Abstract:
The maximum possible number of non-overlapping unit spheres that can touch a unit sphere in $n$ dimensions is called kissing number. The problem for finding kissing numbers is closely connected to the more general problems of finding bounds for spherical codes and sphere packings. We survey old and recent results on the kissing numbers keeping the generality of spherical codes.

Abstract:
We present a comprehensive survey of constructions of the real numbers (from either the rationals or the integers) in a unified fashion, thus providing an overview of most (if not all) known constructions ranging from the earliest attempts to recent results, and allowing for a simple comparison-at-a-glance between different constructions.

Abstract:
The Lyubeznik numbers are invariants of a local ring containing a field that capture ring-theoretic properties, but also have numerous connections to geometry and topology. We discuss basic properties of these integer-valued invariants, as well as describe some significant results and recent developments (including certain generalizations) in the area.

Abstract:
A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of an algebraic integer are also discussed. Some generalisations are given too, though not to Mahler measure of polynomials in more than one variable.

Abstract:
The bondage number of a nonempty graph $G$ is the cardinality of a smallest edge set whose removal from $G$ results in a graph with domination number greater than the domination number of $G$. This lecture gives a survey on the bondage number, including the known results, problems and conjectures. We also summarize other types of bondage numbers.

Abstract:
Purpose: In European Union countries the legalisation forbids the production, processing and use of cadmium. By January 2008 at the latest, all articles and products containing cadmium will either have to be withdrawn from sale or an appropriate substitute for this heavy metal will have to be found.Design/methodology/approach: The present technology of production of fuses in Slovenian firm ETI Elektroelement, and the action thereof are adapted to the existing ecologically harmful alloy of tin and cadmium SnCd20, which ought to be replaced by one or more ecologically safe alloys with technological and application properties as similar as possible to the existing ones.Findings: In the frame of the presented investigation work we have found that practically all stated problems can be successfully solved by the low melting alloy of tin, bismuth and antimony named ETI-Sn-Bi-Sb.Research limitations/implications: Alloy ETI-Sn-Bi-Sb is ecologically safe, and by its technical and physical properties (melting point, conductivity, wettability) corresponds to the requirements of the use for fusible elements of low voltage fuses.Practical implications: Practical implications of our common work is in the introduction of new ecologically safe material for fusible elements, without cadmium in the existing technology of low voltage fuses.Originality/value: High value and originality and of our engineering work is confirmed by European Union patent and two Slovenian national patents for the ecologically safe low melting alloy named ETI-Sn-Bi-Sb, which received authors of this paper and Slovenian firm ETI Elektroelement.

Abstract:
The domination number of a graph is the smallest number of vertices which dominate all remaining vertices by edges of . The bondage number of a nonempty graph is the smallest number of edges whose removal from results in a graph with domination number greater than the domination number of . The concept of the bondage number was formally introduced by Fink et al. in 1990. Since then, this topic has received considerable research attention and made some progress, variations, and generalizations. This paper gives a survey on the bondage number, including known results, conjectures, problems, and some comments, also selectively summarizes other types of bondage numbers. 1. Introduction For terminology and notation on graph theory not given here, the reader is referred to Xu [1]. Let be a finite, undirected, and simple graph. We call and the order and size of and denote them by and , respectively, unless otherwise specified. Through this paper, the notations , , and always denote a path, a cycle, and a complete graph of order , respectively, the notation denotes a complete -partite graph with and , with , and is a star. For two vertices and in a connected graph , we use to denote the distance between and in . For a vertex in , let be the open set of neighbors of and the closed set of neighbors of . For a subset , , and , where . Let be the set of edges incident with in ; that is, . We denote the degree of by . The maximum and the minimum degrees of are denoted by and , respectively. A vertex of degree zero is called an isolated vertex. An edge incident with a vertex of degree one is called a pendant edge. The bondage number is an important parameter of graphs which is based upon the well-known domination number. A subset is called a dominating set of if ; that is, every vertex in has at least one neighbor in . The domination number of , denoted by , is the minimum cardinality among all dominating sets; that is, A dominating set is called a -set of if . The domination is such an important and classic conception that it has become one of the most widely studied topics in graph theory and also is frequently used to study property of networks. The domination, with many variations and generalizations, is now well studied in graph and networks theory. The early vast literature on domination includes the bibliography compiled by Hedetniemi and Laskar [2] and a thorough study of domination appears in the books by Haynes et al. [3, 4]. However, the problem determining the domination number for general graphs was early proved to be NP-complete (see GT2 in Appendix in

Abstract:
This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a construction of natural numbers as a special kind of ordinal. In any case, the natural numbers can be understood as composing a free algebra in a certain signature, {0,s}. The paper here culminates in a construction of, for each algebraic signature S, a class ON_S that is to the class of ordinals as S is to {0,s}. In particular, ON_S has a subclass that is a free algebra in the signature S.

From our standpoint, a school teacher should be
acquainted, in a deeper way, with the content of teaching themes which are on school
curricula. In case of primary teachers and their preparation for teaching mathematics,
they should have a solid knowledge of the properties of the system N of natural numbers and an understanding
of its position as being a basis upon which all other number systems are built.
Up to some degree, these teachers should also be acquainted with further
extensions of number systems going along the line as it is done in school: natural—positive rational—integer—rational—real
numbers. These extensions are enlightened by Peacock’s principle of invariance
of the form—a rule derived for natural numbers, when expressed in general form (as
a literal relation) continues to hold true in all extended systems. In Section
2 of this survey, a precise terminology is fixed which is needed for the study
of the system N and in particular, for
making a difference between syntactic and semantic concepts. The Cantor principle
which expresses the dependence of conception of number on perception of set is also
formulated and largely exploited in this paper. In Section 3, several rules are
derived when different expressions denoting two different groupings of elements
of a set are equated. Forgetting that the variables are bound to N,all these rules also express the properties of the extended number systems,
as well as they are algebraic laws or their derivatives. At the end, discovering
of rules of correspondence of sequences given by a number of their initial
terms is considered as a type of exercises which help the development of the idea
of variable. Some cases of finding formulae for sums of consecutive natural numbers
are also included. This paper is intended to be a paradigmatic example how a mathematical
content has to be elaborated to serve best the school teachers to deepen their
knowledge of subject matter.