If the bias is small, we can let U − E ≈ φM in the expression for κ, where φM, the work function, gives the minimum energy needed to bring an electron from an occupied level, the highest of which is at the Fermi level ( for metals at T = 0 kelvins ), to vacuum level.

If V is a standard model of ZFC and κ is an inaccessible in V, then: V < sub > κ </ sub > is one of the intended models of Zermelo – Fraenkel set theory ; and Def ( V < sub > κ </ sub >) is one of the intended models of Von Neumann – Bernays – Gödel set theory ; and V < sub > κ + 1 </ sub > is one of the intended models of Morse – Kelley set theory.

If α is an ordinal, the cardinal number κ is called α-indescribable if for every formula φ and every subset U of V < sub > κ </ sub >

( If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.

If this were the case, then we could identify κ with some collection of 0-1 sequences of length λ.

If κ is measurable and p ∈ V < sub > κ </ sub > and M ( the ultrapower of V ) satisfies ψ ( κ, p ), then the set of α < κ such that V satisfies ψ ( α, p ) is stationary in κ ( actually a set of measure 1 ).

If λ is any ordinal, κ is λ-strong means that κ is a cardinal number and there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ and

If λ is any ordinal, κ is λ-supercompact means that there exists an elementary embedding j from the universe V into a transitive inner model M with critical point κ, j ( κ )> λ and

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 our SAC bombers were, today, capable of surviving a surprise missile attack and because of infinite dispersion or long endurance had the capability to strike at Russia again, and again, and again, those bombers would unquestionably assure our military dominance.

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 set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

Jarry once wrote, expressing some of the bizarre logic of ' pataphysics, " If you let a coin fall and it falls, the next time it is just by an infinite coincidence that it will fall again the same way ; hundreds of other coins on other hands will follow this pattern in an infinitely unimaginable fashion ".

If f is also surjective and therefore bijective ( since f is already defined to be injective ), then S is called countably infinite.

If one chooses an infinite number of distinct points in the unit interval, then there must be some accumulation point in that interval.

If one rejects that axiom, the situation is more complicated, with additional infinite cardinals that are not alephs.

If at most one of the two terms in ( 2 ) is infinite, then ( 1 ) is the same as ( 2 ).

If the rational number's denominator has any prime factors other than 2 or 5, it cannot be expressed as a finite decimal fraction, and has a unique eventually repeating infinite decimal expansion.

If the expression that defines the DFT is evaluated for all integers k instead of just for, then the resulting infinite sequence is a periodic extension of the DFT, periodic with period N.

Linear Diophantine equations take the form ax + by = c. If c is the greatest common divisor of a and b then this is Bézout's identity, and the equation has an infinite number of solutions.

If Good Times was not stopped in time, an infected computer would enter an " nth-complexity infinite binary loop " ( a meaningless term ), damaging the processor.

If a partially ordered set does not contain any infinite descending chains, it is called well-founded or, in some case, Artinian ; it is then said to satisfy the descending chain condition.

If V is a nonempty finite set then V * is a countably infinite set.

If the underlying manifold is allowed to be infinite dimensional ( for example, a Hilbert manifold ), then one arrives at the notion of an infinite-dimensional Lie group.

# If A is a disjoint union of countably many disjoint Lebesgue measurable sets, then A is itself Lebesgue measurable and λ ( A ) is equal to the sum ( or infinite series ) of the measures of the involved measurable sets.

If there were a large ( possibly infinite ) number of universes, each with possibly different physical laws ( or different fundamental physical constants ), some of these universes, even if very few, would have the combination of laws and fundamental parameters that are suitable for the development of matter, astronomical structures, elemental diversity, stars, and planets that can exist long enough for life to emerge and evolve.

( If the machine is deterministic, the possible computations are the prefixes of a single, possibly infinite, path.

If the universe is static and populated by an infinite number of stars, any sight line from Earth must end at the ( very bright ) surface of a star, so the night sky should be completely bright.

A number a is a root of P if and only if the polynomial x − a ( of degree one in x ) divides P. It may happen that x − a divides P more than once: if ( x − a )< sup > 2 </ sup > divides P then a is called a multiple root of P, and otherwise a is called a simple root of P. If P is a nonzero polynomial, there is a highest power m such that ( x − a )< sup > m </ sup > divides P, which is called the multiplicity of the root a in P. When P is the zero polynomial, the corresponding polynomial equation is trivial, and this case is usually excluded when considering roots: with the above definitions every number would be a root of the zero polynomial, with undefined ( or infinite ) multiplicity.

:" If the pantheist starts with the belief that the one great reality, eternal and infinite, is God, he sees everything finite and temporal as but some part of God.

If adding that detail only altered things slightly then it would not have been too bad, but disaster struck when it was found that the simple correction mentioned above led to infinite probability amplitudes.