Category

Category, plural categories, may refer to:

Philosophy and general uses

Mathematics

Linguistics

Lusternik–Schnirelmann category

In mathematics, the Lyusternik–Schnirelmann category (or, Lusternik–Schnirelmann category, LS-category) of a topological space is the homotopical invariant defined to be the smallest integer number such that there is an open covering of with the property that each inclusion map is nullhomotopic. For example, if is the circle, this takes the value two.

Sometimes a different normalization of the invariant is adopted, which is one less than the definition above Such a normalization has been adopted in the definitive monograph by Cornea, Lupton, Oprea, and Tanré (see below).

In general it is not easy to compute this invariant, which was initially introduced by Lazar Lyusternik and Lev Schnirelmann in connection with variational problems. It has a close connection with algebraic topology, in particular cup-length. In the modern normalization, the cup-length is a lower bound for LS category.

It was, as originally defined for the case of X a manifold, the lower bound for the number of critical points that a real-valued function on X could possess (this should be compared with the result in Morse theory that shows that the sum of the Betti numbers is a lower bound for the number of critical points of a Morse function).

Grammatical category

A grammatical category is a property of items within the grammar of a language; it has a number of possible values (sometimes called grammemes), which are normally mutually exclusive within a given category. Examples of frequently encountered grammatical categories include tense (which may take values such as present, past, etc.), number (with values such as singular, plural, and sometimes dual), and gender (with values such as masculine, feminine and neuter).

Although terminology is not always consistent, a distinction should be made between these grammatical categories (tense, number, etc.) and lexical categories, which are closely synonymous with the traditional parts of speech (noun, verb, adjective, etc.), or more generally syntactic categories. Grammatical categories are also referred to as (grammatical) features.

The name given to a grammatical category (as an uncountable noun) is generally also used (as a countable noun) to denote any of the possible values for that category. For example, the values available in a given language for the category "tense" are called "tenses", the values available for the category "gender" are called "genders", and so on.

Apple TV

Apple TV (marketed stylized as tv) is a digital media player and a microconsole developed and sold by Apple Inc. It is a small network appliance and entertainment device that can receive digital data from a number of sources and stream it to a capable TV for playing on the TV screen.

Apple TV is an HDMI-compliant source device. To use it for viewing, it has to be connected to an enhanced-definition or high-definition widescreen television via an HDMI cable to the TV's HDMI port, while the TV is in HDMI mode. The device has no integrated controls and can only be controlled externally, either by an Apple Remote control device (with which it is shipped) using its infrared capability or by the iTunes Remote app (downloadable from App Store) on iOS devices, such as the iPhone, iPod Touch, iPad, and Apple Watch, using its Wi-Fi capability.

Its Wi-Fi capability is also used to receive digital content from the iTunes app using AirPlay or directly from iTunes Store, which is then streamed to the TV. It also plays digital content from the iTunes Store, Netflix, Hulu Plus, Now TV (UK only), YouTube and Vevo, along with the TV Everywhere portals of several cable and broadcast networks, and the video subscription portals of three of the four major North American sports leagues: MLB.tv, NBA League Pass and NHL GameCenter. It plays content from any OS X or Windows computer running iTunes.

ITV2

ITV2, is a 24-hour, free-to-air entertainment television channel in the United Kingdom, Isle of Man and the Channel Islands owned by ITV Digital Channels Ltd, a division of ITV plc. It was launched on 7 December 1998 and is available digitally via satellite, cable, IPTV and terrestrial (Freeview) platforms. For a number of years, it had the largest audience share after the five analogue terrestrial stations, a claim now held by its sister service ITV3 both of which are freely available to a majority of households.

The channel is known for American programming such as Gossip Girl and The Vampire Diaries; repeats of recently aired episodes of soap operas and other entertainment programming from ITV such as Coronation Street and Emmerdale; spin-off shows such as I'm a Celebrity...Get Me Out of Here! NOW!, The Xtra Factor and Britain's Got More Talent, and original programming such as Celebrity Juice and The Only Way Is Essex (now on ITVBe).

History

Prior to the launch of Channel 4 in 1982, the name "ITV2" had sometimes been used to refer informally to an envisioned second commercial network in the UK. However, the name resurfaced in the late 1990s for very different reasons. The launch of digital terrestrial television services in the UK saw each existing analogue terrestrial broadcaster given a slice of bandwidth with which to carry their existing service after analogue switch-off with space left over for new channels. Whilst strictly speaking this space belonged to each regional contractor for use within their own region, ITV had undergone a series of buy-outs earlier on in the decade, the three players operating the majority of the network, Granada, Carlton and United News & Media jointly launched ITV2 in 1998 to be broadcast to most of the country as a uniform service. Whilst free-to-air, it was marketed alongside their own subscription based ONdigital platform. Other ITV licensees, SMG, UTV and GMTV launched their own services in the space (see below).

ITV 50

ITV 50 was a special event around September 2005, celebrating the 50th anniversary of the ITV terrestrial television network's launch; the official anniversary was stated as being 22 September, the anniversary of the first ITV region, Associated-Rediffusion, beginning broadcasts.

Other regions, though not launched in 1955, were also involved in the celebrations, broadcasting network and local ITV50 celebration programming. All ITV plc regions received ITV50 presentational material including branded graphics and station idents. STV (then Scottish and Grampian) and UTV screened much of the networked ITV 50 content, using some of the ITV 50 presentation.

Introduction

In September 2005, the ITV network celebrated its fiftieth anniversary with a season of ITV 50 programming that was run on the network, including a run down of ITV's 50 top programmes, a World of Sport retrospective, a seven-week Gameshow Marathon presented by Ant & Dec, the launch of an "Avenue Of The Stars", and most notably a five-part documentary series made by Melvyn Bragg, which chronicled ITV's history. The regional companies owned by ITV plc also aired special regional retrospectives (even though none of them were themselves fifty years old), as well as using special ITV 50 station identification. While Scottish Television, Grampian Television, and UTV aired the network ITV 50 programming, they did not themselves air regional programmes of this sort, nor did they use the special identification. ITN also celebrated its fiftieth anniversary with special features in its programming.