Rose (topology)

In mathematics, a rose (also known as a bouquet of n circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topology, where they are closely related to free groups.

Definition

A rose is a wedge sum of circles. That is, the rose is the quotient space C/S, where C is a disjoint union of circles and S a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, and one edge for each circle. This makes it a simple example of a topological graph.

A rose with n petals can also be obtained by identifying n points on a single circle. The rose with two petals is known as the figure eight.

Relation to free groups

The fundamental group of a rose is free, with one generator for each petal. The universal cover is an infinite tree, which can be identified with the Cayley graph of the free group. (This is a special case of the presentation complex associated to any presentation of a group.)

Rose

A rose (/ˈroʊz/) is a woody perennial flowering plant of the genus Rosa, in the family Rosaceae. There are over 100 species and thousands of cultivars. They form a group of plants that can be erect shrubs, climbing or trailing with stems that are often armed with sharp prickles. Flowers vary in size and shape and are usually large and showy, in colours ranging from white through yellows and reds. Most species are native to Asia, with smaller numbers native to Europe, North America, and northwest Africa. Species, cultivars and hybrids are all widely grown for their beauty and often are fragrant. Rose plants range in size from compact, miniature roses, to climbers that can reach seven meters in height. Different species hybridize easily, and this has been used in the development of the wide range of garden roses.

The name rose comes from French, itself from Latin rosa, which was perhaps borrowed from Oscan, from Greek ρόδον rhódon (Aeolic βρόδον wródon), itself borrowed from Old Persian wrd- (wurdi), related to Avestan varəδa, Sogdian ward, Parthian wâr.

Rose (symbolism)

Roses have been long used as symbols in a number of societies. Roses are ancient symbols of love and beauty. "Rose" means pink or red in a variety of languages (such as the Romance languages and Greek).

The Classical Era

The rose was sacred to a number of goddesses including Isis, whose rose appears in the late classical allegorical novel The Golden Ass as "the sweet Rose of reason and virtue" that saves the hero from his bewitched life in the form of a donkey. The ancient Greeks and Romans identified the rose with the goddess of love, Aphrodite (Greek name) and Venus (Roman name).

In Rome a wild rose would be placed on the door of a room where secret or confidential matters were discussed. The phrase sub rosa, or "under the rose", means to keep a secret — derived from this ancient Roman practice.

Islam and Sufism

The cultivation of geometrical gardens, in which the rose has often held pride of place, has a long history in Iran and surrounding lands. In the lyric ghazal, it is the beauty of the rose that provokes the longing song of the nightingale - an image prominent, for example, in the poems of Hafez.

Rosé

A rosé (from French rosé; also known as rosado in Portugal and Spanish-speaking countries and rosato in Italy) is a type of wine that incorporates some of the color from the grape skins, but not enough to qualify it as a red wine. It may be the oldest known type of wine, as it is the most straightforward to make with the skin contact method. The pink color can range from a pale "onion"-skin orange to a vivid near-purple, depending on the varietals used and winemaking techniques. There are three major ways to produce rosé wine: skin contact, saignée and blending. Rosé wines can be made still, semi-sparkling or sparkling and with a wide range of sweetness levels from bone-dry Provençal rosé to sweet White Zinfandels and blushes. Rosé wines are made from a wide variety of grapes and can be found all around the globe.

When rosé wine is the primary product, it is produced with the skin contact method. Black-skinned grapes are crushed and the skins are allowed to remain in contact with the juice for a short period, typically one to three days. The must is then pressed, and the skins are discarded rather than left in contact throughout fermentation (as with red wine making). The longer that the skins are left in contact with the juice, the more intense the color of the final wine.

Topological space

In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, that satisfy a set of axioms relating points and neighbourhoods. The definition of a topological space relies only upon set theory and is the most general notion of a mathematical space that allows for the definition of concepts such as continuity, connectedness, and convergence. Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology.

Definition

The utility of the notion of a topology is shown by the fact that there are several equivalent definitions of this structure. Thus one chooses the axiomatisation suited for the application. The most commonly used, and the most elegant, is that in terms of open sets, but the most intuitive is that in terms of neighbourhoods and so we give this first. Note: A variety of other axiomatisations of topological spaces are listed in the Exercises of the book by Vaidyanathaswamy.

Topology (musical ensemble)

Topology is an indie classical quintet from Australia, formed in 1997. A leading Australian new music ensemble, they perform throughout Australia and abroad and have to date released 12 albums, including one with rock/electronica band Full Fathom Five and one with contemporary ensemble Loops. They were formerly the resident ensemble at the University of Western Sydney. The group works with composers including Tim Brady in Canada, Andrew Poppy, Michael Nyman, and Jeremy Peyton Jones in the UK, and Terry Riley, Steve Reich, Philip Glass, Carl Stone, and Paul Dresher in the US, as well as many Australian composers.

In 2009, Topology won the Outstanding Contribution by an Organisation award at the Australasian Performing Right Association (APRA) Classical Music Awards for their work on the 2008 Brisbane Powerhouse Series.

Members

Bernard studied viola at the Queensland Conservatorium (B.Mus 1987) and at Michigan State University (Master of Music 1993) with John Graham and Robert Dan. He studied in summer schools with Kim Kashkashian (Aldeborough), the Alban Berg Quartet and the Kronos Quartet. While in the US, he played with the Arlington Quartet, touring the US and UK. He was a violist in the Queensland Philharmonic Orchestra from 1994-2000, and is now Associate Principal Violist of the Queensland Orchestra, playing solo parts in works such as the sixth Brandenburg Concerto. He has directed several concerts for the Queensland Philharmonic’s Off the Factory Floor chamber series.

Topology (chemistry)

In chemistry, topology provides a convenient way of describing and predicting the molecular structure within the constraints of three-dimensional (3-D) space. Given the determinants of chemical bonding and the chemical properties of the atoms, topology provides a model for explaining how the atoms ethereal wave functions must fit together. Molecular topology is a part of mathematical chemistry dealing with the algebraic description of chemical compounds so allowing a unique and easy characterization of them.

Topology is insensitive to the details of a scalar field, and can often be determined using simplified calculations. Scalar fields such as electron density, Madelung field, covalent field and the electrostatic potential can be used to model topology.

Each scalar field has its own distinctive topology and each provides different information about the nature of chemical bonding and structure. The analysis of these topologies, when combined with simple electrostatic theory and a few empirical observations, leads to a quantitative model of localized chemical bonding. In the process, the analysis provides insights into the nature of chemical bonding.