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 this seems arbitrary, its effect was to treat citizens of the District of Columbia equally with citizens of the states -- at the expense of expanding a troublesome jurisdiction.

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 source is located at an arbitrary source point, denoted by the vector and the field point is located at the point, then we may represent the scalar Green's function ( for arbitrary source location ) as:

If the subject could so judge his own superior, then all lawful superior authority could lawfully be overthrown by the arbitrary judgement of an inferior, and thus all law was under constant threat.

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.

If the former, then justice is arbitrary ; if the latter, then morality exists on a higher order than God, who becomes little more than a passer-on of moral knowledge.

One can use that instruction, which is usually faster than a comparison against some arbitrary value ( requiring a subtraction ), to implement the command " If i ≤ 0, then exit the loop ".

If the inclination is zero or 180 degrees ( π radians ), the azimuth is arbitrary.

If the radius is zero, both azimuth and inclination are arbitrary.

" If we placed a living organism in a box ... one could arrange that the organism, after any arbitrary lengthy flight, could be returned to its original spot in a scarcely altered condition, while corresponding organisms which had remained in their original positions had already long since given way to new generations.

If the strategy ever called for moving on the square already chosen, the first player can then make another arbitrary move.

If a feature was intended to be located from the center of the part, the operator needs to locate it from the center of the model, not, perhaps, from a more convenient edge or an arbitrary point, as he could when using " dumb " solids.

If we consider an arbitrary constant, then average availability is represented as

If one allows arbitrary temperature dependence of the prefactor, the Arrhenius description becomes overcomplete, and the inverse problem ( i. e., determining the prefactor and activation energy from experimental data ) becomes singular.

If ( for some arbitrary reason ) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio k,

If the arbitrary area for excavation is wisely chosen, the sequence should be revealed and excavation can return to a truly stratigraphic method.

The reasoning behind existential elimination (∃ E ) is as follows: If it is given that there exists an element for which the proposition function is true, and if a conclusion can be reached by giving that element an arbitrary name, that conclusion is necessarily true, as long as it does not contain the name.

must be true for all values of c over the same domain X ; else, the logic does not follow: If c is not arbitrary, and is instead a specific element of the domain of discourse, then stating P ( c ) might unjustifiably give more information about that object.

If data of arbitrary length is to be encrypted, a simple strategy is to split the data into blocks each matching the cipher's block size, and encrypt each block separately using the same key.

If a function is defined on an interval and is an antiderivative of, then the set of all antiderivatives of is given by the functions, where C is an arbitrary constant.

If n is arbitrary, i. e., if the system is insensitive to any rotation, then every component of L is conserved ; in short, angular momentum is conserved.

* If four arbitrary points A, B, C, D are given that do not form an orthocentric system, then the nine-point circles of ABC, BCD, CDA and DAB concur at a point.

If man is actually the product of his environment and if science can discover the laws of human nature and the ways in which environment determines what people do, then someone -- a someone probably standing outside traditional systems of values -- can turn around and develop completely efficient means for controlling people.

If we try to study T using characteristic values, we are confronted with two problems.

If it is assumed that all color values are premultiplied by their alpha values (), we can rewrite the equation for output color as:

( If these values had been rounded they would have become 33K, 66K, and 131K, respectively.

If communities are developed based on something they share in common, whether that be location or values, then one challenge for developing communities is how to incorporate individuality and differences.

If F is clear from context then Ω < sub > F </ sub > may be denoted simply Ω, although different prefix-free universal computable functions lead to different values of Ω.

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 the initial value of a polynomial ( and of its finite differences ) is calculated by some means for some value of X, the difference engine can calculate any number of nearby values, using the method generally known as the method of finite differences.

If people then experience other cultures that have different values and normal behaviors, they will find that the thought patterns appropriate to their birth culture and the meanings their birth culture attaches to behaviors are not appropriate for the new cultures.

If the values are, then the growth rate between measurement and is.

If attributes are not mentioned, default values are used in each case.

If the particles have the same physical properties, the n < sub > j </ sub >' s run over the same range of values.

If either the form or the parameters of F ( x ) are determined from the data X < sub > i </ sub > the critical values determined in this way are invalid.

If there is an additional magnitude of value or a loss of value after transformation compared with before then the relation between values ( proportional to labor ) and prices ( proportional to total capital advanced ) is incomplete.

If " this statement is false " is denoted by A and its truth value is being sought, it is necessary to find a condition that restricts the choice of possible truth values of A.

If there is an even number of observations, then there is no single middle value ; the median is then usually defined to be the mean of the two middle values.

If only the second and third conditions of the definition of measure above are met, and takes on at most one of the values, then is called a signed measure.

If the desired output values are often incorrect ( because of human error or sensor errors ), then the learning algorithm should not attempt to find a function that exactly matches the training examples.

If the feature vectors include features of many different kinds ( discrete, discrete ordered, counts, continuous values ), some algorithms are easier to apply than others.

If the feedback network is made of components with relatively constant, stable values, the variability of the op-amp's open loop response does not seriously affect the circuit's performance.

If c, h, and e were all changed so that the values they have in metric ( or any other ) units were different when we looked them up in our tables of physical constants, but the value of α remained the same, this new world would be observationally indistinguishable from our World.

If these quantities are signed, the length of the arc between any two points on the parabola is always shown by the difference between their values of

If the random variable is real-valued ( or more generally, if a total order is defined for its possible values ), the cumulative distribution function gives the probability that the random variable is no larger than a given value ; in the real-valued case it is the integral of the density.

* If asked to imagine the graph of the relationship y = ax < sup > 2 </ sup >, one typically visualizes a range of values of x, but only one value of a.