If numerical iterative methods have to be employed, the aim is to iterate until full machine accuracy is obtained ( the best that is possible with a finite word length on the computer, and within the mathematical and / or physical approximations made ).

If we consider the thesis and its converse as definition, then the hypothesis is an hypothesis about the application of the mathematical theory developed from the definition.

If one agrees that set theory is an appealing foundation of mathematics, then all mathematical objects must be defined as sets of some sort.

If a mathematical surface can be intersected by a ray, it can be rendered using ray casting.

If T is a theory whose objects of discourse can be interpreted as natural numbers, we say T is arithmetically sound if all theorems of T are actually true about the standard mathematical integers.

" If we mistakenly confuse precision with accuracy, then we might be misled into thinking that an explanation expressed in precise mathematical or graphical terms is somehow more rigorous or useful than one that takes into account particulars of history, institutions or business strategy.

If the two loose ends of an overhand knot are joined together ( without creating additional crossings ), this becomes equivalent to the trefoil knot of mathematical knot theory.

If mathematics is a part of logic, then questions about mathematical objects reduce to questions about logical objects.

If propositions are sets of possible worlds, however, then all mathematical truths are the same set ( the set of all possible worlds ).

If the names of the academic papers are reversed, they describe real mathematical events.

If modern mathematical practices are what distinguish modern professional mathematicians from older ideas of folk mathematics.

If mathematics has been informally used throughout history, in numerous cultures and continents, then it could be argued that " mathematical practice " is the practice, or use, of mathematics in everyday life.

If the first block has index 1, the mathematical formula for CBC encryption is

Describing this situation after many decades, Ambartsumian said, " If an astronomer publishes an article with a mathematical content in a physics journal, then the most likely thing that will happen to it is oblivion.

If Göttingen had been the “ Mecca ” of mathematics in the 1920s and early ‘ 30s, Princeton, following the decimation of German mathematics under the Nazis, had become the center of the mathematical world in the 1940s.

If one inserts a more realistic barrier model into the simplest form of the Schrödinger equation, then an awkward mathematical problem arises over the resulting differential equation: it is known to be mathematically impossible in principle to solve this equation exactly in terms of the usual functions of mathematical physics, or in any simple way.

If more exact values for v ( f ) are required, then provides formulae that give values for v ( f ) to an absolute mathematical accuracy of better than 8 × 10 < sup >− 10 </ sup >.

# If a human m is a sufficiently skillful mathematical logician ( equipped with a sufficiently powerful computer if necessary ) then if m is given L ( m ), he or she can construct T ( L ( m )) and

If an ideal, mathematical point source of light is placed on-axis in the input plane of the first lens, then there will be a uniform, collimated field produced in the output plane of the first lens.

If any mathematical methods can aid in the growth of such wisdom, then catastrophe theory will be part of them.

If we define a decimal expansion such that some digit is dependent on some yet unsolved mathematical problem, we know beforehand that we cannot tell if this number is the same as some other decimal expansion which is independent of this problem.

If one is to use Gödel's technique to prove the proposition that T cannot prove, one must first prove ( the mathematical statement representing ) the consistency of T, a daunting and perhaps impossible task.

If you could see all your opponents ' cards, you would always be able to calculate the correct decision with mathematical certainty.

If Af is the change per unit volume in Gibbs function caused by the shear field at constant P and T, and **yr is the density of the fluid, then the total potential energy of the system above the reference height is Af.

If a union cannot perform this function, then collective bargaining is being palmed off by organizers as a gigantic fraud.

If the Greek letter is used, it is assumed to be a Fourier transform of another function,

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of X.

If the function R is well-defined, its value must lie in the range, with 1 indicating perfect correlation and − 1 indicating perfect anti-correlation.

If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.

If we define the function f ( n ) = A ( n, n ), which increases both m and n at the same time, we have a function of one variable that dwarfs every primitive recursive function, including very fast-growing functions such as the exponential function, the factorial function, multi-and superfactorial functions, and even functions defined using Knuth's up-arrow notation ( except when the indexed up-arrow is used ).

If f is not a function, but is instead a partial function, it is called a partial operation.

The tensor product X ⊗ Y from X and Y is a K-vector space Z with a bilinear function T: X × Y → Z which has the following universal property: If T ′: X × Y → Z ′ is any bilinear function into a K-vector space Z ′, then only one linear function f: Z → Z ′ with exists.

If your side has two aces and a void, then you are not at risk of losing the first two tricks, so long as ( a ) your void is useful ( i. e., does not duplicate the function of an ace that your side holds ) and ( b ) you are not vulnerable to the loss of the first two tricks in the fourth suit ( because, for instance, one of the partnership hands holds a singleton in that suit or the protected king, giving your side second round control ).

If evolutionary processes are blind to the difference between function F being performed by conscious organism O and non-conscious organism O *, it is unclear what adaptive advantage consciousness could provide.

If the wave function is regarded as ontologically real, and collapse is entirely rejected, a many worlds theory results.

If wave function collapse is regarded as ontologically real as well, an objective collapse theory is obtained.

If we take the simple valence bond structure and mix in all possible covalent and ionic structures arising from a particular set of atomic orbitals, we reach what is called the full configuration interaction wave function.

If we take the simple molecular orbital description of the ground state and combine that function with the functions describing all possible excited states using unoccupied orbitals arising from the same set of atomic orbitals, we also reach the full configuration interaction wavefunction.

If the circumstances are faced frankly it is not reasonable to expect this to be true.

If his dancers are sometimes made to look as if they might be creatures from Mars, this is consistent with his intention of placing them in the orbit of another world, a world in which they are freed of their pedestrian identities.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

If they avoid the use of the pungent, outlawed four-letter word it is because it is taboo ; ;

If Wilhelm Reich is the Moses who has led them out of the Egypt of sexual slavery, Dylan Thomas is the poet who offers them the Dionysian dialectic of justification for their indulgence in liquor, marijuana, sex, and jazz.

If he is the child of nothingness, if he is the predestined victim of an age of atomic wars, then he will consult only his own organic needs and go beyond good and evil.

If it is an honest feeling, then why should she not yield to it??

If he thus achieves a lyrical, dreamlike, drugged intensity, he pays the price for his indulgence by producing work -- Allen Ginsberg's `` Howl '' is a striking example of this tendency -- that is disoriented, Dionysian but without depth and without Apollonian control.

If love reflects the nature of man, as Ortega Y Gasset believes, if the person in love betrays decisively what he is by his behavior in love, then the writers of the beat generation are creating a new literary genre.

If he is good, he may not be legal ; ;

If the man on the sidewalk is surprised at this question, it has served as an exclamation.

If the existent form is to be retained new factors that reinforce it must be introduced into the situation.

If we remove ourselves for a moment from our time and our infatuation with mental disease, isn't there something absurd about a hero in a novel who is defeated by his infantile neurosis??

If many of the characters in contemporary novels appear to be the bloodless relations of characters in a case history it is because the novelist is often forgetful today that those things that we call character manifest themselves in surface behavior, that the ego is still the executive agency of personality, and that all we know of personality must be discerned through the ego.

If he is a traditionalist, he is an eclectic traditionalist.

If our sincerity is granted, and it is granted, the discrepancy can only be explained by the fact that we have come to believe hearsay and legend about ourselves in preference to an understanding gained by earnest self-examination.

If to be innocent is to be helpless, then I had been -- as are we all -- helpless at the start.