On Power Integral Bases for Certain Pure Number Fields Defined by x 36 − m

Let K = ℚ(α) be a number field generated by a complex root a of a monic irreducible polynomial ƒ (x) = x36 − m, with m ≠ ±1 a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≠ ±1 (mod 9) then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡±1 (mod 9), then the number field K is not monogenic.

On monogenity of certain pure number fields defined by xpr − m

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

A measure of innovation performance: the Innovation Patent Index

PurposeThe measure of companies' Innovation Performance is fundamental for enhancing the value and decision-making processes of firms. The purpose of this paper is to present a new measure of Innovation Performance, called Innovation Patent Index (IPI), which makes it possible to quantitatively summarize different aspects of firms' innovation.Design/methodology/approachIn order to define the IPI, a secondary source, i.e. patent data, has been used. The five dimensions of IPI, i.e. efficiency, time, diversification, quality and internationalization have been defined both analyzing the literature and applying three different machine learning algorithms (regularized least squares, deep neural networks and decision trees), considering patent forward citations as a proxy of the innovation performance.FindingsResults show that the IPI index is a very useful tool, simple to use and very promptly. In fact, it is possible to get important results without making time consuming analysis with primary sources. It is a tool that can be used by managers, businessmen, policymakers, organizations, patent experts and financiers to evaluate and plan future activities, to enhance the innovation capability, to find financing and to support and improve innovation.Research limitations/implicationsPatent data are not widely used in all the sectors. Moreover, the pure number of forward citations is not the only forward looking indicator suggested by the literature.Originality/valueThe demand for a useable Innovation Performance tool, as well as the lack of tools able to grasp different aspects of the innovation, highlight the need to develop new instruments. In fact, although previous studies provide several measures of Innovation Performance, these are often difficult for managers to use, do not appreciate different aspects of the innovation and are not forward looking.

Epistemological considerations about mathematical concepts

Purpose Categories (particular (P) and general (V)) constitute a bipole with epistemological implications. The mutual categorical implication of this bipole is embodied in ordinary notions. It follows that a concept because it forms an element of concrete, sensible-rational, practical-theoretical activity has to unite the two inseparable poles, the general and the particular. If the concept of a physical quantity is abstract in relation to the physical object, it is concrete in comparison with mathematical quantity. This product of a secondary abstraction covers the background of physical qualities to extract the pure number, legitimately named abstract number. Both kinds of numbers are mutually exclusive: either the numbers are attached to a unit name and the number is concrete or nothing is attached and the number is abstract. However, in addition to their coordination in extension, they involve each other in comprehension: in fact, the pure number is the general pole V and concrete numbers form the particular pole of the dialectical concept of number K. The purpose of this paper is to provide a model for epistemological issues that arise in the context of meaning, concepts and use of words. Design/methodology/approach A dialectical theory of the binomial comprehension-extension of mathematical magnitudes. Findings The findings provide an objection to the traditional deductive order being also true in mathematics, and also that the reverse order cannot be considered as characteristic of mathematics, but show dialectic as universal. This opens the way to the special scientific deduction (mathematical, physical, biological, etc). going from the general to individual. Originality/value The structure of the mathematical concepts is elaborated.