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

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 example, the chairman of the U. S. Federal Reserve Bank is appointed by the President of the U. S. ( all nominees for this post are recommended by the owners of the Federal Reserve, as are all the board members ), his choice must be confirmed by the Congress, and he must appear and testify before congress twice a year.

For the beginning of the next congress, six more seats from Massachusetts would be reapportioned to Maine.

For the first time the party could hold its congress openly.

For his opposition to some arbitrary measures of President Mosquera he was arrested with others, by order of Mosquera, when the congress was dissolved in March 1867.

For the 1998 election cycle, the American Heritage Party fielded two candidates for congress.

For example: the plan repeals the Hyde Amendment to allow abortion funding through federal sources by using congress ; the negative runs a courts counterplan that repeals the hyde amendment and runs a politics disadvantage that says the plan will drain the political capital of the president which causes a certain bill not to be passed ; the affirmative would claim that the " perm shields the link " because congress would claim that the courts made them repeal the hyde amendment, therefore no political capital would be lost.

For part of his time in congress he served as chairman of the Committee on the District of Columbia.

Before the 21st CPN party congress MLC published the appeal ' For the Unity of the Communist Movement '.

For each specific consistent effectively represented axiomatic system for the natural numbers, such as Peano arithmetic, there exists a constant N such that no bit of Ω after the Nth can be proven to be one or zero within that system.

For instance, we could define natural number as follows ( after Peano ):

For example, a number-theoretic statement might be expressible in the language of arithmetic ( i. e. the language of the Peano Axioms ) and a proof might be given that appeals to topology or complex analysis.

For the classic Peano and Hilbert space-filling curves, where two subcurves intersect ( in the technical sense ), there is self-contact without self-crossing.

For this reason, the name Peano ’ s Interlingua might be regarded as the most accurate for the particular standard by Peano.

For example, the set of ( codes for ) formulas of first-order Peano arithmetic that are true in N is definable by a formula in second order arithmetic.

For weeks I wandered about this neighborhood of warehouses and garages, truck terminals and taxi repair shops, gasoline pumps and longshoremen's lunch counters, yet never did I cease to feel myself a stranger there.

For Rachel, conceded to be the prettiest of the Szold girls -- and she did make a pretty picture sitting in the grape-arbor strumming her guitar and singing in her silvery tones -- there was no particular March counterpart ; ;

For centuries the North Atlantic nations dominated the world and as long as they did they could afford the luxury of fighting each other.

For if it did, the plane of L and l' would contain two generators of Af, which is impossible.

For what I express in my remark is something going on in me at the time, and that of course did not exist until I did come on the scene.

For the use of students and future restorers, a full, day-by-day record was kept of all three undertakings, complete technical reports on what we found and what we did.

For example, the unwed mothers expressed their frustration with males who did not indicate more explicitly `` what it is they really want from a girl so one can act accordingly ''.

For example, in 1693 the Philadelphia Yearly Meeting of Friends declared that its members should emancipate their slaves and in 1776 it determined to exclude from membership all who did not comply.

For many years people lived in the village and then in 1845 Famine struck in Achill as it did in the rest of Ireland.

For various reasons, however, poverty and personal inclination among others, he did not take a prominent part in the military operations of this period.

For nearly 150 years after, Venice did not own an inch of soil on the mainland of Greece, except the Ionian dependencies of Parga and Butrinto, but of her insular dominions Cyprus, Crete, Tenos and 6 Ionian islands still remained.

For the third movement, Beethoven used variation techniques ; he also did this in the second movement of his Quartet op.

For many years, early Anglo-Saxon history was essentially a retelling of the Historia, but recent scholarship has focused as much on what Bede did not write as what he did.

For example, in 2006 more than 60 percent of Fortune 500 companies did not have a COO, and in 2007 almost 58 percent of Fortune 500 companies did not have a COO.

Young did not walk a batter and was later quoted: " For my part, I think it was the greatest game of ball I ever took part in.

For temporal secular reasons, the kings did nothing to dispel this attitude.

For example, one begins to answer the question " Why did China not develop modern science and capitalism?

For example, for a long period it was assumed that Imperial China had no system of civil law because the law codes did not have explicit provisions for civil lawsuits.

For example, Gorbachev quickly became the party's de facto Second Secretary, even though Chernenko did not support him.

For example, in the political controversy over anthropogenic climate change that is prevalent in the United States-it has been thought that those who are opposed to the scientific consensus did so because of a lack of evidence.

For example Cyril writes “ I gave my back to those who beat me and my cheeks to blows ; and my face I did not shield from the shame of spitting ”.

For instance, in Nanook of the North Flaherty did not allow his subjects to shoot a walrus with a nearby shotgun, but had them use a harpoon instead.