Category
Category, plural categories, may refer to:
Philosophy and general uses
Mathematics
- Abelian category
- Category of small categories
- Category of sets
- Category (topology) in the context of Baire spaces
- Lusternik–Schnirelmann category, sometimes called LS-category or simply category
- Monoidal category
- Preadditive category
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.
Kao
Kao may refer to:
Places
People
- Kao (musician), Kaoru's alternative name
- Kao (painter), 15th Century Japanese painter
- Nicholas Kao Se Tseien (1897–2007), Catholic priest, supercentenarian
- Ping-Tse Kao (1888–1970), Taiwanese astronomer, after whom the crater is named
- R. N. Kao (1918–2002), Founder Director of Research and Analysis Wing (RAW)
- Charles Kao, Father of fiber optics, awarded 2009 Nobel Prize in Physics
Kaoru (musician)
Kaoru (薫, Kaoru, born February 17, 1974, in Hyōgo) is a Japanese musician, best known as the leader and one of the guitarists of the visual kei metal band Dir En Grey. He has been with the group since its inception in 1997 and was previously a member of La:Sadie's. He has composed most of Dir en grey's songs (at least up to the Vulgar album, at which individual credits for the music were dropped). His contributions display a fair amount of variety, ranging from slow ballads such as "Zakuro" to upbeat numbers like "Jessica". He performs many solos for the band, which vary strongly in feeling and tempo, similar to his compositions. Kaoru cites hide as his biggest influence and the reason he plays guitar, saying "If I had never known him, I wouldn’t be who I am today."
Equipment
Kaoru is endorsed by ESP Guitars in Japan, who provides him with most of his guitars, picks, and straps. In the past, Kaoru primarily used ESP Custom Shop guitars from his signature-series, Ganesa. He stopped using the Ganesa guitars around 2005 and switched to the Viper series, in which he has his own custom model, the D-KV-420. Recently a limited edition Edwards version of his D-KV is being sold.
Kao (painter)
Kao (active early 15th century) was a Japanese priest-painter whose work is considered typical of early Muromachi painting.
He is especially known for his depiction of the legendary monk Kensu (Hsien-tzu in Chinese) at the moment he achieved enlightenment.
This type of painting was executed with quick brush strokes and a minimum of detail.
Podcasts:
