The set of dynamically special points

We call a point ??dynamically special?? if it has a dynamical property, which no other point has. We prove that, for continuous self maps of the real line, all dynamically special points are in the closure of the union of the full orbits of periodic points, critical points and limits at infinity. We completely describe the set of dynamically special points of real polynomial functions. The following characterization for the set of special points is also obtained: A subset of ${\mathbb{R}}$ is the set of dynamically special points for some continuous self map of ${\mathbb{R}}$ if and only if it is closed.     

Special polynomials associated with rational solutions of some hierarchies

New special polynomials associated with rational solutions of the Painlevé hierarchies are introduced. The Hirota relations for these special polynomials are found. Differential-difference hierarchies to find special polynomials are presented. These formulae allow us to search special polynomials associated with the hierarchies. It is shown that rational solutions of the Caudrey–Dodd–Gibbon, the Kaup–Kupershmidt and the modified hierarchy for these ones can be obtained using new special polynomials.     

Efficiently solvable special cases of hard combinatorial optimization problems

We survey some recent advances in the field of polynomially solvable special cases of hard combinatorial optimization problems like the travelling salesman problem, quadratic assignment problems and Steiner tree problems. Such special cases can be found by considering special cost structures, the geometry of the problem, the special topology of the underlying graph structure or by analyzing special algorithms. In particular we stress the importance of recognition algorithms. We comment on open problems in this area and outline some lines for future research in this field. This research has been supported by the Spezialforschungsbereich F 003 “Optimierung und Kontrolle”, Projektbereich Diskrete Optimierung.     

Homology stability for orthogonal groups over algebraically closed fields

We give the best ranges of stability, for homology of orthogonal groups and special orthogonal groups, over an algebraically closed field, of characteristic different from 2. This answers affirmatively a conjecture asserted in a previous paper. We find distinct ranges of stability for orthogonal and special orthogonal groups, and Milnor K-theory appears as an obstruction to stability for special orthogonal groups. The main results are formulated, more generally, for infinite Pythagorean fields.     

Radicals of <Emphasis Type="Italic">l</Emphasis>-rings and special classes of <Emphasis Type="Italic">l</Emphasis>-modules

A special class of lattice-ordered modules is studied. We show that for any special class of l-modules we can define a special class of l-rings. The special radical of an l-ring R can be represented as the intersection of the l-annihilators of l-modules over R belonging to the special class. The prime radical of an l-ring R can be represented as the intersection of the l-annihilators of l-prime l-modules over R.     

Classes of Almost Clean Rings

A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS (i.e. right extending) and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite AW ^*-algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring R is special clean (special almost clean) if each element a can be decomposed as the sum of a unit (regular element) u and an idempotent e with aR?∩?eR?=?0. The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart. If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasi-continuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean. Finally, we adapt some of our results to rings with involution: a *-ring is *-clean (almost *-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) *-clean ring is similarly defined by replacing “idempotent” with “projection” in the appropriate definition. We show that an abelian *-ring is a Rickart *-ring if and only if it is special almost *-clean, and that an abelian *-ring is *-regular if and only if it is special *-clean.     

Sister Celine's technique and its generalizations

Sister Celine Fasenmyer's technique for obtaining pure recurrence relations for hypergeometric polynomials is formalized and generalized in various directions. Applications include algorithms for verifying any given binomial coefficients identity and any identities involving sums and integrals of products of special functions. This is shown to lead to a new approach to the theory of special functions which allows a natural definition of special functions of several variables.     

Construction of polynomials of a special kind and examples of their application in the theory of power series and in the wavelet theory

In this paper we construct polynomials of a special type. We consider examples of their application for studying the convergence of special power series, for determining the upper and lower Riesz bounds for a basis consisting of B-splines, and for studying the convergence of a sequence of Battle-Lemarié scaling functions.     

A Note on Bernoulli Numbers and Shintani Generalized Bernoulli Polynomials

Generalized Bernoulli polynomials were introduced by Shintani in 1976 in order to express the special values at non-positive integers of Dedekind zeta functions for totally real numbers. The coefficients of such polynomials are finite combinations of products of Bernoulli numbers which are difficult to get hold of. On the other hand, Zagier was able to get the explicit formula for the special values in cases of real quadratic number fields.

In this paper, we shall improve Shintani's formula by proving that the special values can be determined by a finite set of polynomials. This provides a convenient way to evaluate the special values of various types of Dedekind functions. Indeed, a much broader class of zeta functions considered by the author [4] admits a similar formula for its special values. As a consequence, we are able to find infinitely many identities among Bernoulli numbers through identities among zeta functions. All these identities are difficult to prove otherwise.

     

Combinatorial invariance of Kazhdan-Lusztig polynomials on intervals starting from the identity

We show that for Bruhat intervals starting from the identity in Coxeter groups the conjecture of Lusztig and Dyer holds, that is, the R-polynomials and the Kazhdan-Lusztig polynomials defined on [e,u] only depend on the isomorphism type of [e,u]. To achieve this we use the purely poset-theoretic notion of special matching. Our approach is essentially a synthesis of the explicit formula for special matchings discovered by Brenti and the general special matching machinery developed by Du Cloux.     

Non-reversibility and self-joinings of higher orders for ergodic flows

By studying the weak closure of multidimensional off-diagonal self-joinings, we provide a sufficient condition for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid automorphisms. In particular, we apply the criterion to special flows over irrational rotations, providing a large class of non-reversible flows, including some analytic reparametrizations of linear flows on $\mathbb{T}^2$ , so-called von Neumann flows and some special flows with piecewise polynomial roof functions. A topological counterpart is also developed with the full solution of the problem of the topological self-similarity of continuous special flows over irrational rotations. This yields examples of continuous special flows over irrational rotations without non-trivial topological self-similarities and having all non-zero real numbers as scales of measure-theoretic self-similarities.     

Ruled Special Lagrangian 3-Folds In C3

This is the fifth in a series of papers constructing explicitexamples of special Lagrangian submanifolds in C^m. A submanifoldof C^m is ruled if it is fibred by a family of real straightlines in C^m. This paper studies ruled special Lagrangian 3-foldsin C³, giving both general theory and families of examples.Our results are related to previous work of Harvey and Lawson,Borisenko, and Bryant. Special Lagrangian cones in C³ are automaticallyruled, and each ruled special Lagrangian 3-fold is asymptoticto a unique special Lagrangian cone. We study the family ofruled special Lagrangian 3-folds N asymptotic to a fixed specialLagrangian cone N0. We find that this depends on solving a linearequation, so that the family of such N has the structure ofa vector space. We also show that the intersection of N0 withthe unit sphere S⁵ in C³ is a Riemann surface, and constructa ruled special Lagrangian 3-fold N asymptotic to N0 for eachholomorphic vector field w on . As corollaries of this we writedown two large families of explicit special Lagrangian 3-foldsin C³ depending on a holomorphic function on C, which includemany new examples of singularities of special Lagrangian 3-folds.We also show that each special Lagrangian T2-cone N0 can beextended to a 2-parameter family of ruled special Lagrangian3-folds asymptotic to N0, and diffeomorphic to T²xR. 2000 Mathematical Subject Classification: 53C38, 53D12.     

(T, E)-Korovkin Closures in Normed Spaces and BKW-Operators

We give a characterization of BKW-operators on special normed spaces and determine completely a class of BKW-operators from a function space on [0, 1] into C(Ω) for special test functions.     

Program Explanation and Higher-Order Properties

Our aim in this paper is to evaluate Frank Jackson and Philip Pettit’s ‘program explanation’ framework as an account of the autonomy of the special sciences. We argue that this framework can only explain the autonomy of a limited range of special science explanations. The reason for this limitation is that the framework overlooks a distinction between two kinds of properties, which we refer to as ‘higher-level’ and ‘higher-order’ properties. The program explanation framework can account for the autonomy of special science explanations that appeal to higher-level properties but it does not account for the autonomy of most of those explanations that appeal to higher-order properties.     

Norm equalities and inequalities for three circulant operator matrices

In this paper, three new circulant operator matrices, scaled circulant operator matrices, diag-circulant operator matrices and retrocirculant operator matrices, are given respectively. Several norm equalities and inequalities for these operator matrices are proved. We show the special cases for norm equalities and inequalities, such as the usual operator norm and the Schatten p-norm. Pinching type inequality is also given for weakly unitarily invariant norms. These results are closely related to the nice structure of these special operator matrices. Furthermore, some special cases and specific examples are also considered.     

Generalization of the simplest equation method for nonlinear non-autonomous differential equations

It is known that the simplest equation method is applied for finding exact solutions of autonomous nonlinear differential equations. In this paper we extend this method for finding exact solutions of non-autonomous nonlinear differential equations (DEs). We applied the generalized approach to look for exact special solutions of three Painlevé equations. As ODE of lower order than Painlevé equations the Riccati equation is taken. The obtained exact special solutions are expressed in terms of the special functions defined by linear ODEs of the second order.     

On some sheaves of special groups

Using sheaves of special groups, we show that a general local-global principle holds for every reduced special group whose associated space of orderings only has a finite number of accumulation points. We also compute the behaviour of the Boolean hull functor applied to sheaves of special groups. The research leading to this note was carried out with the partial support of the European RTN Networks HPRN-CT-2002-00287 “Algebraic K-Theory, Linear Algebraic Groups and Related Structures”, and HPRN-CT-2001-00271 “Real Algebraic and Analytic Geometry”     

SPECIAL RADICALS OF NEAR-RING MODULES

Abstract

Equiprime near-rings, which generalize the concept of prime-ness in rings, were defined by the present authors, together with S. Veldsman. This concept was shown in subsequent work to lead to a very satisfactory theory of special radicals for near-rings. In the current paper, we define equiprime N-groups for a near-ring N. It is shown that an ideal A of N is equiprime if and only if it is the annihilator of an equiprime TV-group G. Special classes of near-ring modules are defined, and a module-theoretic characterization of special radicals of near-rings is established, similar to that given by Andrunakievich and Rjabuhin for special radicals of rings.     

Advanced computing solutions for health care and medicine

This guest editorial introduces the special issue on “Advanced Computing Solutions for Health Care and Medicine”. The goal of this special issue was to collect high quality papers describing the application of computer science methods and techniques to main health care and clinical problems, resulting in high performance applications or prototypes for medical and clinical environments. The special issue touched different health informatics hot topics and is organized in four sections: (i) clinical decision support systems; (ii) biomedical imaging; (iii) high performance computing and biomedical simulations; (iv) bioinformatics data analysis.     

Four-point conditions for the TSP: The complete complexity classification

The combinatorial optimization literature contains a multitude of polynomially solvable special cases of the traveling salesman problem (TSP) which result from imposing certain combinatorial restrictions on the underlying distance matrices. Many of these special cases have the form of so-called four-point conditions: inequalities that involve the distances between four arbitrary cities.In this paper we classify all possible four-point conditions for the TSP with respect to computational complexity, and we determine for each of them whether the resulting special case of the TSP can be solved in polynomial time or whether it remains NP-hard.