Efficient three-step iterative methods with sixth order convergence for nonlinear equations

In this paper, we present two new three-step iterative methods for solving nonlinear equations with sixth convergence order. The new methods are obtained by composing known methods of third order of convergence with Newton’s method and using an adequate approximation for the derivative, that provides high order of convergence and reduces the required number of functional evaluations per step. The first method is obtained from Potra-Pták’s method and the second one, from Homeier’s method, both reaching an efficiency index of 1.5651. Our methods are comparable with the method of Parhi and Gupta (Appl Math Comput 203:50–55, 2008). Methods proposed by Kou and Li (Appl Math Comput 189:1816–1821, 2007), Wang et al. (Appl Math Comput 204:14–19, 2008) and Chun (Appl Math Comput 190:1432–1437, 2007) reach the same efficiency index, although they start from a fourth order method while we use third order methods and simpler arithmetics. We prove the convergence results and check them with several numerical tests that allow us to compare the convergence order, the computational cost and the efficiency order of our methods with those of the original methods.     

A genetic algorithm for a global optimization problem arising in the detection of gravitational waves

The detection of gravitational waves is a long-awaited event in modern physics and, to achieve this challenging goal, detectors with high sensitivity are being used or are under development. In order to extract gravitational signals emitted by coalescing binary systems of compact objects (neutron stars and/or black holes), from noisy data obtained by interferometric detectors, the matched filter technique is generally used. Its computational kernel is a box-constrained global optimization problem with many local solutions and a highly nonlinear and expensive objective function, whose derivatives are not available. To tackle this problem, we designed a real-coded genetic algorithm that exploits characteristic features of the problem itself; special attention was devoted to the choice of the initial population and of the recombination operator. Computational experiments showed that our algorithm is able to compute a reasonably accurate solution of the optimization problem, requiring a much smaller number of function evaluations than the grid search, which is generally used to solve this problem. Furthermore, the genetic algorithm largely outperforms other global optimization algorithms on significant instances of the problem.     

AG codes from vector bundles

We discuss the properties of algebraic-geometric codes defined by a vector bundle on a curve X over a finite field. We investigate the correction capacities of the codes defined from vector bundles constructed by means of finite coverings of X.     

Generalizations of spectrally multiplicative surjections between uniform algebras

Let $ A $ A and ℬ be unital semisimple commutative Banach algebras. It is shown that if surjections S,T: $ A $ A → ℬ with S(1)=T(1)= 1 and α ∈ ℂ \ {0} satisfy r(S(a)T(b) − α)= r(ab− α) for all a,b ∈ $ A $ A , then S=T and S is a real algebra isomorphism, where r(a) is the spectral radius of a. Let I be a nonempty set, A and B be uniform algebras. Let ρ, τ: I → A and S,T: I → B be maps satisfying σ _π (S(p)T(q)) ⊂ σ _π (ρ(p) τ(q)) for all p,q ∈ I, where σ _π (f) is the peripheral spectrum of f. Suppose that the ranges ρ(I), τ(I) ⊂ A and S(I),T(I) ⊂ B are closed under multiplication in a sense, and contain peaking functions “enough”. There exists a homeomorphism ϕ: Ch(B)→Ch(A) such that S(p)(y)= ρ(p)(ϕ(y)) and T(p)(y)= τ(p)(ϕ(y)) for every p ∈ I and y ∈ Ch(B), where Ch(A) is the Choquet boundary of A.     

A new relaxation framework for quadratic assignment problems based on matrix splitting

Quadratic assignment problems (QAPs) are known to be among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on various matrix splitting schemes. Then we introduce the so-called symmetric mappings that can be used to derive strong cuts for the proposed relaxation model. We show that the bounds based on the new models are comparable to some strong bounds in the literature. Promising experimental results based on the new relaxations are reported.     

Solutions in Hölder spaces of boundary-value problems for parabolic equations with nonconjugate initial and boundary data

The first and second one-dimensional boundary-value problems for parabolic equations are investigated in the case where the conjugation conditions for all required orders are not satisfied. The existence and uniqueness are proved. Estimates of solutions in classical and weighted Hölder spaces are obtained. We prove that the violation of conjugation for the given functions on the boundary of the domain at the initial-time moment causes the appearance of singular solutions. The order of singularity (as a power of t) is found for the singular solutions for t = 0.     

Weighted Hardy operators and commutators on Morrey spaces

The operator norms of weighted Hardy operators on Morrey spaces are worked out. The other purpose of this paper is to establish a sufficient and necessary condition on weight functions which ensures the boundedness of the commutators of weighted Hardy operators (with symbols in BMO(ℝ ⁿ )) on Morrey spaces.     

On <Emphasis Type="Italic">g</Emphasis>-measures in symbolic dynamics

Finitary Markov processes are described in G. Morvai and B. Weiss, Prediction for discrete time series, Probability Theory and Related Fields 132 (2005), 1–12. The transition functions of finitary Markov processes are residually locally constant g-functions that can be extended by continuity to their maximal domain of definition. The study of their associated symbolic dynamics leads one to the D-shifts as introduced in W. Krieger, On g-functions for subshifts, Institute of Mathematical Statistics Lecture Notes-Monograph Series, Vol. 48, Dynamics & Stochastics, arXiv:math.DS/0608259, (2006), 306–316, We study the phenomena that can arise in residually locally constant and locally constant maximally defined g-functions on D-shifts, Markov shifts and synchronizing systems with respect to future measures and g-measures     

Composition-diamond lemma for modules

We investigate the relationship between the Gröbner-Shirshov bases in free associative algebras, free left modules and “double-free” left modules (that is, free modules over a free algebra). We first give Chibrikov’s Composition-Diamond lemma for modules and then we show that Kang-Lee’s Composition-Diamond lemma follows from it. We give the Gröbner-Shirshov bases for the following modules: the highest weight module over a Lie algebra sl ₂, the Verma module over a Kac-Moody algebra, the Verma module over the Lie algebra of coefficients of a free conformal algebra, and a universal enveloping module for a Sabinin algebra. As applications, we also obtain linear bases for the above modules.     

Simultaneous estimation and variable selection in median regression using Lasso-type penalty

We consider the median regression with a LASSO-type penalty term for variable selection. With the fixed number of variables in regression model, a two-stage method is proposed for simultaneous estimation and variable selection where the degree of penalty is adaptively chosen. A Bayesian information criterion type approach is proposed and used to obtain a data-driven procedure which is proved to automatically select asymptotically optimal tuning parameters. It is shown that the resultant estimator achieves the so-called oracle property. The combination of the median regression and LASSO penalty is computationally easy to implement via the standard linear programming. A random perturbation scheme can be made use of to get simple estimator of the standard error. Simulation studies are conducted to assess the finite-sample performance of the proposed method. We illustrate the methodology with a real example.