For those of the Cardinal, Melanchthon, and Dürer's final major work, a drawn portrait of the Nuremberg patrician Ulrich Starck, Dürer depicted the sitters in profile, perhaps reflecting a more mathematical approach.

For the proof of the equivalence of the four approaches the reader is referred to mathematical expositions like or.

For the mathematical formulation of the model, is also assumed to be twice continuously differentiable, so that differential equations describing the motion may be formulated.

For the mathematical definition, see Decimal representation.

For this, and other, reasons mathematical historian Kurt Vogel writes: “ Diophantus was not, as he has often been called, the father of algebra.

For the mathematical constant, see e ( mathematical constant ).

For the mathematical congress, Peano did not speak, but Padoa's memorable presentation has been frequently recalled.

For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Chaldeans from Babylonia.

For the purpose of facilitating jointly sponsored activities and jointly pursuing the objectives of the IMU, multinational mathematical societies and professional societies can join IMU as an Affiliate Member.

For instance, the inverse of the sine function is typically called the arcsine function, written as arcsin, which is, like sin, conventionally denoted in roman type and not in italics ( note that software libraries of mathematical functions often use the name < tt > asin </ tt >):

In 1991 he was awarded the Balzan Prize for Genetics and Evolution " For his powerful analysis of evolutionary theory and of the role of sexual reproduction as a critical factor in evolution and in the survival of species ; for his mathematical models applying the theory of games to evolutionary problems " ( motivation of the Balzan General Prize Committee ).

For example, if A is some mathematical statement that an intuitionist has not yet proved or disproved, then that intuitionist will not assert the truth of " A or not A ".

For games that directly involve mathematics in their play, see mathematical puzzle.

For instance, in the original version, the RSA algorithm was used to encrypt session keys ; RSA's security depends upon the one-way function nature of mathematical integer factoring.

For example, Spearman is a non-parametric test as it is computed from the order of the data regardless of the actual values, whereas Pearson is a parametric test as it is computed directly from the data and can be used to derive a mathematical relationship.

For example, Mancala is a mathematical game, because mathematicians can study it using combinatorial game theory, even though no mathematics is necessary in order to play it.

For a brief period in fact, complete normed linear spaces where referred to as " Banach-Wiener " spaces in mathematical literature, based on terminology introduced by Wiener himself.

For a given physical process, the selection of suitable independent local non-equilibrium macroscopic state variables for the construction of a thermodynamic description calls for qualitative physical understanding, rather than being a simply mathematical problem concerned with a uniquely determined thermodynamic description.

For example, devices such as cameras and scanners produce essentially continuous-tone raster graphics that are impractical to convert into vectors, and so for this type of work, an image editor will operate on the pixels rather than on drawing objects defined by mathematical expressions.

For example, the first 15 digits of the mathematical constant pi ( 3. 14159265358979 ) can be encoded as " Now I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics ".

For sound waves, Lord Rayleigh published a full mathematical analysis of propagation modes in his seminal work, “ The Theory of Sound ”.

For this work he was presented in 1826 with the Gold Medal of the Royal Astronomical Society ( which he won again in 1836 ), and with the Lalande Medal of the French Academy of Sciences in 1825, while in 1821 the Royal Society bestowed upon him the Copley Medal for his mathematical contributions to their Transactions.

For a precise mathematical description refer to the original paper by Erné et al.

For purposes of this explanation, this percentage is referred to as the State's `` unadjusted Federal share ''.

For purposes of sample selection only ( individual tests were given later ) we obtained group test scores of reading achievement and intelligence from school records of the entire third-grade population in each school system.

For expository purposes, this is best treated as a model which spells out the conditions under which an important industry affected with the public interest would find it profitable to raise wages even in the absence of union pressures for higher wages.

For the purposes of setting the product price, the industry behaves as a single entity.

For the purposes of this discussion, the problem of relative prices is encompassed in these two variables, since GNP includes other prices.

For our present purposes we assume that the sole subject of bargaining is the basic wage rate ( not including productivity improvement factors or cost-of-living adjustments ), and it is this basic wage rate which determines the level of costs.

'' For present purposes it may be assumed that this charge so narrowly limited speech as to violate the federal Constitution.

For all practical purposes, the West stands disunited, undedicated, and unprepared for the tasks of world leadership.

For their purposes, oxygen and other gases were not drawn from the surrounding atmosphere through the medium of lungs but sustained artificially by solution in their shells.

For the purposes of the treaty system, Antarctica is defined as all of the land and ice shelves south of 60 ° S latitude.

For geopolitical purposes, the Sinai Peninsula of Egypt – east of the Suez Canal – is often considered part of Africa.

For socket numbers, a few well-known numbers were reserved for special purposes specific to the AppleTalk protocol itself.

For example, letters 1, 5, and 8 contain a discussion on the question, whether the use of a piece of metal with the figure of a lion, as a talisman, is permitted by Jewish law for medicinal purposes, or is prohibited as idolatrous.

For administrative purposes the archipelago constitutes the comune of Favignana in the Province of Trapani.

For comparison purposes that is approximately the same size as Aylesbury, Carlisle, Guildford or Scunthorpe urban areas.

For commercial purposes, it is derived primarily from Gelidium amansii.

For statistical purposes, Baker is grouped with the United States Minor Outlying Islands.

For purposes of scoring and reference, each player is identified by one of the points of the compass and thus North and South play against East and West.

For calendric purposes, Bede made a new calculation of the age of the world since the creation, which he dated as 3952 BC.

For scoring purposes, a standing eight count is treated as a knockdown.

For all practical purposes that was the Baltic language until 1919.

For most purposes, most jurisdictions, including the U. S. federal system and most states, have merged the two courts.

For purposes of international communication and trade, the official names of the chemical elements both ancient and more recently recognized are decided by the International Union of Pure and Applied Chemistry ( IUPAC ), which has decided on a sort of international English language, drawing on traditional English names even when an element's chemical symbol is based on a Latin or other traditional word, for example adopting " gold " rather than " aurum " as the name for the 79th element ( Au ).

For international banks, including the 55 member central banks of the Bank for International Settlements, the threshold is 8 % ( see the Basel Capital Accords ) of risk-adjusted assets, whereby certain assets ( such as government bonds ) are considered to have lower risk and are either partially or fully excluded from total assets for the purposes of calculating capital adequacy.

For enumeration purposes the country is demarcated into 4, 042 enumeration areas.