Continuous approximation for a 1D analog of the Gol’dshtik model for separated flows of an incompressible fluid

A modification of a 1D analog of the Gol’dshtik mathematical model for separated flows of an incompressible fluid is considered. The model is a nonlinear differential equation with a boundary condition. Nonlinearity in the equation is continuous and depends on a small parameter. When this parameter tends to zero, we have a discontinuous nonlinearity. The results of the solutions are in agreement with the results obtained for the 1D analog of the Gol’dshtik model for separated flows of an incompressible fluid.     

Gol’dberg’s constants

We study two extremal problems of geometric function theory introduced by A. A. Gol’dberg in 1973. For one problem we find the exact solution, and for the second one we obtain partial results. In the process, we study the lengths of hyperbolic geodesics in the twice punctured plane, prove several results about them, and make a conjecture. Gol’dberg’s problems have important applications to control theory.     

Set-valued approximations and Newton’s methods

n to R^m. Under the assumption of semi-smoothness of the mapping, we prove that the approximation can be obtained through the Clarke generalized Jacobian, Ioffe-Ralph generalized Jacobian, B-subdifferential and their approximations. As an application, we propose a generalized Newton’s method based on the point-based set-valued approximation for solving nonsmooth equations. We show that the proposed method converges locally superlinearly without the assumption of semi-smoothness. Finally we include some well-known generalized Newton’s methods in our method and consolidate the convergence results of these methods. Received October 2, 1995 / Revised version received May 5, 1998 Published online October 9, 1998     

Rate of convergence of Euler’s approximations for SDEs with non-Lipschitz coefficients

We prove that Euler＇s approximations for stochastic differential equations driven by infinite many Brownian motions and with non-Lipschitz coefficients converge almost surely. Moreover, the rate of convergence is obtained.     

New degrees of freedom for high-order Whitney approximations of Darcy’s flows

We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. We propose a high-order discretisation based on Whitney finite elements, namely, Raviart-Thomas finite elements of degree r +?1 for the discharge and discontinuous piecewise polynomial finite elements of degree r for the pressure, with r ≥?0. We comment on the use of new degrees of freedom that have a clear physical meaning, the so-called weights on the small simplices, for the involved discharge and pressure fields. We describe a new numerical strategy to solve the discrete problem based on a tree-cotree block-decomposition of the unknowns that is natural when considering these new degrees of freedom. Preliminary numerical tests in two dimensions confirm the stability of the adopted method and the effectiveness of the new degrees of freedom.

     

Asymptotic expansions and convergence rates for Euler’s approximations of semigroups

We provide optimal bounds for errors in Euler’s approximations of semigroups in Banach algebras and of semigroups of operators in Banach spaces. Furthermore, we construct asymptotic expansions for such approximations with optimal bounds for remainder terms. The sizes of errors are controlled by smoothness properties of semigroups. In this paper we use Fourier–Laplace transforms and a reduction of the problem to the convergence rates and asymptotic expansions in the Law of Large Numbers. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09. This paper was written in 2004. In the interim, several related articles were published; let us mention [14, 13, 15].     

New applications of Picard?s successive approximations

The iterative method of successive approximations, originally introduced by Émile Picard in 1890, is a basic tool for proving the existence of solutions of initial value problems regarding ordinary first order differential equations. In the present paper, it is shown that this method can be modified to get estimates for the growth of solutions of linear differential equations of the typef(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=0 with analytic coefficients. A short comparison to the growth results in the literature, obtained by means of different methods, is also given. It turns out that many known results can be proved by applying Picard?s successive approximations in an effective way. Self-contained considerations are carried out in the complex plane and in the unit disc, and some remarks about solutions of real linear differential equations are made.     

Continuous selections and convergence of best Lp –approximations in subspaces of spline functions

The chief purpose of this paper is to study the problem of existence of continuous selections for the metric projection and of convergence of best L_p–approximations in subspaces of polynomial spline functions defined on a real compact interval I. Nürnberger-Sommer [8] have shown that there exists a continuous selection s if and only if the numberof knots k is less than or equal to the order m of the splines. Using their construction of s the author [12] has proved that the sequence of best L_p–approximations of f converges to s(f) as ρ→∞ for every continuous function f. The main results of this paper say that also in the case when k>m there exists always a continuous selection s (it is even pointwise-Lipschitz-continuous and quasi-linear) provided that the approximation problem is restricted to certain subsets I_epsilon; of I. In addition it is shown that anologously as for k≤m the sequence of best L_papproximations of f converges to s(f) for every continuous function f on Iε     

Generalized Debreu’s Open Gap Lemma and Continuous Representability of Biorders

We prove a generalization of Debreu’s Open Gap Lemma. Given any subset of the real line, this lemma guarantees the existence of a strictly increasing real function such that all the gaps of the image of the subset are open. Now we extend it to the case of n subsets of the real line and study the existence of a strictly increasing real function such that all the gaps of the image of each set are open. This function does not exist in general so, we characterize the cases in which it exists. This generalization is not equivalent to Debreu’s lemma working on the union of the n subsets, neither on the intersection. Then we use it in order to obtain new results on continuous representations of biorders. We also generalize the concept of biorder.     

Solvability of termoviscoelasticity problem for certain Oskolkov’s model

In this paper we investigate the existence of a weak solution for initial boundary-value problem of thermoviscoelasticity in certain Oskolkov’s mathematical model describing a motion of linearly elastic-delayed Voigt fluid.     

Nonlinear dynamics and synchronization of saline oscillator’s model

The Okamura model equation of saline oscillator is refined into a non-autonomous ordinary differential equation whose coefficients are related to physical parameters of the system. The dependence of the oscillatory period and amplitude on remarkable physical parameters are computed and compared to experimental results in order to test the model. We also model globally coupled saline oscillators and bring out the dependence of coupling coefficients on physical parameters of the system. We then study the synchronization behaviors of coupled saline oscillators by the mean of numerical simulations carried out on the model equations. These simulations agree with previously reported experimental results.     

Verification of Raz’s model for vibrational even-even nuclei

Raz has used the collective model with interparticle interactions included, to examine the modifications of the vibrational spectra as the strength of (a) the two-particle interaction and (b) the vibration-particle interaction is varied. Raz’s comparison of the experimentally observed energy ratios for Ti⁴⁸ with the theoretical predictions for D=1·00 has been extended for other even-even nuclei. His predictions are found to be in good agreement with the observed data only for some of the even-even nuclei.     

Improved King’s methods with optimal order of convergence based on rational approximations

In this paper, we present three-point and four-point methods for solving nonlinear equations. The methodology is based on King’s family of fourth order methods [R.F. King, A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (1973) 876–879] and further developed by using rational function approximations. The three-point method requires four function evaluations and has the order of convergence eight, whereas the four-point method requires five function evaluations and has the order of convergence sixteen. Therefore, the methods are optimal in the sense of Kung–Traub hypothesis. The proposed schemes are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified in the examples.     

A Bernstein function related to Ramanujan’s approximations of exp(n)

Ramanujan’s sequence θ(n),n=0,1,2,…?, is defined by $\frac{e^{n}}{2}=\sum_{j=0}^{n-1}\frac{n^{j}}{j!}+\frac{n^{n}}{n!} \theta(n)$ . It is possible to define, in a simple manner, the function θ(x) for all nonnegative real numbers x. We show that the function $\lambda(x):=x (\theta(x)-\frac{1}{3} )$ is a Bernstein function on [0,∞), that is, λ(x) is nonnegative with completely monotonic derivative on [0,∞). This implies some earlier results concerning complete monotonicity of the function θ(x) on [0,∞).     

Sharp constant in Jackson’s inequality with modulus of smoothness for uniform approximations of periodic functions

It is proved that, in the space C_2π, for all k, n ∈ ?,n > 1, the following inequalities hold: where e _n?1(f) is the value of the best approximation of f by trigonometric polynomials and ω ₂(f, h) is the modulus of smoothness of f. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.     

On controllable variants of Richardson’s model in political science

It is proved that the family of bi-Lipschitz classes of Delone sets in Euclidean space of dimension at least 2 has the cardinality of the continuum.