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If and valid
If the direct-sum decomposition ( A ) is valid, how can we get hold of the projections Af associated with the decomposition??
If T is a linear operator on an arbitrary vector space and if there is a monic polynomial P such that Af, then parts ( A ) and ( B ) of Theorem 12 are valid for T with the proof which we gave.
If the exponent r is even, then the inequality is valid for all real numbers x.
If the client remains connected to a network for which this IP is valid, the server may grant the request.
If time, space, and energy are secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic system might resolve the EPR paradox ( although Bell's theorem would still be valid ).
If this understanding were valid, then it would it would follow:
If this understanding were valid, then it would it would follow:
If this understanding were valid, then it would it would follow:
If this triangulated graph is colorable using four colors or less, so is the original graph since the same coloring is valid if edges are removed.
If on the other hand Theorem 2 holds and φ is valid in all structures, then ¬ φ is not satisfiable in any structure and therefore refutable ; then ¬¬ φ is provable and then so is φ, thus Theorem 1 holds.
If the wavepacket has a relatively large frequency spread, or if the dispersion has sharp variations ( such as due to a resonance ), or if the packet travels over very long distances, this assumption is not valid.
If the works were renewed, the copyrights would still be valid in the United States.
If we neglect the possibility of overlapping states, which is valid if the temperature is high, then the number of times we count each state is approximately N < nowiki >!</ nowiki >.
If a player cannot make a valid move after first move of 6 they must pass the die to the next player.
If the defendant asserts his right to remain silent all interrogation must immediately stop and the police may not resume the interrogation unless the police have " scrupulously honored " the defendant ’ s assertion and obtain a valid waiver before resuming the interrogation.
If the defendant does reinitiate contact, a valid waiver must be obtained before interrogation may resume.
If his reasoning is valid, he goes on to argue, either it is not very immoral to value small luxuries over saving many lives, or such affluent people are very immoral.
If one player cannot make a valid move, play passes back to the other player.
If the substitution of an expression with its value is valid only at a certain point in the execution of the program, then the expression is not referentially transparent.
If an argument form is valid, the conjunction of the premises will be logically equivalent to the conclusion and this can be clearly seen in a truth table ; it is displayed.
# If they do have a demonstrable genetic relationship, do they form a valid linguistic taxon?
If the server is case sensitive and < tt >< nowiki > http :// en. example. org / wiki / URL </ nowiki ></ tt > is correct, < tt >< nowiki > http :// en. example. org / WIKI / URL </ nowiki ></ tt > or < tt >< nowiki > http :// en. example. org / wiki / url </ nowiki ></ tt > will display an HTTP 404 error page, unless these URLs point to valid resources themselves.
If the deviations from ideality are not too strong, Raoult's law will still be valid in a narrow concentration range when approaching x = 1 for the majority phase ( the solvent ).
If the number of bits changed is even, the check bit will be valid and the error will not be detected.

If and element
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 numbering does not start at 0, the constant B may not be the address of any element.
If the minimum legal value for every index is 0, then B is the address of the element whose indices are all zero.
If the concentration of element or compound in a sample is too high for the detection range of the technique, it can simply be diluted in a pure solvent.
If the keys match, then a matching element has been found so its index, or position, is returned.
If an element has isotopes that are not radioactive, they are termed " stable.
If A is a fixed element of a ring ℜ, the first additional relation can also be interpreted as a Leibniz rule for the map given by B.
If G is a group, and g is a fixed element of G, then the conjugation map
If a diatomic molecule consists of two atoms of the same element, such as H < sub > 2 </ sub > and O < sub > 2 </ sub >, then it is said to be homonuclear, but otherwise it is heteronuclear.
If no bulk flow occurs in an element of length dx, the rates of diffusion of two gases A and B must be equal and opposite, that is.
If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity.
If R is a commutative ring, and a and b are in R, then an element d of R is called a common divisor of a and b if it divides both a and b ( that is, if there are elements x and y in R such that d · x = a and d · y = b ).
If there is an edge from a vertex x to a vertex y, then the element is 1 ( or in general the number of xy edges ), otherwise it is 0.
If f is an element of G ( x, y ) then x is called the source of f, written s ( f ), and y the target of f ( written t ( f )).
* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if z is an element of V not in the closure of U, then there exists a continuous linear map with ψ ( x ) = 0 for all x in U, ψ ( z ) = 1, and || ψ || = 1 / dist ( z, U ).
If the suggested letter does not occur in the word, the other player draws one element of the hangman diagram as a tally mark.
If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.
Then an element e of S is called a left identity if e * a = a for all a in S, and a right identity if a * e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.
If r represents an arbitrary element of R, f can be viewed as a right R-homomorphism so that, or f can also be viewed as a left R module homomorphism, where.
In a flat d-dimensional space M, at any given time, the configuration of two identical particles can be specified as an element of M × M. If there is no overlap between the particles, so that they do not interact ( at the same time, we are not referring to time delayed interactions here, which are mediated at the speed of light or slower ), then we are dealing with the space
# If A is Lebesgue measurable and x is an element of R < sup > n </ sup >, then the translation of A by x, defined by A + x =
If two elements form more than one compound between them, then the ratios of the masses of the second element which combine with a fixed mass of the first element will be ratios of small whole numbers.

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