If an abelian group is expressed as a direct product, and the dual group likewise decomposed, and the elements of each sorted in lexicographic order, then the character table of the product group is the Kronecker product ( tensor product ) of the character tables for the two component groups, which is just a statement that the value of a product homomorphism on a product group is the product of the values:

If his circumspection in regard to Philip's sensibilities went so far that he even refused to grant a dispensation for the marriage of Amadee's daughter, Agnes, to the son of the dauphin of Vienne -- a truly peacemaking move according to thirteenth-century ideas, for Savoy and Dauphine were as usual fighting on opposite sides -- for fear that he might seem to be favoring the anti-French coalition, he would certainly never take the far more drastic step of ordering the return of Gascony to Edward, even though, as he admitted to the English ambassadors, he had been advised that the original cession was invalid.

If the moments of the aligned and anti-aligned ions balance completely so as to have zero net magnetization, despite the magnetic ordering, then it is an antiferromagnet.

If the entire volume of the observable universe ( taken to be 3 × 10 < sup > 80 </ sup > m < sup > 3 </ sup >) were packed solid with fine dust particles about 1. 5 micrometres in size, then the number of different ways of ordering these particles ( that is, assigning the number 1 to one particle, then the number 2 to another particle, and so on until all particles are numbered ) would be approximately one googolplex.

In computer science, a heap is a specialized tree-based data structure that satisfies the heap property: If A is a parent node of B then key ( A ) is ordered with respect to key ( B ) with the same ordering applying across the heap.

If the temperature rise is too high, thermal motion will destroy the delicate cooperative ordering of the LC phase, pushing the material into a conventional isotropic liquid phase.

If a thing is to be well established or founded, its arche or static point must be secure, and the most secure foundations are those provided by the gods: the indestructible, immutable and eternal ordering of things.

That means, define E ≤ F if E is an initial segment of F and the ordering of the members in E is the same as their ordering in F. If E is a chain in A, then the union of the sets in E can be ordered in a way that makes it a continuation of any set in E ; this ordering is a well ordering, and therefore, an upper bound of E in A.

On his deathbed, he committed a manuscript to his chaplain, ordering him to consult with Libert Fromondus, a theology professor at Leuven, and Henri Calenus, a canon at the metropolitan church, and to publish the manuscript if they agreed it should be published, adding " If, however, the Holy See wishes any change, I am an obedient son, and I submit to that Church in which I have lived to my dying hour.

If the move ordering for the search is optimal ( meaning the best moves are always searched first ), the number of leaf node positions evaluated is about O ( b * 1 * b * 1 *...* b ) for odd depth and O ( b * 1 * b * 1 *...* 1 ) for even depth, or.

If the input values to the spreadsheet change, all of the remaining values of the spreadsheet may be recomputed with a single evaluation per cell, by topologically ordering the cells and re-evaluating each cell in this order.

If one is only interested in stochastic ordering of the two populations ( i. e., the concordance probability P ( Y > X )), the U test can be used even if the shapes of the distributions are different.

If n is the degree of P, then the splitting type is a partition Π of n. Considering also the Galois group G of K over Q, each g in G is a permutation of the roots of P in K ; in other words by choosing an ordering of α and its algebraic conjugates, G is faithfully represented as a subgroup of the symmetric group S < sub > n </ sub >.

* If the axiom of constructibility holds then there is a subset of the product of the Baire space with itself which is and is the graph of a well ordering of Baire space.

If the axiom holds then there is also a well ordering of Cantor space.

( If there should be infinitely many indeterminates, this convention is incompatible with the condition of being a well ordering, and one would be forced to use the opposite ordering ; however the case of polynomials in infinitely many variables is rarely considered.

If the a < sub > i </ sub > are rationally independent numbers ( so in particular none of them are zero and all fractions are irrational ) then a tie can never occur, and the weight vector itself specifies a monomial ordering.

At the outbreak of the Civil War, he sent a telegram to the Treasury agents in New Orleans ordering that: " If any one attempts to haul down the American flag, shoot him on the spot.

Deviously ordering someone to accuse Lao Ai of a crime punishable by castration, Lü also privately told the queen dowager, " If we can fake the castration, we can make him a servant in the harem.

If the result consists of nodes, then duplicates are eliminated and the nodes are returned in document order, an ordering defined in terms of the relative positions of the nodes in the original XML tree.

If F is an ordered field ( not just orderable, but a definite ordering is fixed as part of the structure ), the Artin – Schreier theorem states that F has an algebraic extension, called the real closure K of F, such that K is a real closed field whose ordering is an extension of the given ordering on F, and is unique up to a unique isomorphism of fields ( note that every ring homomorphism between real closed fields automatically is order preserving, because x ≤ y if and only if ∃ z y = x + z < sup > 2 </ sup >).

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 he found data that fitted his general plan, he used it and counted his sources trustworthy.

If they are used, the walls and roof should be 10 inches thick to give the same protection as the 8-inch solid concrete blocks.

If your state has no provisions for the numbering of pleasure boats, you must apply for a number from the U.S. Coast Guard for any kind of boat with mechanical propulsion rated at more than 10 horsepower before it can be used on Federal waterways.

Note: If 1/2-inch panel board is used inside and out, or 5/8-inch one side and 3/8-inch the other, and 1/8-inch glass is used, stock lumber in Af, Af, and Af can be used in making the glass panels.

If only state funds were used to pay for the vocational education, it could be argued that the state should not have to bear the cost of vocational training which would benefit employers in other states.

If the master of scops who was most responsible for the poem ever used kennings that were traditional, he was at least partly deprived of free will and not inclined towards shrewd and sophisticated misuse of speech elements.

If the one- or two-digit address of an index word or electronic switch is used or is included in the operand of an XRESERVE or SRESERVE statement ( see page 99 ), the corresponding index word or electronic switch is reserved.

If the voice is just a shade less glorious than it used to be, it is still a beautiful instrument, controlled and flexible.

If, for instance, someone says " I buried $ 100 in the bank ", most people would not think someone used a shovel to dig in the mud.

If " biweekly " is used in a conversation about a meeting schedule, it may be difficult to infer which meaning was intended.

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

If the latter, Amaryllis would be the correct name for the genus Hippeastrum, and a different name would have to be used for the genus discussed here.

If an alpha channel is used in an image, it is common to also multiply the color by the alpha value, to save on additional multiplications during compositing.

If this article / noun pair is used as the object of a verb, it ( usually ) changes to the accusative case, which entails an article shift in German – Ich sehe den Wagen.

If, conversely, tin is used to coat steel, when a breach of the coating occurs it actually accelerates oxidation of the iron.

If the amount in the sample is below an instrument's range of measurement, the method of addition can be used.

If an ESN / MIN Pair is intercepted, it could then be cloned onto a different phone and used in other areas for making calls without paying.

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 used for drinking, mineral supplements or mineralization is necessary.

If the assets used are not identical ( so a price divergence makes the trade temporarily lose money ), or the margin treatment is not identical, and the trader is accordingly required to post margin ( faces a margin call ), the trader may run out of capital ( if they run out of cash and cannot borrow more ) and go bankrupt even though the trades may be expected to ultimately make money.

If chosen, the SysOp could then initiate a text-to-text chat with the caller ; similar to what commercial websites have used to sell and support products.

If either one die or the other but not both can be moved, the higher must be used.

If F. tularensis were used as a weapon, the bacteria would likely be made airborne for exposure by inhalation.