Explicit Expressions Research Articles

A Novel Explicit Canonical Dynamic Modeling Method for Multi-Rigid-Body Mechanisms Considering Joint Friction

Friction is an inevitable phenomenon in mechanical systems that affects the dynamic characteristics of systems. To reduce the modeling complexity of complex multi-rigid-body mechanisms, a novel explicit canonical dynamic modeling method considering joint friction is proposed. Based on the explicit dynamic modeling theory that we have proposed, the solution of the constraint force required by the joint friction modeling of multi-rigid-body mechanisms is derived and improved, which greatly simplifies the solution of the constraint force. According to the obtained explicit expression of the constraint force equations, two joint friction models of the Coulomb–viscous effect and Stribeck effect are derived in analytical form. Moreover, the Stribeck effect of the joint is experimentally analyzed. A five-axis tree-chain mechanism and a three-loop closed-chain mechanism are chosen to demonstrate the method and compared with ADAMS software. Moreover, the proposed model is analyzed and compared with other methods.

Quasinormal modes in noncommutative Schwarzschild black holes

We investigate the quasinormal modes of a massless scalar field in a Schwarzschild black hole, which is deformed due to noncommutative corrections. We introduce the deformed Schwarzschild black hole solution, which depends on the noncommutative parameter Θ. We then extract the master equation as a Schrödinger-like equation, giving the explicit expression of the effective potential which is modified due to the noncommutative corrections. After that, we solve the master equation numerically. The significance of these results is twofold. Firstly, our results can be related to the detection of gravitational waves by the near future gravitational wave detectors, such as LISA, which will have a significantly increased accuracy. In particular, these observed gravitational waves produced by binary strong gravitational systems have oscillating modes which can provide valuable information. Secondly, our results can serve as an additional tool to test the predictions of GR, as well as to examine the possible detection of this kind of gravitational corrections.

Singular layer physics informed neural network method for plane parallel flows

We construct in this article the semi-analytic Physics Informed Neural Networks (PINNs), called singular layer PINNs (or sl-PINNs), that are suitable to predict the stiff solutions of plane-parallel flows at a small viscosity. Recalling the boundary layer analysis, we first find the corrector for the problem which describes the singular behavior of the viscous flow inside boundary layers. Then, using the components of the corrector and its curl, we build our new sl-PINN predictions for the velocity and the vorticity by either embedding the explicit expression of the corrector (or its curl) in the structure of PINNs or by training the implicit parts of the corrector (or its curl) together with the PINN predictions. Numerical experiments confirm that our new sl-PINNs produce stable and accurate predicted solutions for the plane-parallel flows at a small viscosity.

Impact of [formula omitted] theory on the geometry of charged stellar objects

The main objective of this article is to study the viable charged compact stellar structures in f(R,φ,χ) theory, where R, φ and χ denote the Ricci scalar, scalar field and kinetic term, respectively. For this purpose, a static spherical metric with anisotropic matter configuration is considered to analyze the compact stellar structures. We consider a particular model of this modified theory to formulate the explicit expressions of energy density and pressure components that govern the relationship between matter and geometry. The unknown constants in the Finch–Skea solutions are determined by the smooth matching of interior and exterior spacetimes to analyze the configuration of the proposed stellar structures. Physical parameters such as matter variables, energy bounds and equation of state parameters are analyzed to examine the viability of the considered stellar objects. Further, we use sound speed and adiabatic index methods to examine the stability of the compact objects. The rigorous analysis and satisfaction of necessary conditions lead to the conclusion that the stellar objects studied in this framework are viable and stable in the presence of charge.

A distributed state and fault estimation scheme for state-saturated systems with quantized measurements over sensor networks

This article explores a new framework of distributed state and fault estimation (DSFE) for the state-saturated systems over sensor networks. To this aim, the upper bound on estimation error covariance (EEC) is ensured and the explicit expression of the corresponding estimator gains is given with both quantization effects and state saturations. Further, a feasible upper bound is located on EEC and minimized by parameterizing the estimator gain. The matrix simplification technique is adopted to deal with the sensor network topology’s sparseness problem. Additionally, the estimation performance is first analyzed and then ensured by conducting a sufficient condition. At last, experiments are carried out to verify the feasibility of the developed DSFE method.

Wright's Hierarchical F-Statistics.

This perspective article offers a meditation on FST and other quantities developed by Sewall Wright to describe the population structure, defined as any departure from reproduction through random union of gametes. Concepts related to the F-statistics draw from studies of the partitioning of variation, identity coefficients, and diversity measures. Relationships between the first two approaches have recently been clarified and unified. This essay addresses the third pillar of the discussion: Nei's GST and related measures. A hierarchy of probabilities of identity-by-state provides a description of the relationships among levels of a structured population with respect to genetic diversity. Explicit expressions for the identity-by-state probabilities are determined for models of structured populations undergoing regular inbreeding and recurrent mutation. Levels of genetic diversity within and between subpopulations reflect mutation as well as migration. Accordingly, indices of the population structure are inherently locus-specific, contrary to the intentions of Wright. Some implications of this locus-specificity are explored.

Estimation and comparison of Weibull-Normal distribution with some other probability models using Bayesian method of estimation

In statistical applications the Normal distribution is adjudged to be the best. Recent studies Terna (2017) using classical method indicated that Weibull-Normal distribution outperformed the Normal distribution. In this study we used the non-classical Bayesian method of estimation to estimate and compare the Weibull-Normal distribution with some other distributions including Normal and Gamma-Normal distributions. This study derived explicit expressions for basic statistical properties such as moments, moment generating function, the characteristic function, reliability analysis and the distribution of order statistics. It looks at estimation of confidence intervals for the parameters of the Weibull-Normal distribution and estimated the parameters of the new distribution using a non-classical approach for the purpose of theoretical comparisons. The two other distributions whose parameters were also estimated by using Bayesian estimation are the normal distribution and gamma distribution as well as the combination Gamma-Normal distribution. Based on the analyses and interpretations of the results obtained it was found that the parameters and other general properties of Normal distribution gives a better fit than other distributions. R-software was used; the models were written as an R code in R program using the rjags library, the distribution parameters were obtained from a Gibbs sampling of a Bayesian Fit for data set I and data set II. The criteria used in R for comparisons were the negative log-likelihood, AIC (Akaike information criterion), CAIC (Consistent Akaike Information Criterion) and BIC (Bayesian information Criterion).

Expressive Talking Avatars.

Stylized avatars are common virtual representations used in VR to support interaction and communication between remote collaborators. However, explicit expressions are notoriously difficult to create, mainly because most current methods rely on geometric markers and features modeled for human faces, not stylized avatar faces. To cope with the challenge of emotional and expressive generating talking avatars, we build the Emotional Talking Avatar Dataset which is a talking-face video corpus featuring 6 different stylized characters talking with 7 different emotions. Together with the dataset, we also release an emotional talking avatar generation method which enables the manipulation of emotion. We validated the effectiveness of our dataset and our method in generating audio based puppetry examples, including comparisons to state-of-the-art techniques and a user study. Finally, various applications of this method are discussed in the context of animating avatars in VR.

Evolutionary neural networks for learning turbulence closure models with explicit expressions

Developing physical closure models with explicit expressions based on a given dataset is essential to science and engineering. For such symbolic regression tasks, biology-inspired evolutionary algorithms are most widely used. However, typical evolutionary algorithms do not utilize any structural information inherent in training data, which limits their performance in finding accurate model structures and coefficients. By combining one evolutionary algorithm, gene expression programing (GEP), with an artificial neural network (ANN) for symbolic regression, we propose a novel evolutionary neural network method, in which candidate expressions are specifically designed so that they can be transformed between the GEP and ANN structures during training iterations. By combining the GEP's global searching and the ANN's gradient optimization capabilities, efficient and robust convergence to accurate models can be achieved. In addition, sparsity-enhancing strategies have been introduced to improve the interpretability of the trained models. The present method has been tested for finding different physical laws and then applied to turbulence modeling problems with different configurations, showing advantages compared to the existing GEP and ANN methods.

An analytical study on generation of waves due to the rolling of flexible porous barriers

Complete analytical solution for generation of waves due to rolling of a partially immersed vertical elastic porous plate or a tensioned membrane is obtained. The originally defined boundary value problem in the half-plane has been decomposed first into a couple of flexible porous wave potential problems in the quarter-plane. Then, a pair of integral relations are introduced to establish a connection among the flexible porous wave potentials, solid wave potentials, and auxiliary wave potentials. By utilizing the connection, the flexible porous wave potentials can be determined in terms of the solid and the auxiliary wave potentials. The problems for the latter two types of potentials are easier to handle mathematically than the problems for the flexible porous wave potentials because of the involved structural boundary condition. Thus, the solid and the auxiliary wave potentials have been obtained analytically by the existing methods of solutions. Explicit analytical expressions are derived for the radiated wave amplitudes and velocity potentials. These radiated wave amplitudes are numerically computed and analyzed for different parameters involved. The results are validated with those derived for the problem involving a solid plate.

EFTofLSS meets simulation-based inference: σ 8 from biased tracers

Cosmological inferences typically rely on explicit expressions for the likelihood and covariance of the data vector, which normally consists of a set of summary statistics. However, in the case of nonlinear large-scale structure, exact expressions for either likelihood or covariance are unknown, and even approximate expressions can become very cumbersome, depending on the scales and summary statistics considered. Simulation-based inference (SBI), in contrast, does not require an explicit form for the likelihood but only a prior and a simulator, thereby naturally circumventing these issues. In this paper, we explore how this technique can be used to infer σ 8 from a Lagrangian effective field theory (EFT) based forward model for biased tracers. The power spectrum and bispectrum are used as summary statistics to obtain the posterior of the cosmological, bias and noise parameters via neural density estimation. We compare full simulation-based inference with cases where the data vector is drawn from a Gaussian likelihood with sample and analytical covariances. We conclude that, for k max = 0.1hMpc-1 and 0.2hMpc-1, the form of the covariance is more important than the non-Gaussianity of the likelihood, although this conclusion is expected to depend on the cosmological parameter inferred, the summary statistics considered and range of scales probed.

A semi-analytical time-domain model with explicit fluid force expressions for fluidelastic vibration of a tube array in crossflow

It is widely acknowledged that fluidelastic instability (FEI), among other mechanisms, is of the greatest concern in the flow-induced vibration (FIV) of tube bundles in steam generators and heat exchangers. A range of theoretical models have been developed for FEI analysis, and, in addition to the earliest semi-empirical Connors' model, the unsteady model, the quasi-steady model and the semi-analytical model are believed to be three advanced models predominant in the literature. The unsteady and the quasi-static models share the merits of having explicit fluid force expressions and ease of being implemented but require more experimental inputs, whereas the semi-analytical model requires fewer parameters due to its analytical nature but is hard, if not prohibitive, to derive explicit fluid force expressions. Since the fluid force formulations set in the core of development of FEI models, the understanding and in particular the implementation of the semi-analytical model has been impaired by the nonexistence of explicit fluid force expression. This issue becomes more profound in time-domain analysis whereby the simple harmonic assumption is discarded. Here we report a new semi-analytical time-domain (SATD) FEI model with explicit fluid force expressions. The new model allows a consistent frequency-domain stability analysis and more importantly a truly time-domain response analysis. The theory was validated by calculating linear stability thresholds of two typical tube array patterns and comparing against reported experimental data. We then present a nonlinear time-domain analysis of a single loosely-supported tube with piece-wise linear stiffness. The nonlinear and nonsmooth dynamics was probed in details by utilizing various techniques, playing an emphasis on characterizing and distinguishing the chaotic vibration. We found that the system follows a quasi-periodic route to chaos. Such an in-depth study of the nonlinear dynamics of tubes in crossflow has never been reported in the context of SATD model. Our results enrich the theory and provide a different approach for linear and nonlinear dynamics of tube bundles, which are essential for the subsequent fretting wear analysis.

Bayes estimator for the inequality-constrained regression model*

This article studies the Bayesian estimation of the inequality-constrained regression model under a balanced loss function. The usual Bayes estimator is often involved in some complex integrals and has no explicit expression, which causes the corresponding properties be difficult to describe. Based on the idea of linear Bayes method suggested by Rao (1973), the present paper constructs a linear Bayes estimator for the regression parameters without specify their specific prior distributions. It is proved that the proposed linear Bayes estimator outperforms the corresponding least squares estimator, and numerical simulations show that when there is collinearity in design matrix, the linear Bayes estimator performs better than the usual Bayes estimator.

The effects of additional food and environmental stochasticity on the asymptotic properties of a nutrient–phytoplankton model

In this paper, we investigate the deterministic and stochastic features of an additional food provided nutrient–phytoplankton model, and discuss the effects of additional food and environmental stochasticity (noise) on its asymptotic properties. A detailed global dynamics analysis of the deterministic model is firstly provided, and then the critical conditions for persistence and extinction of phytoplankton is found. For the stochastic model, we establish some sufficient conditions for the exponential extinction of phytoplankton and the existence of stationary distribution. By solving the Fokker–Planck equation, the explicit expression of local probability density of the stochastic model is further derived. With the help of probability density, we obtain the mean, variance, skewness and kurtosis of the biomass of phytoplankton, and numerically find the additional food and stochasticity have complex effects on these statistical properties. Both theoretical and numerical findings indicate that the additional food source is favorable for the growth of phytoplankton, whereas the stochasticity is adverse to its growth. Moreover, the additional food source and stochasticity play important roles in the formation of phytoplankton bloom.

Explicit evaluation of triple convolution sums of the divisor functions

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums [Formula: see text] for odd integers [Formula: see text] and [Formula: see text], where [Formula: see text] is the sum of the [Formula: see text]th powers of the positive divisors of [Formula: see text]. We consider four cases, namely (i) [Formula: see text], (ii) [Formula: see text]; [Formula: see text] (iii) [Formula: see text]; [Formula: see text] and (iv) [Formula: see text], and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer [Formula: see text] by certain positive definite quadratic forms. The existing formulas for [Formula: see text] (in [20]), [Formula: see text] (in [7]), [Formula: see text] (in [35]), [Formula: see text], [Formula: see text] (in [30]) and [Formula: see text] (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.

Canonical vs. Grand Canonical Ensemble for Bosonic Gases under Harmonic Confinement.

We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose-Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit.

The enumeration of equivalent classes of minimal general dihedral group codes

A group code is a linear code which can be realized as a two-sided ideal of a group algebra over a finite field. When the characteristic of the field is prime to the order of the group, we will give explicit expressions for primitive central idempotents in the group algebra, which enables us to determine the number of equivalent classes of minimal group codes. Then, we apply our formula to calculate the number of equivalent classes of minimal general dihedral group codes.

Bifurcation of the travelling wave solutions in a perturbed (1 + 1)-dimensional dispersive long wave equation via a geometric approach

Choosing ${\kappa }$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$ -dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa }\in \left (0,1-\frac {\sqrt 3}{3}\right )\cup \left (1+\frac {\sqrt 3}{3},2\right )$ , while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt {15}/3$ for $\kappa =0$ or $\kappa =2$ . The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.

Understanding the dynamics of hepatitis B transmission: A stochastic model with vaccination and Ornstein-Uhlenbeck process

In this paper, we explore and analyze a stochastic epidemiological model with vaccination and two mean-reverting Ornstein-Uhlenbeck processes to describe the dynamics of HBV transmission. Our study begins by deriving a sufficient condition for the extinction of hepatitis B. Additionally, we determine the presence of a stationary distribution using the Markov process stability theory. Next, we utilize algebraic equation theory and the associated Fokker-Planck equation to derive an explicit expression for the unique probability density function. To validate our theoretical findings, we conduct several numerical simulations using hepatitis B data provided by the World Health Organization (WHO). The results of these simulations support and complement our analytical approach.

Orbital stability of the black soliton for the quintic Gross–Pitaevskii equation

In this work, a proof of the orbital stability of the black soliton solution of the quintic Gross–Pitaevskii equation in one spatial dimension is obtained. We first build and show explicitly black and dark soliton solutions and we prove that the corresponding Ginzburg–Landau energy is coercive around them by using some orthogonality conditions related to perturbations of the black and dark solitons. The existence of suitable perturbations around black and dark solitons satisfying the required orthogonality conditions is deduced from an implicit function theorem. In fact, these perturbations involve dark solitons with sufficiently small speeds and some proportionality factors arising from the explicit expression of their spatial derivative.