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 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 mathematical purposes, elementary particles are normally treated as point particles, although some particle theories such as string theory posit a physical dimension.

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 a detailed presentation of the various points of view around the definition of " algorithm " see Algorithm characterizations.

For Lie algebras the definition is slightly different.

For planets, comets and asteroids a different definition of absolute magnitude is used which is more meaningful for nonstellar objects.

For example, one definition of bandwidth could be the range of frequencies beyond which the frequency function is zero.

For a rigorous treatment of the Dirac inner product of non-normalizable states see the definition given by D. Carfì in and.

For a rigorous definition of basis with a continuous set of indices and consequently for a rigorous definition of position and momentum basis see.

For example, they might interpret a tomato as a vegetable — according to the English definition of tomato — even though the ideal Blissymbol of vegetable was restricted by Bliss to just vegetables growing underground.

For example, in Virginia, the definition of the conduct that constitutes the crime of robbery exists only in the common law, and the robbery statute only sets the punishment.

For example, jīngjì ( 经济 ／ 經濟, keizai ), which in the original Chinese meant " the workings of the state ", was narrowed to " economy " in Japanese ; this narrowed definition was then re-imported into Chinese.

For alternative ( equivalent ) formulations of the definition in terms of a bijective function or a surjective function, see the section Formal definition and properties below.

In his preface to Jürgen Osterhammel's Colonialism: A Theoretical Overview, Roger Tignor says, " For Osterhammel, the essence of colonialism is the existence of colonies, which are by definition governed differently from other territories such as protectorates or informal spheres of influence.

For example, in formal languages like mathematics, a ' stipulative ' definition guides a specific discussion.

For this reason, Locking adds that a definition of a term must not comprise of terms which are synonymous with it.

For terms such as these it is not possible and indeed not necessary to state a definition ; rather, one simply comes to understand the use of the term.

For square matrices with entries in a non-commutative ring, for instance the quaternions, there is no unique definition for the determinant, and no definition that has all the usual properties of determinants over commutative rings.

For example, should animal experimentation become illegal in a society, it will no longer be an ethical issue on Hoy's definition.

For example, the statement " A society is free if and only if liberty is maximized and people are required to take responsibility for their actions " is true or paradoxical, depending on the individual's definition of liberty.

For a long time this definition was used with a derogatory intention.

For all intents and purposes, the 1947 ICC definition confirmed the 1895 MCC definition and gave it international recognition and usage.

For a planar graph, the crossing number is zero by definition.