On simultaneous approximation to (α,α) with α+kα−1=0

We show that the modified Jacobi-Perron algorithm gives the best simultaneous approximation to (α,α²) with α³+kα−1=0. We claim the following facts:

(1)

the limit set of become an ellipse, where (p_n,q_n,r_n) is the nth convergent (p_n/q_n,r_n/q_n) of (α,α²) by the modified Jacobi-Perron algorithm,

(2)

the limit set of belongs to outside of the ellipse in (1).

     

A 3D non-linear k–ε turbulent model for prediction of flow and mass transport in channel with vegetation

The results from a 3D non-linear k–ε turbulence model with vegetation are presented to investigate the flow structure, the velocity distribution and mass transport process in a straight compound open channel and a curved open channel. The 3D numerical model for calculating flow is set up in non-orthogonal curvilinear coordinates in order to calculate the complex boundary channel. The finite volume method is used to disperse the governing equations and the SIMPLEC algorithm is applied to acquire the coupling of velocity and pressure. The non-linear k–ε turbulent model has good useful value because of taking into account the anisotropy and not increasing the computational time. The water level of this model is determined from 2D Poisson equation derived from 2D depth-averaged momentum equations. For concentration simulation, an expression for dispersion through vegetation is derived in the present work for the mixing due to flow over vegetation. The simulated results are in good agreement with available experimental data, which indicates that the developed 3D model can predict the flow structure and mass transport in the open channel with vegetation.     

Computational study of decaying annular vortex flow using the Rε/k − ε turbulence model

In this article, we present three dimensional CFD study of turbulent vortex flow in an annular passage using OpenFOAM 1.6. The vortex flow is generated by introducing the flow through a tangential entry to the passage. For the analysis presented in this article, turbulence was modeled using the R_ε/k − ε model, in addition, a comparison between such model with the standard k − ε model was conducted and discussed. The main characteristics of the flow such as vortex structure and recirculation zone were investigated. It was found that flow is subjected to Rankine vortex structure with three forced vortex regimes and a free vortex region near to the outer wall. The phenomenon of vortex decay was investigated by depicting the swirl number trend along the axial direction of the flow domain. It was found that the vortex decay is subjected to an exponential decay behavior. New coefficients for the exponential decay correlation were derived based on local values of velocity components in different radial planes.     

α Regularization of the POD-Galerkin dynamical systems of the Kuramoto-Sivashinsky equation

In order to improve the dynamics and stability of the POD-Galerkin models of strongly-stiff systems, an α-like regularization is suggested and assessed in the present article. In this method, the POD eigenmodes of the non-linear terms are replaced by their Helmholtz filtered counterparts, while the other terms are remained unchanged. As an example, the method is applied to the POD-Galerkin models of the one-dimensional Kuramoto-Sivashinsky (KS) equation in a full chaotic regime; and the fidelity of the original and regularized models to the direct numerical simulations (DNS) are investigated. Moreover, the effects of regularization on the dynamics of various terms, and whole of the systems, are analyzed via eigenvalue analysis of each term separately, and the total dynamical system as a whole. The numerical experiments show definite effectiveness of the method and excellent improvements in the predicted dynamics and stability, by minimum number of free parameters.     

Weak convergence of approximate solutions of stochastic equations with applications to random differential and integral equations ∗

In this paper, we considerably extend our earlier result about convergence in distribution of approximate solutions: of random operator equations, where the stochastic inputs and the underlying deterministic equation are simultaneously approximated. As a by-product, we obtain convergence results for approximate solutions of equations between spaces of probability measures. We apply our results to random Fredholm integral equations of the second kind and to a random [nbar]onlinear elliptic boundary value problem.     

Approximation of signals of class Lip(α, p) by linear operators

Mittal, Rhoades [5], [6], [7] and [8] and Mittal et al. [9] and [10] have initiated a study of error estimates E_n(f) through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrix T does not have monotone rows. In this paper we continue the work. Here we extend two theorems of Leindler [4], where he has weakened the conditions on {p_n} given by Chandra [2], to more general classes of triangular matrix methods. Our Theorem also partially generalizes Theorem 4 of Mittal et al. [11] by dropping the monotonicity on the elements of matrix rows, which in turn generalize the results of Quade [15].     

Counter-examples of regularity behavior for σ-evolution equations

In this paper, we mainly discuss the infinite loss of regularity and μ-loss for a σ-evolution type model with oscillating in time coefficients. On the one hand, an explicit counter-example has been constructed in the frequency space to show the precise infinite loss of regularity. On the other hand, due to the finite propagation speed property for σ∈(0,1], we construct the counter-example of a sequence of solutions in R by applying state of art techniques.     

Asymptotic behavior of solutions to Euler–Poisson equations for bipolar hydrodynamic model of semiconductors

In this paper we study the Cauchy problem for 1-D Euler–Poisson system, which represents a physically relevant hydrodynamic model but also a challenging case for a bipolar semiconductor device by considering two different pressure functions and a non-flat doping profile. Different from the previous studies (Gasser et al., 2003 [7], Huang et al., 2011 [12], Huang et al., 2012 [13]) for the case with two identical pressure functions and zero doping profile, we realize that the asymptotic profiles of this more physical model are their corresponding stationary waves (steady-state solutions) rather than the diffusion waves. Furthermore, we prove that, when the flow is fully subsonic, by means of a technical energy method with some new development, the smooth solutions of the system are unique, exist globally and time-algebraically converge to the corresponding stationary solutions. The optimal algebraic convergence rates are obtained.     

Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise: itô's case

In this article we prove new results concerning the long-time behaviour of random fields that are solutions in some generalized sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise. Our main results state that these random fields eventually converge with probability. one to a global attractor represented by a single random variable whose properties we investigate in detail. We analyze the partial differential equations of this article in light Itô's stochastic calculus and thereby obtain stabilization and stability results which are substantially different from our earlier results concerning their interpretation in the sense of Stratonovitch. In particular, the asymptotic properties of the random fields that we investigate here exhibit no recurrence and oscillatory properties     

How many miles to βX? II — Approximations to βX versus cofinal types of sets of metrics

Kada, Tomoyasu and Yoshinobu proved that the Stone-?ech compactification of a locally compact separable metrizable space is approximated by the collection of d-many Smirnov compactifications, where d is the dominating number. By refining the proof of this result, we will show that the collection of compatible metrics on a locally compact separable metrizable space has the same cofinal type, in the sense of Tukey relation, as the set of functions from ω to ω with respect to eventually dominating order.     

π and the hypergeometric functions of complex argument

Text

In this article we derive some new identities concerning π, algebraic radicals and some special occurrences of the Gauss hypergeometric function ₂F₁ in the analytic continuation. All of them have been derived by tackling some elliptic or hyperelliptic known integral, and looking for another representation of it by means of hypergeometric functions like those of Gauss, Appell or Lauricella. In any case we have focused on integrand functions having at least one couple of complex-conjugate roots. Founding upon a special hyperelliptic reduction formula due to Hermite (1876) [6], π is obtained as a ratio of a complete elliptic integral and the four-variable Lauricella function. Furthermore, starting with a certain binomial integral, we succeed in providing as a ratio of a linear combination of complete elliptic integrals of the first and second kinds to the Appell hypergeometric function of two complex-conjugate arguments. Each of the formulae we found theoretically has been satisfactorily tested by means of Mathematica^®.

Video

For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=rQqtVtAf-RQ.     

Lexicographic α-robustness: An alternative to min–max criteria

Robustness in Operations Research/Decision Aid is often associated with min–max and min–max regret criteria. This common approach to determine robust solutions consists in finding a solution which minimizes the maximal cost or regret. Nevertheless, these criteria are known to be too conservative. In this paper, we present and study a new robustness approach, called lexicographic α-robustness, which compensates for this major drawback and many others. Furthermore, we establish a link between lexicographic α-robustness and a third robustness approach called p-robustness.     

Heat kernel bounds, ancient κ solutions and the Poincaré conjecture

We establish certain Gaussian type upper bound for the heat kernel of the conjugate heat equation associated with 3-dimensional ancient κ solutions to the Ricci flow. As an application, using the W entropy associated with the heat kernel, we give a different and much shorter proof of Perelman's classification of backward limits of these ancient solutions. The method is partly motivated by Cao (2007) [1] and Sesum (2006) [27]. The current paper or Chow and Lu (2004) [6] combined with Chen and Zhu (2006) [4] and Zhang (2009) [31] lead to a simplified proof of the Poincaré conjecture without using reduced distance and reduced volume.     

η Forms and determinant lines

We construct a canonical zeta function (Quillen) connection on the determinant line bundle for a family of APS elliptic boundary value problems and show that its curvature is the 2-form component of a relative eta form.     

The hyperelliptic integrals and π

Text

A class of hyperelliptic integrals are expressed through hypergeometric functions, like those of Gauss, Lauricella and Appell, namely multiple power series. Whenever they can on their own be reduced to elliptic integrals through an algebraic transformation, we obtain a two-fold representation of the same mathematical object, and then several completely new π determinations through the above special functions and/or Euler integrals. All our π formulae have been successfully tested by means of convenient Mathematica_®'s packages and enter in a wide historical/sound context of π-formulae quite far from being exhausted. Due to their structure, the formulae's practical value does not lie in computing π, but in allowing, through π, a benchmark for computing the involved special functions, particularly those less elementary.

Video

For a video summary of this paper, please visit http://www.youtube.com/watch?v=UHZIHgAeCUc.     

On the existence and long-time behavior of solutions to stochastic three-dimensional Navier–Stokes–Voigt equations

We consider the 3D stochastic Navier–Stokes–Voigt equations in bounded domains with the homogeneous Dirichlet boundary condition and infinite-dimensional Wiener process. First, we prove the existence and uniqueness of solutions to the problem. Then we investigate the mean square exponential stability and the almost sure exponential stability of the stationary solutions.