Linear algebra with applications

Lecture slides and notes for a PhD level course on linear algebra for electrical engineers and computer scientists. This course is given in in the framework of the School of Electronics and Computer Science Mathematics Training Courses https://secure.ecs.soton.ac.uk/notes/pg_maths/ (ECS password required)

Alternating sums of binomial coefficients with unit fraction arithmetic sequence coefficients

Expressions for finite sums involving the binomial coefficients with unit fraction coefficients whose denominators form an arithmetic sequence are determined.

Tests of concrete beams reinforced with unit frames.

Thesis (B.S.)--University of Illinois at Urbana-Champaign, 1910.

A treatise on universal algebra, with applications.

Available on demand as hard copy or computer file from Cornell University Library.

An introduction to algebra : with notes and observations : designed for the use of schools and places of public education : to which is added an appendix on the application of algebra to geometry /

Mode of access: Internet.

Algebra, with Arithmetic and mensuration, from the Sanscrit of Brahmegupta and Bháscara.

The 12th and 18th chapters, Gaṇitādhyāya and Kuṭṭakādhyāya, of Brahmagupta's Brahmasiddhānta ; and the first two parts, Līlāvatī and Bījagaṇita, of Bhāskara's Siddhāntaśirḿanṇi.

The exact sample complexity of PAC-learning problems with unit VC dimension

The Vapnik-Chervonenkis (VC) dimension is a combinatorial measure of a certain class of machine learning problems, which may be used to obtain upper and lower bounds on the number of training examples needed to learn to prescribed levels of accuracy. Most of the known bounds apply to the Probably Approximately Correct (PAC) framework, which is the framework within which we work in this paper. For a learning problem with some known VC dimension, much is known about the order of growth of the sample-size requirement of the problem, as a function of the PAC parameters. The exact value of sample-size requirement is however less well-known, and depends heavily on the particular learning algorithm being used. This is a major obstacle to the practical application of the VC dimension. Hence it is important to know exactly how the sample-size requirement depends on VC dimension, and with that in mind, we describe a general algorithm for learning problems having VC dimension 1. Its sample-size requirement is minimal (as a function of the PAC parameters), and turns out to be the same for all non-trivial learning problems having VC dimension 1. While the method used cannot be naively generalised to higher VC dimension, it suggests that optimal algorithm-dependent bounds may improve substantially on current upper bounds.

Powers and Logarithms

There are applied power mappings in algebras with logarithms induced by a given linear operator D in order to study particular properties of powers of logarithms. Main results of this paper will be concerned with the case when an algebra under consideration is commutative and has a unit and the operator D satisfies the Leibniz condition, i.e. D(xy) = xDy + yDx for x, y ∈ dom D. Note that in the Number Theory there are well-known several formulae expressed by means of some combinations of powers of logarithmic and antilogarithmic mappings or powers of logarithms and antilogarithms (cf. for instance, the survey of Schinzel S[1].

Students' engagement with Facebook in a university undergraduate policing unit

Web 2.0 technologies are increasingly being used to support teaching in higher education courses. However, preliminary research has shown that students are using such technologies primarily for social purposes, rather than as a means of further engaging with academic content. This study examines a cohort of tertiary students' use of a Facebook page, which was created for a second year university policing unit at the Queensland University of Technology in Brisbane, Australia. Results from content analysis of the Facebook "wall" and a survey of student users and non-users showed that although students only demonstrated very little active engagement with academic content posted on the site (that is, they were reluctant to interact with unit materials in a way that would leave a digital trace), they reported that Facebook had increased their ability to engage with and critically analyse the unit content. In alignment with other research in this area, students also reported the usefulness of the Facebook page for increasing communication with their peers and with the teaching staff. This paper concludes by offering a number of best practice guidelines for the use of Facebook in tertiary education.

Algebra, or universal mathematics: revived 1738 with notes and additions : manuscript, [after 1738]

Instructional book in algebra with exercises.

Pseudo-merohedral twinning in the structure of the hydrated 1:1 proton-transfer compound of 5-sulfosalicylic acid with 4-aminopyridine

The structure of the pseudo-merohedrally twinned crystal of the 1:1 proton-transfer compound of 5-sulfosalicylic acid (3-carboxy-4-hydroxybenzenesulfonic acid) with 4-aminopyridine: 4-aminopyridinium 3-carboxy-4-hydroxybenzenesulfonate sesquihydrate has been determined at 180 K and the hydrogen-bonding pattern is described. Crystals of the compound are monoclinic with space group P21/c, with unit cell dimensions a = 35.2589(8), b = 7.1948(1), c = 24.5851(5) Å, β = 110.373(2)o, and Z = 16. The monoclinic asymmetric unit comprises four cation-anion pairs and six water molecules of solvation with only the pyridinium cations having pseudo-symmetry as a result of inter-cation aromatic ring π-π stacking effects. Extensive hydrogen bonding gives a three-dimensional framework structure.