If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

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 so, they may have originally comprised residual Celtic elements in central eastern Europe such as the Cotini, who formed a Celtic enclave in the Germanic-speaking zone and are described by Tacitus as iron-ore miners working as tributaries of the powerful Quadi Germanic people.

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 we assume the controller C, the plant P, and the sensor F are linear and time-invariant ( i. e., elements of their transfer function C ( s ), P ( s ), and F ( s ) do not depend on time ), the systems above can be analysed using the Laplace transform on the variables.

If so, it is unclear how much Tatian may have borrowed from this previous author in determining his own narrative sequence of Gospel elements.

If ~ is an equivalence relation on X, and P ( x ) is a property of elements of X, such that whenever x ~ y, P ( x ) is true if P ( y ) is true, then the property P is said to be well-defined or a class invariant under the relation ~.

If is an equivalence relation on, and is a property of elements of such that whenever, is true if is true, then the property is said to be an invariant of, or well-defined under the relation.

If found, it removes it by concatenating all the elements prior to it with all the elements after it.

) If a set is finite, its elements may be written as a sequence:

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 R is an integral domain then any two gcd's of a and b must be associate elements, since by definition either one must divide the other ; indeed if a gcd exists, any one of its associates is a gcd as well.

If the Gospel of Barnabas is seen as an attempted synthesis of elements from both Christianity and Islam, then 16th and 17th century parallels can be suggested in Morisco and anti-Trinitarian writings.

If so, these huge stars would have quickly consumed their supply of fuel and became supernovae, releasing heavy elements into the interstellar medium.

If a game has a board, the board is usually irregular rather than uniform or symmetric ( like Risk rather than chess or Scrabble ); the board is often random ( like Settlers of Catan ) or has random elements ( like Tikal ).

If the 3 generations are then put in a 3x3 hermitian matrix with certain additions for the diagonal elements then these matrices form an exceptional ( grassman -) Jordan algebra, which has the symmetry group of one of the exceptional Lie groups ( F < sub > 4 </ sub >, E < sub > 6 </ sub >, E < sub > 7 </ sub > or E < sub > 8 </ sub >) depending on the details.

If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are " spread apart " from each other while leaving the arithmetic mean unchanged — then the geometric mean always decreases.

If a set of non-identical numbers is subjected to a mean-preserving spread — that is, two or more elements of the set are " spread apart " from each other while leaving the arithmetic mean unchanged — then the harmonic mean always decreases.

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.

If R is the direct sum of the rings R < sub > 1 </ sub >,..., R < sub > n </ sub >, then the identity elements of the rings R < sub > i </ sub > are central idempotents in R, pairwise orthogonal, and their sum is 1.

If the figure comprises more than three elements, it is said to be rekit “ extended ”.

If a particular KR is inadequate in some way, that set of problematic KR elements may be transformed by importing them into a KBS, modified and operated on to eliminate the problematic elements or augmented with additional knowledge imported from other sources, and then exported into a different, more appropriate KR.

If the list is a statistical sample, we call the resulting statistic a sample mean.

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

If are independent and identically distributed random variables, each with a standard Cauchy distribution, then the sample mean has the same standard Cauchy distribution ( the sample median, which is not affected by extreme values, can be used as a measure of central tendency ).

If this critical sample is obtained at the time of hypoglycemia, before it is reversed, it can provide information that would otherwise require a hospital admission and unpleasant starvation testing.

If the degree of underrepresentation is small, the sample can be treated as a reasonable approximation to a random sample.

If entire segments of the population are excluded from a sample, then there are no adjustments that can produce estimates that are representative of the entire population.

If a sample of cobalt-60 atoms is magnetized so that they spin counterclockwise around some axis, the beta radiation resulting from their nuclear decay will be preferentially directed opposite that axis.

If the data set were based on a series of observations obtained by sampling from a statistical population, the arithmetic mean is termed the " sample mean " () to distinguish it from the " population mean " ( or < sub > x </ sub >).

If an upregulated gene is observed by an abundance of mRNA on the northern blot the sample can then be sequenced to determine if the gene is known to researchers or if it is a novel finding.

If we assemble a deck of 52 playing cards with no jokers, and draw a single card from the deck, then the sample space is a 52-element set, as each card is a possible outcome.

If we do not alter the system, then the parameter λ is unchanged from measurement to measurement ; if, on the other hand, we modulate the system by replacing the sample with a more radioactive one, then the parameter λ would increase.

If the parameter " dopefish " is added to the executable, a sample of burping is heard and Scott's Mystical Head is seen spinning in circles on the screen.

If subgroups ' variances differ significantly and the data need to be stratified by variance, then there is no way to make the subgroup sample sizes proportional ( at the same time ) to the subgroups ' sizes within the total population.

If the tip is moved across the sample in the x-y plane, the changes in surface height and density of states cause changes in current.

If the beam enters the sample perpendicular to the surface, then the activated region is uniform about the axis of the beam and a certain number of electrons " escape " from within the sample.

If all of the are drawn from the same distribution and the integer weights indicate frequency of occurrence in the sample, then the unbiased estimator of the weighted population variance is given by

If the collected sample contains too few sperm cells to likely have a spontaneous fertilization, intracytoplasmic sperm injection can be performed.

If it is desired to sample more of the exponential electron portion of the characteristic, an asymmetric double probe may be used, with one electrode larger than the other.

If the patient is infected with H. pylori, the breath sample will contain radioactive carbon dioxide.