamental significance in nuclear reactor physics.

Entanglement asymmetry in CFT and its relation to non-topological defects

The entanglement asymmetry is an information based observable that quantifies the degree of symmetry breaking in a region of an extended quantum system. We investigate this measure in the ground state of one dimensional critical systems described by a CFT. Employing the correspondence between global symmetries and defects, the analysis of the entanglement asymmetry can be formulated in terms of partition functions on Riemann surfaces with multiple non-topological defect lines inserted at their branch cuts. For large subsystems, these partition functions are determined by the scaling dimension of the defects. This leads to our first main observation: at criticality, the entanglement asymmetry acquires a subleading contribution scaling as log ℓ/ℓ for large subsystem length ℓ. Then, as an illustrative example, we consider the XY spin chain, which has a critical line described by the massless Majorana fermion theory and explicitly breaks the U(1) symmetry associated with rotations about the z-axis. In this situation the corresponding defect is marginal. Leveraging conformal invariance, we relate the scaling dimension of these defects to the ground state energy of the massless Majorana fermion on a circle with equally-spaced point defects. We exploit this mapping to derive our second main result: the exact expression for the scaling dimension associated with n defects of arbitrary strengths. Our result generalizes a known formula for the n = 1 case derived in several previous works. We then use this exact scaling dimension to derive our third main result: the exact prefactor of the log ℓ/ℓ term in the asymmetry of the critical XY chain.

Analysis of a detailed multi-stage model of stochastic gene expression using queueing theory and model reduction

We introduce a biologically detailed, stochastic model of gene expression describing the multiple rate-limiting steps of transcription, nuclear pre-mRNA processing, nuclear mRNA export, cytoplasmic mRNA degradation and translation of mRNA into protein. The processes in sub-cellular compartments are described by an arbitrary number of processing stages, thus accounting for a significantly finer molecular description of gene expression than conventional models such as the telegraph, two-stage and three-stage models of gene expression. We use two distinct tools, queueing theory and model reduction using the slow-scale linear-noise approximation, to derive exact or approximate analytic expressions for the moments or distributions of nuclear mRNA, cytoplasmic mRNA and protein fluctuations, as well as lower bounds for their Fano factors in steady-state conditions. We use these to study the phase diagram of the stochastic model; in particular we derive parametric conditions determining three types of transitions in the properties of mRNA fluctuations: from sub-Poissonian to super-Poissonian noise, from high noise in the nucleus to high noise in the cytoplasm, and from a monotonic increase to a monotonic decrease of the Fano factor with the number of processing stages. In contrast, protein fluctuations are always super-Poissonian and show weak dependence on the number of mRNA processing stages. Our results delineate the region of parameter space where conventional models give qualitatively incorrect results and provide insight into how the number of processing stages, e.g. the number of rate-limiting steps in initiation, splicing and mRNA degradation, shape stochastic gene expression by modulation of molecular memory.

Analysis of the Dynamics in Linear Chain Models by means of Generalized Langevin Equations

We analyse the motion of one particle in a polymer chain. For this purpose, we use the framework of the exact (non-stationary) generalized Langevin equation that can be derived from first principles via the projection-operator method. Our focus lies on determining memory kernels from either exact expressions for autocorrelation functions or from simulation data. We increase the complexity of the underlying system starting out from one-dimensional harmonic chains and ending with a polymer driven through a polymer melt. Here, the displacement or the velocity of an individual particle in the chain serves as the observable. The central result is that the time-window in which the memory kernels show structure before they rapidly decay decreases with increasing complexity of the system.

Extreme value statistics of jump processes.

We investigate extreme value statistics (EVS) of general discrete time and continuous space symmetric jump processes. We first show that for unbounded jump processes, the semi-infinite propagator G_{0}(x,n), defined as the probability for a particle issued from zero to be at position x after n steps whilst staying positive, is the key ingredient needed to derive a variety of joint distributions of extremes and times at which they are reached. Along with exact expressions, we extract universal asymptotic behaviors of such quantities. For bounded, semi-infinite jump processes killed upon first crossing of zero, we introduce the strip probability μ_{0,[under x]̲}(n), defined as the probability that a particle issued from zero remains positive and reaches its maximum x on its nth^{} step exactly. We show that μ_{0,[under x]̲}(n) is the essential building block to address EVS of semi-infinite jump processes, and obtain exact expressions and universal asymptotic behaviors of various joint distributions.