Trigonometric functions

In mathematics, the trigonometric functions (also called the circular functions) are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

The most familiar trigonometric functions are the sine, cosine, and tangent. In the context of the standard unit circle (a circle with radius 1 unit), where a triangle is formed by a ray originating at the origin and making some angle with the x-axis, the sine of the angle gives the length of the y-component (the opposite to the angle or the rise) of the triangle, the cosine gives the length of the x-component (the adjacent of the angle or the run), and the tangent function gives the slope (y-component divided by the x-component). More precise definitions are detailed below. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

Broca's area

Broca's area or the Broca area /broʊˈkɑː/ or /ˈbroʊkə/ is a region in the frontal lobe of the dominant hemisphere (usually the left) of the hominid brain with functions linked to speech production.

Language processing has been linked to Broca's area since Pierre Paul Broca reported impairments in two patients. They had lost the ability to speak after injury to the posterior inferior frontal gyrus of the brain. Since then, the approximate region he identified has become known as Broca's area, and the deficit in language production as Broca's aphasia, also called expressive aphasia. Broca's area is now typically defined in terms of the pars opercularis and pars triangularis of the inferior frontal gyrus, represented in Brodmann's cytoarchitectonic map as areas 44 and 45 of the dominant hemisphere. Studies of chronic aphasia have implicated an essential role of Broca's area in various speech and language functions. Further, fMRI studies have also identified activation patterns in Broca's area associated with various language tasks. However, slow destruction of the Broca's area by brain tumors can leave speech relatively intact suggesting its functions can shift to nearby areas in the brain.

Transaction authentication number

A transaction authentication number (TAN) is used by some online banking services as a form of single use one-time passwords to authorize financial transactions. TANs are a second layer of security above and beyond the traditional single-password authentication.

TANs provide additional security because they act as a form of two-factor authentication. Should the physical document or token containing the TANs be stolen, it will be of little use without the password; conversely, if the login data are obtained, no transactions can be performed without a valid TAN.

Classic TAN

An outline of how TANs function:

Film (Polish magazine)

Film is a monthly Polish magazine devoted to cinema. It has been in publication since 1946, originally as a bimonthly publication. The founders were Jerzy Giżycki, Zbigniew Pitera, Tadeusz Kowalski, and Leon Bukowiecki.

Since September 2012, the editor-in-chief has been Tomasz Raczek. Previous editors have included Maciej Pawlicki, Lech Kurpiewski, Igor Zalewski and Robert Mazurek, Agnieszka Różycka, Marcin Prokop and Jacek Rakowiecki.

In January 2007, Film was purchased by Platforma Mediowa Point Group (PMPG).

Editorial staff

References

External links

Official website (Polish)

Film (Iranian magazine)

Film (Persian:فیلم) is an Iranian film review magazine published for more than 30 years. The head-editor is Massoud Mehrabi.

References

External links

Lubrication theory

In fluid dynamics, lubrication theory describes the flow of fluids (liquids or gases) in a geometry in which one dimension is significantly smaller than the others. An example is the flow above air hockey tables, where the thickness of the air layer beneath the puck is much smaller than the dimensions of the puck itself.

Internal flows are those where the fluid is fully bounded. Internal flow lubrication theory has many industrial applications because of its role in the design of fluid bearings. Here a key goal of lubrication theory is to determine the pressure distribution in the fluid volume, and hence the forces on the bearing components. The working fluid in this case is often termed a lubricant.

Free film lubrication theory is concerned with the case in which one of the surfaces containing the fluid is a free surface. In that case the position of the free surface is itself unknown, and one goal of lubrication theory is then to determine this. Surface tension may then be significant, or even dominant. Issues of wetting and dewetting then arise. For very thin films (thickness less than one micrometre), additional intermolecular forces, such as Van der Waals forces or disjoining forces, may become significant.