Network models for nonlocal traffic flow

We present a network formulation for a traffic flow model with nonlocal velocity in the flux function. The modeling framework includes suitable coupling conditions at intersections to either ensure maximum flux or distribution parameters. In particular, we focus on 1-to-1, 2-to-1 and 1-to-2 junctions. Based on an upwind type numerical scheme, we prove the maximum principle and the existence of weak solutions on networks. We also investigate the limiting behavior of the proposed models when the nonlocal influence tends to infinity. Numerical examples show the difference between the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richards network model.

An Inferential Aptness of a Weibull Generated Distribution and Application

In this article an attempt has been made to develop a flexible single parameter continuous distribution using Weibull distribution. The Weibull distribution is most widely used lifetime distributions in both medical and engineering sectors. The exponential and Rayleigh distribution is particular case of Weibull distribution. Here in this study we use these two distributions for developing a new distribution. Important statistical properties of the proposed distribution is discussed such as moments, moment generating and characteristic function. Various entropy measures like Rényi, Shannon and cumulative entropy are also derived. The kthkt⁢h order statistics of pdf and cdf also obtained. The properties of hazard function and their limiting behavior is discussed. The maximum likelihood estimate of the parameter is obtained that is not in closed form, thus iteration procedure is used to obtain the estimate. Simulation study has been done for different sample size and MLE, MSE, Bias for the parameter λλ has been observed. Some real data sets are used to check the suitability of model over some other competent distributions for some data sets from medical and engineering science. In the tail area, the proposed model works better. Various model selection criterion such as -2LL, AIC, AICc, BIC, K-S and A-D test suggests that the proposed distribution perform better than other competent distributions and thus considered this as an alternative distribution. The proposed single parameter distribution is found more flexible as compare to some other two parameter complicated distributions for the data sets considered in the present study.

QUANTUM ERGODICITY FOR COMPACT QUOTIENTS OF IN THE BENJAMINI–SCHRAMM LIMIT

We study the limiting behavior of Maass forms on sequences of large-volume compact quotients of $\operatorname {SL}_d({\mathbb R})/\textrm {SO}(d)$ , $d\ge 3$ , whose spectral parameter stays in a fixed window. We prove a form of quantum ergodicity in this level aspect which extends results of Le Masson and Sahlsten to the higher rank case.

Analysis of the limiting behavior of a biological neurons system with delay

In this work an analytical and numerical analysis of the limiting behaviors of a system consisting of a pair of biological neurons was carried out. In this case connection between neurons will occur with a delay. As a neuron model, the FitzHugh-Nagumo model was chosen as a model that can reproduce many dynamic behaviors of a real neuron and, at the same time, is not very complex computationally.

On warped string vacuum profiles and cosmologies. Part I. Supersymmetric strings

We investigate in detail solutions of supergravity that involve warped products of flat geometries of the type Mp+1× R × TD−p−2 depending on a single coordinate. In the absence of fluxes, the solutions include flat space and Kasner-like vacua that break all supersymmetries. In the presence of a symmetric flux, there are three families of solutions that are characterized by a pair of boundaries and have a singularity at one of them, the origin. The first family comprises supersymmetric vacua, which capture a universal limiting behavior at the origin. The first and second families also contain non-supersymmetric solutions whose behavior at the other boundary, which can lie at a finite or infinite distance, is captured by the no-flux solutions. The solutions of the third family have a second boundary at a finite distance where they approach again the supersymmetric backgrounds. These vacua exhibit a variety of interesting scenarios, which include compactifications on finite intervals and p + 1-dimensional effective theories where the string coupling has an upper bound. We also build corresponding cosmologies, and in some of them the string coupling can be finite throughout the evolution.

Translates of homogeneous measures associated with observable subgroups on some homogeneous spaces

In the present article, we study the following problem. Let $\boldsymbol {G}$ be a linear algebraic group over $\mathbb {Q}$ , let $\Gamma$ be an arithmetic lattice, and let $\boldsymbol {H}$ be an observable $\mathbb {Q}$ -subgroup. There is a $H$ -invariant measure $\mu _H$ supported on the closed submanifold $H\Gamma /\Gamma$ . Given a sequence $(g_n)$ in $G$ , we study the limiting behavior of $(g_n)_*\mu _H$ under the weak- $*$ topology. In the non-divergent case, we give a rather complete classification. We further supplement this by giving a criterion of non-divergence and prove non-divergence for arbitrary sequence $(g_n)$ for certain large $\boldsymbol {H}$ . We also discuss some examples and applications of our result. This work can be viewed as a natural extension of the work of Eskin–Mozes–Shah and Shapira–Zheng.

The low-frequency limiting behavior of ambipolar diffusive models of impedance spectroscopy

The Poisson–Nernst–Planck (PNP) diffusional model is a successful theoretical framework to investigate the electrochemical impedance response of insulators containing ionic impurities to an external ac stimulus. Apparent deviations of the experimental spectra from the predictions of the PNP model in the low frequency region are usually interpreted as an interfacial property. Here, we provide a rigorous mathematical analysis of the low-frequency limiting behavior of the model, analyzing the possible origin of these deviation related to bulk properties. The analysis points toward the necessity to consider a bulk effect connected with the difference in the diffusion coefficients of cations and anions (ambipolar diffusion). The ambipolar model does not continuously reach the behavior of the one mobile ion diffusion model when the difference in the mobility of the species vanishes, for a fixed frequency, in the cases of ohmic and adsorption–desorption boundary conditions. The analysis is devoted to the low frequency region, where the electrodes play a fundamental role in the response of the cell; thus, different boundary conditions, charged to mimic the non-blocking character of the electrodes, are considered. The new version of the boundary conditions in the limit in which one of the mobility is tending to zero is deduced. According to the analysis in the dc limit, the phenomenological parameters related to the electrodes are frequency dependent, indicating that the exchange of electric charge from the bulk to the external circuit, in the ohmic model, is related to a surface impedance, and not simply to an electric resistance.

Asymptotics of Reinforcement Learning with Neural Networks

We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution that is the solution of the Bellman equation, thus giving the optimal control for the problem. In addition, we study the convergence of the limit differential equation to the stationary solution. As a by-product of our analysis, we obtain the limiting behavior of single-layer neural networks when trained on independent and identically distributed data with stochastic gradient descent under the widely used Xavier initialization.