A functional method for linear sets,II.

Various exceptional sets (such as Helson sets and sets of multiplicity) are constructed by means of a functional method using differentiable mappings. An erratum to this article is available at .     

A closed form type reduction method for piecewise linear interval type-2 fuzzy sets

In this study, a new centroid type reduction method is proposed for piecewise linear interval type-2 fuzzy sets based on geometrical approach. The main idea behind the proposed method relies on the assumption that the part of footprint of uncertainty (FOU) of an interval type-2 fuzzy set (IT2FS) has a constant width where the centroid is searched. This constant width assumption provides a way to calculate the centroid of an IT2FS in closed form by using derivative based optimization without any need of iterations. When the related part of FOU is originally constant width, the proposed method finds the accurate centroid of an IT2FS; otherwise, an enhancement can be performed in the algorithm in order to minimize the error between the accurate and the calculated centroids. Moreover, only analytical formulas are used in the proposed method utilizing geometry. This eliminates the need of using discretization of an IT2FS for the type reduction process which in return naturally improves the accuracy and the computation time. The proposed method is compared with Enhanced Karnik–Mendel Iterative Procedure (EKMIP) in terms of the accuracy and the computation time on seven test fuzzy sets. The results show that the proposed method provides more accurate results with shorter computation time than EKMIP.     

Solution-free sets for linear equations

We study the maximum size and the structure of sets of natural numbers which contain no solution of one or more linear equations. Thus, for every natural i and k?2, we find the minimum α=α(i,k) such that if the upper density of a strongly k-sum-free set is at least α, then A is contained in a maximal strongly k-sum-free set which is a union of at most i arithmetic progressions. We also determine the maximum density of sets of natural numbers without solutions to the equation x=y+az, where a is a fixed integer.     

A characterization of vNM-stable sets for linear production games

We discuss linear production games or market games with a continuum of players which are represented as minima of finitely many nonatomic measures.?Within this context we consider vNM-Stable Sets according to von Neumann and Morgenstern. We classify or characterize all solutions of this type which are convex polyhedra, i.e., which are the convex hull of finitely many imputations. Specifically, in each convex polyhedral vNM-Stable Set (and not only in the symmetric ones), the different types of traders must organize themselves into cartels. The vNM-Stable Set is then the convex hull of the utility distributions of the cartels.?Using the results from the continuum, we obtain a similar characterization also for finite glove market games. Received December 1998/Revised version June 1999     

Stability of sets for impulsive functional differential equations via Razumikhin method

In this paper, we consider the functional differential equation with impulsive perturbations

A taylor collocation method for solving high‐order linear pantograph equations with linear functional argument

A numerical method based on the Taylor polynomials is introduced in this article for the approximate solution of the pantograph equations with constant and variable coefficients. Some numerical examples, which consist of the initial conditions, are given to show the properties of the method. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27:1628–1638, 2011     

A procedure for designing linear functional observers

We present low-order functional observer existence conditions; the proposed existence conditions always lead to a design of functional observers of order less than or equal to that of a Luenberger observer. The proposed low-order observer always exists if a Luenberger observer exists.     

A geometric characterisation of linear k-blocking sets

In this paper we prove that any linear k-blocking set is either a canonical subgeometry or a projection of some canonical subgeometry. Received 6 February 2001.     

A Taylor method for numerical solution of generalized pantograph equations with linear functional argument

This paper is concerned with a generalization of a functional differential equation known as the pantograph equation which contains a linear functional argument. In this paper, we introduce a numerical method based on the Taylor polynomials for the approximate solution of the pantograph equation with retarded case or advanced case. The method is illustrated by studying the initial value problems. The results obtained are compared by the known results.     

A sign-based linear method for horizontal linear complementarity problems

Numerical Algorithms - For the horizontal linear complementarity problem, we establish a linear method based on the sign patterns of the solution of the equivalent modulus equation under the...     

Convex vNM-stable sets for linear production games

We characterize convex vNM-Stable Sets according to von Neumann and Morgenstern for orthogonal linear production games with a continuum of players. The results of Rosenmüller and Shitovitz (Int J Game Theory 29, 39–61, 2000) are thereby substantially improved. Simultaneously, this is a corrigendum concerning the proof of a lemma in our first paper.     

Attainable sets for linear stochastic control systems

Let x_t^u(w) be the solution process of the n-dimensional stochastic differential equation dx_t^u = [A(t)x_t^u + B(t) u(t)] dt + C(t) dW_t, where A(t), B(t), C(t) are matrix functions, W_t is a n-dimensional Brownian motion and u is an admissable control function. For fixed ? ? 0 and 1 ? δ ? 0, we say that x?Rⁿ is (?, δ) attainable if there exists an admissable control u such that P{x_t^u?S_?(x)} ? δ, where S_?(x) is the closed ?-ball in Rⁿ centered at x. The set of all (?, δ) attainable points is denoted by $\text{A}$(t). In this paper, we derive various properties of $\text{A}$(t) in terms of K(t), the attainable set of the deterministic control system $\text{x}\text{?}\text{= A(t)x + B(t)u}$. As well a stochastic bang-bang principle is established and three examples presented.     

Alternating methods for sets of linear equations

Summary A generalization of alternating methods for sets of linear equations is described and the number of operations calculated. It is shown that the lowest number of arithmetic operations is achieved in the SSOR algorithm.     

A smoothing method for sparse optimization over convex sets

In this paper, we investigate a class of heuristic schemes to solve the NP-hard problem of minimizing $$ell _0$$-norm over a convex set. A well-known appr