SUMMARY / RELATED TOPICS

Topology

In mathematics, topology is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. A topological space is a set endowed with a structure, called a topology, which allows defining continuous deformation of subspaces, more all kinds of continuity. Euclidean spaces, more metric spaces are examples of a topological space, as any distance or metric defines a topology; the deformations that are considered in topology are homotopies. A property, invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; the ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg problem and polyhedron formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed.

The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside. In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once; this result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks. This Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory; the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere.

As with the Bridges of Königsberg, the result does not depend on the shape of the sphere. To deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism; the impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, the hairy ball theorem applies to any space homeomorphic to a sphere. Intuitively, two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist cannot distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence; this is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.

An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends on the font used, on whether the strokes making up the letters have some thickness or are ideal curves with no thickness; the figures here use the sans-serif Myriad font and are assumed to consist of ideal curves without thickness. Homotopy equivalence is a coarser relationship than homeomorphism; the simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent. For example, O fits inside P and the tail of the P can be squished to the "hole" part. Homeomorphism classes are: no holes corresponding with C, G, I, J, L, M, N, S, U, V, W, Z. Homotopy classes are larger, they are: one hole, two holes, no holes. To classify the letters we must show that two letters in the same class are equivalent and two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by selecting points and showing their removal disconnects the letters differently.

For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the fundamental group, is different on the differing classes. Letter topology has practical relevance in stencil

Simple shear

Simple shear is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other. In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value: V x = f V y = V z = 0 And the gradient of velocity is constant and perpendicular to the velocity itself: ∂ V x ∂ y = γ ˙,where γ ˙ is the shear rate and: ∂ V x ∂ x = ∂ V x ∂ z = 0 The displacement gradient tensor Γ for this deformation has only one nonzero term: Γ = Simple shear with the rate γ ˙ is the combination of pure shear strain with the rate of 1/2 γ ˙ and rotation with the rate of 1/2 γ ˙: Γ = ⏟ simple shear = ⏟ pure shear + ⏟ solid rotation The mathematical model representing simple shear is a shear mapping restricted to the physical limits, it is an elementary linear transformation represented by a matrix. The model may represent laminar flow velocity at varying depths of a long channel with constant cross-section.

Limited shear deformation is used in vibration control, for instance base isolation of buildings for limiting earthquake damage. In solid mechanics, a simple shear deformation is defined as an isochoric plane deformation in which there are a set of line elements with a given reference orientation that do not change length and orientation during the deformation; this deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material. When rubber deforms under simple shear, its stress-strain behavior is linear. A rod under torsion is a practical example for a body under simple shear. If e1 is the fixed reference orientation in which line elements do not deform during the deformation and e1 − e2 is the plane of deformation the deformation gradient in simple shear can be expressed as F =. We can write the deformation gradient as F = 1 + γ e 1 ⊗ e 2. {\displaystyle =+\gamma \mathbf _\

John Macklin (academic)

John Joseph Macklin was a Northern Irish scholar of Hispanic studies. He held posts at a number of British universities, from 2001 to 2005 was Principal and Vice-Chancellor of the University of Paisley. At the time of his death he was Professor of Hispanic Studies at the University of Glasgow and Head of the School of Modern Languages and Cultures, a Visiting Professor at the University of Ulster. In 1994, he was made a Commandor of the Order of Isabella the Catholic by King Juan Carlos for his services to Spanish studies. Macklin studied at Queen's University Belfast in Belfast, Northern Ireland, graduating with first class honours in French and Spanish and with a Ph. D. in 1976 on the works of Spanish writer Ramón Pérez de Ayala. Macklin began his teaching career at the University of Hull as a lecturer in Hispanic Studies in 1973, being promoted to Senior Lecturer in 1985, he was made Head of the Department of Hispanic Studies in 1986, in 1988 moved to the University of Leeds as Cowdray Professor of Spanish, a Chair endowed by Weetman Pearson, 1st Viscount Cowdray and founder of Pearson PLC.

He was appointed Dean of the Faculty of Arts in 1992, Dean for Research in Humanities in 1994 and Pro-Vice-Chancellor in 1999, before moving in 2001 to the University of Paisley as Principal and Vice-Chancellor. He was appointed Professor of Spanish at the University of Strathclyde in 2006 and in 2010 became Professor of Hispanic Studies and Head of the School of Modern Languages and Cultures at the University of Glasgow, he was a Visiting Professor at the University of Ulster from 2007. Macklin was married with three children