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.

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

List of Kill Bill characters

The following is a list of characters from the film Kill Bill. Written and directed by Quentin Tarantino, the film was released in two separate parts, Kill Bill: Volume 1 in 2003 and Kill Bill: Volume 2 in 2004. The film takes place after a massacre that killed the fiancé and friends of the main character, The Bride, at the chapel in which she was to be married; she was also thought to have been killed by being shot in the head. However, The Bride survived, but was put into a coma for four years as a result of the attack. Upon finally awakening she plots her vengeance against the killers.

Deadly Viper Assassination Squad

Beatrix Kiddo (Black Mamba, also known as The Bride)

Beatrix Kiddo, a.k.a. The Bride (portrayed by Uma Thurman), is the protagonist of the film. She abandons her life as a hired assassin for the Deadly Viper Assassination Squad (DVAS) upon realizing she is pregnant with Bill's child, denying him the right of fatherhood to preserve her unborn's future. This action provokes the attacks on her and her loved ones and its ensuing revenge, which is the entire basis of the film. Her code name while working for DVAS was Black Mamba.

Bill (weapon)

The bill is a polearm weapon used by infantry in medieval Europe. The bill is similar in size, function and appearance to the halberd, differing mainly in the hooked blade form. Other terms for the bill include English bill, bill hook or bill-guisarme.

Design

Derived originally from the agricultural billhook, the bill consisted of a hooked chopping blade with several pointed projections mounted on a staff. The end of the cutting blade curves forward to form a hook, which is the bill's distinguishing characteristic. In addition, the blade almost universally had one pronounced spike straight off the top like a spear head, and also a hook or spike mounted on the 'reverse' side of the blade. There were many types of bill. English bills tended to be relatively short, with broad chopping heads, while Italian bills (ronche) often had very long thrusting points. The English distinguished between several varieties of bill, including the black, brown and forest bills, but the differences between them are currently not fully understood.

List of unidentified decedents in the United States

An unidentified decedent, or UID, is a deceased person whose legal identity is unable to be determined by law enforcement. Although the majority of UIDs are identified soon after their bodies are recovered, it is not uncommon for bodies to remain unidentified for years or even decades. There are approximately 40,000 unidentified decedents in the United States at any given time.

A deceased body can be identified in several different ways. Most commonly, a relative of the deceased identifies the body by sight. However, if a body is heavily decomposed, skeletal, or unrecognizable due to severe facial trauma at the time of recovery, other methods must be used. Some other common modes of identification include fingerprints, dental records, chest x-rays, and as a last resort, comparing the deceased's DNA to the DNA sequence of a genetically close relative such as a parent or sibling. In some cases, circumstantial evidence can be used to formally identify a body; for example, if the deceased has a driver's license on their person, or is found deceased on their own property after going missing, those context clues can be used to make an identification. Since identification of a deceased individual is a legal matter, officials require a high degree of certainty in order to make a formal identification.