On real eigenvalues of real tridiagonal matrices

Lower bounds for the number of different real eigenvalues as well as for the number of real simple eigenvalues of a class of real irreducible tridiagonal matrices are given. Some numerical implications are discussed.     

Existence of real eigenvalues of real tensors

We use the Brouwer degree to establish the existence of real eigenpairs of higher order real tensors in various settings. Also, we provide some finer criteria for the existence of real eigenpairs of two-dimensional real tensors and give a complete classification of the Brouwer degree zero and ±2 maps induced by general third order two-dimensional real tensors.     

Defining continuity of real functions of real variables

Continuity of a real function of a real variable has been defined in various ways over almost 200 years. Contrary to popular belief, the definitions are not all equivalent, because their consequences for four somewhat pathological functions reveal five essentially different cases. The four defensible ones imply just two cases for continuity on an interval if that is defined by using pointwise continuity at each point. Some authors had trouble: two different textbooks each gave two arguably inconsistent definitions, three more changed their definitions in their second editions, two more claimed continuity at a point for functions not defined there, and one gave a definition implying it for a function with no limit there.     

Typically real functions with singularities on the real axis

This article considers the class ^k of typically real functions in the region G=CK, where K is a closed set on the real axis. An integral representation is given for the class ^k and it is used to establish a number of extremal properties for functions of the class ^k and associated classes of functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 60–71, 1990.     

Strongly real hilbert rings and the real jacobson semisimplicity

In this paper, several false results in reference [1] related to rela Hilbert rings and the ral Jacobson semisimplicity are negated by a counterexample. By introducing the notion of a strongly real Hilbert ring, we characterize those rings of which every finitely generated real extension is real jacobson semisimple. Moreover, the so-called strictly real Hilbert rings are considered.     

Rankin-Selberg method for real analytic cusp forms of arbitrary real weight

Supported by a grant of the Deutsche Forschungsgemeinschaft (DFG)     

Non-special divisors on real algebraic curves and embeddings into real projective spaces

We show that there is a large class of non-special divisors of relatively small degree on a given real algebraic curve. If the real algebraic curve has many real components, such a divisor gives rise to an embedding (birational embedding, resp.) of the real algebraic curve into the real projective space ℙ ^r for r≥3 (r=2, resp.). We study these embeddings in quite some detail. Received: October 17, 2001?Published online: February 20, 2003     

A package for symbolic solution of real functional equations of real variables

Summary This paper discusses the problems associated with the symbolic treatment of functional equations and presents a Mathematica package for the solution of real functional equations of real variables. The package includes a minimal basic database which contains a reduced set of functional equations with its four components: equation, domain, class and the corresponding solution. The word minimal is used in the sense that any equation that is solvable by the system using non-searching methods is excluded from the database. The package incorporates a searching algorithm which can solve functional equations independently of their notation and their algebraic representation. Not only general solutions but particular and candidate solutions are dealt with. This implies a careful analysis of domains and classes. The package includes some methods for solving functional equations, which are used when the input functional equations are not found in the database. Some methods have been implemented internally and some are in an external package. Finally, some examples illustrate the use of the package.     

Colouring the real line

The problem of colouring the real line so that the distance between like coloured numbers does not lie in some specified set D, called the distance set, is discussed. In particular, the minimum number of colours needed for various distance sets are determined.     

Roots of real matrices

An algorithm for computing the roots of a matrix with real elements and the real part of the eigenvalues not zero will be described. The algorithm is based on the sign matrix given in an earlier paper and the projectors of the matrix.     

A real Riemann-Hurwitz theorem

We prove a real version of the Riemann-Hurwitz theorem and apply it to solve a problem of enumerative geometry in the real case: the number of plane projective curves tangent to a line and passing through the appropriate number of points.     

On formally real modules

The aim of this paper is to develop the study of modules in real algebra. By introducing two kinds of reality for modules over commutative rings, we establish some results on semireal modules and real modules. Our results involve real radicals of submodules and orderings on modules.