Efficient General Methods Research Articles

A new class of efficient general linear methods for ordinary differential equations

We describe the construction of a new class of efficient general linear methods of high stage order, for nonstiff and stiff differential systems. Examples of explicit methods with large regions of stability, and implicit methods which are A- and L-stable, up to the order p=4 are presented. It is confirmed by numerical experiments that all methods achieve the expected order of accuracy and no order reduction occurs for stiff systems.

Efficient general linear methods for a class of Volterra integro-differential equations

This paper concerns on the introduction of a method for solving a class of Volterra integro-differential equations (VIDEs) of the second kind. It is based on the combination of special general linear methods for ordinary differential equations and Gregory quadrature rule and equipped with a starting procedure. The convergence and stability of the method are analyzed. Some numerical experiments are given to illustrate the agreement of our implementation with the theoretical convergence orders and show the capability of the method in solving nonlinear VIDEs.

Reaction Development on π- and σ-Conjugated Bonds and Creation of Innovative Functions.

Monocarba-closo-dodecaborate (1; [closo-CB11H12](-), or C1-carborane anion) is a symmetrical, stable anionic cluster, which possesses low nucleophilicity/basicity and exhibits three-dimensional aromaticity. In contrast to the rich applications of C2-carborane molecules (C2B10H12), the chemistry of the C1-carborane anion as a platform for functional molecules has not been thoroughly studied thus far due to the lack of its efficient functionalization. In particular, no efficient general methods are available for the introduction of aryl and sp(2)/sp-carbon groups at the carbon vertex of the C1-carborane anion. The unique electronic structure and potential applications of the C1-carborane anion prompted us to investigate methods to functionalize it. We developed a general, efficient C-C cross-coupling reaction of 1 under palladium catalysis which yields a variety of 1-C-functionalized C1-carborane derivatives. The use of copper(I) or lithium species as a transmetalating partner facilitated the cross-coupling process of the sterically hindered C1-carborane anion. The potential application of 1-C-arylated C1-carborane anion derivatives thus obtained were explored, some of which showed potential as pharmacophores and ionic liquid crystal behavior. Furthermore, conjugation between σ- and π-aromatic moieties in 1-C-arylated monocarba-closo-dodecaborate anion derivatives was identified by means of kinetic experimental studies combined with theoretical calculations.

Construction of efficient general linear methods for stiff Volterra integral equations

General linear methods in the Nordsieck form have been introduced for the numerical solution of Volterra integral equations. In this paper, we introduce general linear methods of order p and stage order q=p for the numerical solution of Volterra integral equations in general form, rather than Nordsieck form. A- and V0(α)-stable methods are constructed and applied on stiff problems to show their efficiency.

Search for efficient general linear methods for ordinary differential equations

We analyze implicit general linear methods with s internal stages and r=s+1 external stages of order p=s+1 and stage order q=s or q=s+1. These methods might eventually lead to more efficient formulas than the class of DIMSIMs and the class of general linear methods with inherent Runge–Kutta stability. We analyze also error propagation and estimation of local discretization errors. Examples of such methods which are A- and L-stable are derived up to the stage order q=3 or q=4 and order p=4.

Near-Entropy Hotlink Assignments

Consider a rooted tree T of arbitrary maximum degree d representing a collection of n web pages connected via a set of links, all reachable from a source home page represented by the root of T. Each web page i carries a probability p i representative of the frequency with which it is visited. By adding hotlinks—shortcuts from a node to one of its descendents—we wish to minimize the expected number of steps l needed to visit pages from the home page, expressed as a function of the entropy H(p) of the access probabilities p. This paper introduces several new strategies for effectively assigning hotlinks in a tree. For assigning exactly one hotlink per node, our method guarantees an upper bound on l of 1.141H(p)+1 if d>2 and 1.08H(p)+2/3 if d=2. We also present the first efficient general methods for assigning at most k hotlinks per node in trees of arbitrary maximum degree, achieving bounds on l of at most $\frac{2H(p)}{\log(k+1)}+1$ and $\frac{H(p)}{\log(k+d)-\log d}+1$ , respectively. All our methods are strong, i.e., they provide the same guarantees on all subtrees after the assignment. We also present an algorithm implementing these methods in O(nlog n) time, an improvement over the previous O(n 2) time algorithms. Finally we prove a Ω(nlog n) lower bound on the running time of any strong method that guarantee an average access time strictly better than 2H(p).