Topic
Pencil (mathematics)
About: Pencil (mathematics) is a research topic. Over the lifetime, 3678 publications have been published within this topic receiving 37917 citations.
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Book•
01 Jan 1983
TL;DR: This paper presents a meta-modelling framework for developing and assessing regression models for multivariate and multi-dimensional data distributions and describes the distribution of a set of data.
Abstract: This book present graphical methods for analysing data. Some methods are new and some are old, some require a computer and others only paper and pencil; but they are all powerful data analysis tools. In many situations, a set of data � even a large set- can be adequately analysed through graphical methods alone. In most other situations, a few well-chosen graphical displays can significantly enhance numerical statistical analyses.
1,763 citations
Book•
01 Jan 1952
TL;DR: This chapter discusses the origins and development of chemical knowledge, and the role of atoms and molecules in the development of knowledge.
Abstract: PART 1: THE ORIGINS AND DEVELOPMENT OF GEOMETRICAL KNOWLEDGE PART 2: ABSTRACT PROJECTIVE GEOMETRY
778 citations
Book•
20 Feb 1986
TL;DR: Projective Geometrics Over Finite Fields (OUP, 1979) as mentioned in this paper considers projective spaces of three dimensions over a finite field and examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory.
Abstract: This self-contained and highly detailed study considers projective spaces of three dimensions over a finite field It is the second and core volume of a three-volume treatise on finite projective spaces, the first volume being Projective Geometrics Over Finite Fields (OUP, 1979) The present work restricts itself to three dimensions, and considers both topics which are analogous of geometry over the complex numbers and topics that arise out of the modern theory of incidence structures The book also examines properties of four and five dimensions, fundamental applications to translation planes, simple groups, and coding theory
713 citations
TL;DR: In this article, it has been shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring.
Abstract: Serre [9, ?50] has shown that there is a one-to-one correspondence between algebraic vector bundles over an affine variety and finitely generated projective modules over its coordinate ring. For some time, it has been assumed thlat a similar correspondence exists between topological vector bundles over a compact Hausdorff space X and finitely generated projective modules over the ring of continuous real-valued functions on X. A number of examples of projective modules have been given using this correspondence. However, no rigorous treatment of the correspondence seems to have been given. I will give such a treatment here and then give some of the examples which may be constructed in this way.
598 citations
TL;DR: In this article, it was shown that the genus of a smooth 2-manifold can be characterized by the formula g = (d − 1)(d − 2)/2.
Abstract: 1. Statement of the result The genus of a smooth algebraic curve of degree d in CP is given by the formula g = (d − 1)(d − 2)/2. A conjecture sometimes attributed to Thom states that the genus of the algebraic curve is a lower bound for the genus of any smooth 2-manifold representing the same homology class. The conjecture has previously been proved for d ≤ 4 and for d = 6, and less sharp lower bounds for the genus are known for all degrees [5,6,7,10,14]. In this note we confirm the conjecture. Theorem 1. Let Σ be an oriented 2-manifold smoothly embedded in CP so as to represent the same homology class as an algebraic curve of degree d. Then the genus g of Σ satisfies g ≥ (d − 1)(d − 2)/2.
562 citations