Derivative

The derivative of a function of a real variable measures the sensitivity to change of a quantity (a function value or dependent variable) which is determined by another quantity (the independent variable). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time is advanced.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

Derivative (film)

Derivative (Turkish: Türev) is a 2005 Turkish drama film, written, produced and directed by Ulaş İnaç based on a short story by Miguel de Cervantes, about the complicated relationships between three people who confess their thoughts each evening to the camera for a movie project. The film, which went on nationwide general release across Turkey on November 18, 2005 (2005-11-18), won the Golden Orange for Best Film at the 42nd Antalya "Golden Orange" International Film Festival.

Plot

Nazım (Güçlü Yalçıner) is a successful young Istanbul writer who is about to marry Süreyya (Gülçin Santırcıoğlu). Süreyya is not sure about Nazım's love and so she asks her best friend Burcu (Beste Bereket) to seduce him in order to test his fidelity. Reluctantly Burcu complies and then she falls in love.

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Normal function

In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) iff it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:

Examples

A simple normal function is given by f(α) = 1 + α (see ordinal arithmetic). But f(α) = α + 1 is not normal. If β is a fixed ordinal, then the functions f(α) = β + α, f(α) = β × α (for β ≥ 1), and f(α) = β^α (for β ≥ 2) are all normal.

More important examples of normal functions are given by the aleph numbers which connect ordinal and cardinal numbers, and by the beth numbers .

Properties

If f is normal, then for any ordinal α,

Proof: If not, choose γ minimal such that f(γ) < γ. Since f is strictly monotonically increasing, f(f(γ)) < f(γ), contradicting minimality of γ.

Furthermore, for any non-empty set S of ordinals, we have

Öland

Öland (Swedish pronunciation: [ˈøːland]) is the second largest Swedish island and the smallest of the traditional provinces of Sweden. Öland has an area of 1,342 km² and is located in the Baltic Sea just off the coast of Småland. The island has 25,000 inhabitants. It is separated from the mainland by the Kalmar Strait and connected to it by the 6-km Öland Bridge, which opened on 30 September in 1972.

Administration

The traditional provinces of Sweden no longer serve administrative or political purposes but still exist as historical and cultural entities. Öland is part of the administrative county of Kalmar County (Kalmar län) and consists of the two municipalities of Borgholm Municipality and Mörbylånga Municipality. There was an Öland County in the short period between 1819 and 1826; otherwise, the island has been part of Kalmar County since 1634.

Heraldry

Öland was granted provincial arms in 1560, but it would not be until the 1940s that the province was assigned its proper ones. The arms granted to Öland had been mixed up with the arms granted to Åland and this was not discovered until the 20th century. While Öland changed its coat of arms, Åland, which was now a Finnish (autonomous) province, kept its established but originally unintended coat of arms. The deer is meant to symbolise the status of Öland as a royal game park and the arms are topped by a ducal crown. Blazon: "Azure a Deer Or attired, hoofed and gorged Gules."

Generic top-level domain

Generic top-level domains (gTLDs) are one of the categories of top-level domains (TLDs) maintained by the Internet Assigned Numbers Authority (IANA) for use in the Domain Name System of the Internet. A top-level domain is the last label of every fully qualified domain name. They are called generic for historic reasons; initially, they were contrasted with country-specific TLDs in RFC 920.

The core group of generic top-level domains consists of the com, info, net, and org domains. In addition, the domains biz, name, and pro are also considered generic; however, these are designated as restricted, because registrations within them require proof of eligibility within the guidelines set for each.

Historically, the group of generic top-level domains included domains, created in the early development of the domain name system, that are now sponsored by designated agencies or organizations and are restricted to specific types of registrants. Thus, domains edu, gov, int, and mil are now considered sponsored top-level domains, much like the themed top-level domains (e.g., jobs). The entire group of domains that do not have a geographic or country designation (see country-code top-level domain) is still often referred to by the term generic TLDs.

Sixology

Sixology (Chinese: JJ陸) is Singaporean Mandopop Singer-songwriter JJ Lin's sixth Mandarin studio album. It was released on 18 October 2008 by the Ocean Butterflies Music and contains 14 tracks. It is called this because it is his sixth album. There are three versions for this album. The song "主角" is popular in China, with many performances recently and concerts in many cities.

The album was awarded one of the Top 10 Selling Mandarin Albums of the Year at the 2009 IFPI Hong Kong Album Sales Awards, presented by the Hong Kong branch of IFPI.

Track list