A constructive approach to nodal splines

In the context of local spline interpolation methods, nodal splines have been introduced as possible fundamental functions by de Villiers and Rohwer in 1988. The corresponding local spline interpolation operator possesses the desirable property of reproducing a large class of polynomials. However, it was remarked that their definition is rather intricate so that it seems desirable to reveal the actual origin of these splines. The real source can be found in the Martensenoperator which can be obtained by two-point Hermite spline interpolation problem posed and proved by Martensen [Darstellung und Entwicklung des Restgliedes der Gregoryschen Quadraturformel mit Hilfe von Spline-Funktionen, Numer. Math. 21(1973)70–80]. On the one hand, we will show how to represent the Hermite Martensen spline recursively and, on the other hand, explicitly in terms of the B-spline by using the famous Marsden identity. Having introduced the Martensenoperator, we will show that the nodal spline interpolation operator can be obtained by a special discretization of the occurring derivatives. We will consider symmetric nodal splines of odd degree that can be obtained by our methods in a natural way.     

A new approach to fuzzy initial value problem by improved Euler method

This paper targets to investigate the solution of linear and nonlinear ordinary differential equations with fuzzy initial condition. Here, two improved Euler type methods have been proposed in order to obtain numerical solution of the problem. Along with this, an exact methodology is also discussed. The obtained results are depicted in term of plots to show the efficiency of the proposed methods. The solutions are compared with the known results and are found that those obtained by the proposed methods are tighter than the results from the existing method.     

A new approach to multivariate adaptive regression splines by using Tikhonov regularization and continuous optimization

This paper introduces a model-based approach to the important data mining tool Multivariate adaptive regression splines (MARS), which has originally been organized in a more model-free way. Indeed, MARS denotes a modern methodology from statistical learning which is important in both classification and regression, with an increasing number of applications in many areas of science, economy and technology. It is very useful for high-dimensional problems and shows a great promise for fitting nonlinear multivariate functions. The MARS algorithm for estimating the model function consists of two algorithms, these are the forward and the backward stepwise algorithm. In our paper, we propose not to use the backward stepwise algorithm. Instead, we construct a penalized residual sum of squares for MARS as a Tikhonov regularization problem which is also known as ridge regression. We treat this problem using continuous optimization techniques which we consider to become an important complementary technology and model-based alternative to the concept of the backward stepwise algorithm. In particular, we apply the elegant framework of conic quadratic programming. This is an area of convex optimization which is very well-structured, herewith, resembling linear programming and, hence, permitting the use of powerful interior point methods. Based on these theoretical and algorithmical studies, this paper also contains an application to diabetes data. We evaluate and compare the performance of the established MARS and our new CMARS in classifying diabetic persons, where CMARS turns out to be very competitive and promising.     

A topological spline system approach to even degree polynomial splines

The existence of minimum norm properties for even degree polynomial splines, analogous to the. ones known for odd degree splines, is investigated within the framework of the theory of

topological spline systems. It is shown that such properties cannot exist for even degree splines interpolating functions halfway between the partition points. For another class of even

degree spline functions, however, which hterpolate the local integrals of given functions with respect to the partitions, the seeked minimum norm properties can be proved. This is carried out

by first investigating a generalized problem within the theory of spline systems and then deriving corresponding conclusions. As a corollary the existence of spline systems with respect to differential operators of fractional degree is obtained.     

A new approach to superfractals

V-variable fractals and superfractals have recently been introduced by Barnsley [Barnsley Michael, Hutchinson John, Stenflo Örjan. A fractal valued random iteration algorithm and fractal hierarchy. Fractals 2005;13(2):111–46 [MR2151094 (2006b:28014)]] to the world of mathematics and computer graphics. In this paper, we introduce superior iterates to study the role of contractive and non-contractive operators in relation to superfractals. A modified algorithm along with details of computer implementation is also provided to compute V-variable fractals. A brief discussion about the various aspects of the computed figures indicates usefulness of the study.     

A constructive approach to Euler hydrodynamics for attractive processes. Application to k-step exclusion

We derive by a constructive method the hydrodynamic behavior of attractive processes with irreducible jumps and product invariant measures. Our approach relies on (i) explicit construction of Riemann solutions without assuming convexity, which may lead to contact discontinuities and (ii) a general result which proves that the hydrodynamic limit for Riemann initial profiles implies the same for general initial profiles. The k-step exclusion process provides a simple example. We also give a law of large numbers for the tagged particle in a nearest neighbor asymmetric k-step exclusion process.     

A new approach to agglomeration problems

In this paper, we study a location problem with positive externalities. We define a new transferable utility game, considering there is no restriction on the transfer of benefits between firms. We prove that the core of this game is non-empty, provide an expression for it, and an axiomatic characterization. We also study several core allocations, selected by means of a certain bankruptcy problem.     

A new approach to Matsumoto's theorem

We give a proof of Matsumoto's theorem on K ₂ of a field using techniques from homological algebra. By considering a complex associated to the action of GL(2, F) on P ₁(F) (F a field), we derive the Matsumoto presentation for H ₀ (F ^., H ₂(SL(2, F))) and, by considering the action of GL(n + 1, F) on P ⁿ (F), we prove the stability part of the theorem; namely, that H ₀(F ^., H ₂(SL(2, F))) is isomorphic to H ₂(SL(F)) = K ₂(F).     

A new approach to firm evaluation

In this paper, a method is developed to evaluate firms on the basis of the risks they face. In accordance with the multi-factor method, risk is represented as a vector of sensitivities to unexpected changes of risk factors. Subsequently, the sensitivities themselves are related to firm characteristics.In addition, an application of the method to interfirm comparison is presented. This application is illustrated by a numerical example based on estimates concerning real data. Apart from this application, some other possible future applications are mentioned. Finally, some decision support tools are presented which may enhance the usefulness of the method in practice.     

A new approach to grid generation

An entirely new approach to numerical grid generation, the deformation method, is presented. Each point of an existing grid is moved, in an inter-related manner, to a new position according to a system of n ordinary differential equations (n=spacial dimension). The resulting grid has prescribed mesh sizes.     

A new approach to numerical differentiation

In this paper we consider the numerical differentiation of functions specified by noisy data. A new approach, which is based on an integral equation of the first kind with a suitable compact operator, is presented and discussed. Since the singular system of the compact operator can be obtained easily, TSVD is chosen as the needed regularization technique and we show that the method calls for a discrete sine transform, so the method can be implemented easily and fast. Numerical examples are also given to show the efficiency of the method.     

A new approach to spectral approximation

A new technique for approximating eigenvalues and eigenvectors of a self-adjoint operator is presented. The method does not incur spectral pollution, uses trial spaces from the form domain, has a self-adjoint algorithm, and exhibits superconvergence.     

A new approach to Cagniard's problem

Cagniard problem refers to the class of linear reflection and transmission problem for pulsed line and point sources, which have solution methods leading to exact algebraic representations of the wave fields. All previous methods have relied heavily on integral or differential transforms. We present in this paper a new and direct approach to the problem which involves only the wave equation and its associated characteristic equation. We illustrate the new method by applying it to the problem of the reflection and transmission of acoustic waves radiating from a line source in the vicinity of a plane boundary separating two uniform acoustic media.     

A new approach to strong embeddings

We revisit strong approximation theory from a new perspective, culminating in a proof of the Komlós–Major–Tusnády embedding theorem for the simple random walk. The proof is almost entirely based on a series of soft arguments and easy inequalities. The new technique, inspired by Stein’s method of normal approximation, is applicable to any setting where Stein’s method works. In particular, one can hope to take it beyond sums of independent random variables.     

A new approach to chordal graphs

By a chordal graph is meant a graph with no induced cycle of length ⩾ 4. By a ternary system is meant an ordered pair (W, T), where W is a finite nonempty set, and T ⊆ W × W × W. Ternary systems satisfying certain axioms (A1)–(A5) are studied in this paper; note that these axioms can be formulated in a language of the first-order logic. For every finite nonempty set W, a bijective mapping from the set of all connected chordal graphs G with V(G) = W onto the set of all ternary systems (W, T) satisfying the axioms (A1)–(A5) is found in this paper.