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 this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining m exist.

If a detector was placed at a distance of 1 m, the ion flight times would be X and Y ns.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

If m

If a new player feels that he or she needs assistance in learning the rules of craps, it is recommended to approach an empty craps table at a slow time of day ( for example, between 9 a. m. and 11 a. m .).

If k, m, and n are 1, so that and, then the Jacobian matrices of f and g are.

To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m. If ( x, y ) are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

If m is greater than 1, the figure becomes larger ; if m is between 0 and 1, it becomes smaller.

If the ambient temperature exceeds T < sub > m </ sub >, the homologous pairing of the sticky ends would not be stable because the high temperature disrupts hydrogen bonding.

If m and n are natural numbers and f ( x ) is a smooth ( meaning: sufficiently often differentiable ) function defined for all real numbers x in the interval, then the integral

This can be done for all m of the p < sub > i </ sub >, showing that m ≤ n. If there were any q < sub > j </ sub > left over we would have

* If m is a non-negative integer, then gcd ( m · a, m · b ) = m · gcd ( a, b ).

If it were wholly random and unrelated, it would be 2.0, assuming the five classes were equal in n, which approximately they are.

If denotes the quantum state of a particle ( n ) with momentum p, spin J whose component in the z-direction is σ, then one has

If each node additionally records the size of its subtree ( including itself and its descendants ), then the nodes can be retrieved by index in O ( log n ) time as well.

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

More formally a k-combination of a set S is a subset of k distinct elements of S. If the set has n elements the number of k-combinations is equal to the binomial coefficient

If a is a point in R < sup > n </ sup >, then the higher dimensional chain rule says that:

If n ≥ 1 and is an integer, the numbers coprime to n, taken modulo n, form a group with multiplication as operation ; it is written as ( Z / nZ )< sup >×</ sup > or Z < sub > n </ sub >< sup >*</ sup >.

If the affected bits are independently chosen at random, the probability of a two-bit error being undetected is 1 / n.

If S is an arbitrary set, then the set S < sup > N </ sup > of all sequences in S becomes a complete metric space if we define the distance between the sequences ( x < sub > n </ sub >) and ( y < sub > n </ sub >) to be, where N is the smallest index for which x < sub > N </ sub > is distinct from y < sub > N </ sub >, or 0 if there is no such index.

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.

If you have an annual or regular physical examination program, is it worth what it is costing you??

If Barnett doesn't call a special session in 1961, it will be the first year in the last decade that the Legislature has not met in regular or special session.

# If adiabats and isotherms are graphed severally at regular changes of entropy and temperature, respectively ( like altitude on a contour map ), then as the eye moves towards the axes ( towards the south-west ), it sees the density of isotherms stay constant, but it sees the density of adiabats grow.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

If μ is both inner regular and locally finite, it is called a Radon measure.

If a decision must be reached in a particular match ( e. g., in a tournament ) and it ends in a draw after regular time, there are at maximum two overtimes of 2 × 5 minutes with a 1-minute break each.

If the edge length of a regular octahedron is a, the radius of a circumscribed sphere ( one that touches the octahedron at all vertices ) is

If the workman lost, I would pay his regular wages ; if he made more than the wages, he kept it.

If μ is a complex-valued countably additive Borel measure, μ is regular iff the non-negative countably additive measure | μ | is regular as defined above.

If police were spotted, regular white lights were turned on, signaling that everyone should stop dancing or touching.

If the change of a sequence of fricatives such that one becomes a stop is dissimilation, then such changes as Proto-Germanic * χs to ( spelled x ) in English would count as a regular sound law: PGmc.

If a flat ribbon or strip is tightly folded into a flattened overhand knot, it assumes a regular pentagonal shape.

If a variable undergoes regular oscillations, and a graph of the system is drawn with the oscillating variable as the vertical axis and time as the horizontal axis, the amplitude is visually represented by the vertical distance between the extrema of the curve and the equilibrium value.

If all elements are regular, then the semigroup ( or monoid ) is called regular, and every element has at least one inverse.

If the see is an archbishopric, the correct term is prince-archbishop ; the equivalent in the regular ( monastic ) clergy is prince-abbot.

If, after a regular trial, a captured soldier is found to be a mercenary, then he can expect treatment as a common criminal and may face execution.

If White plays a regular opening and Black responds in an unconventional way, the opening is not categorized A00.

If, after a " fair and regular trial ", the individual in question is found guilty of a crime, then the " unlawful combatant " can be punished by whatever lawful methods are available to the party to the conflict.

If found guilty at a regular trial, they can be punished under the civilian laws of the detaining power.