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 Af denotes the space of N times continuously differentiable functions, then the space V of solutions of this differential equation is a subspace of Af.

If D denotes the differentiation operator and P is the polynomial Af then V is the null space of the operator p (, ), because Af simply says Af.

If we are discussing differentiable complex-valued functions, then Af and V are complex vector spaces, and Af may be any complex numbers.

If and we have for all v, w in V, then we say that B is symmetric.

If a system is composed of two subsystems described in V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces.

If is a ket in V and is a ket in W, the direct product of the two kets is a ket in.

If an epidemic of cholera is suspected, the most common causative agent is V. cholerae O1.

If V. cholerae serogroup O1 is not isolated, the laboratory should test for V. cholerae O139.

If V is finite-dimensional, then V * has the same dimension as V. Given a basis

If the electrode has a positive potential with respect to the SHE, then that means it is a strongly reducing electrode which forces the SHE to be the anode ( an example is Cu in aqueous CuSO < sub > 4 </ sub > with a standard electrode potential of 0. 337 V ).

* If G is a topological group and V is a topological vector space, a continuous representation of G on V is a representation ρ such that the application defined by is continuous.

* If V is a normed vector space with linear subspace U ( not necessarily closed ) and if is continuous and linear, then there exists an extension of φ which is also continuous and linear and which has the same norm as φ ( see Banach space for a discussion of the norm of a linear map ).

* 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 V is a set of strings then V * is defined as the smallest superset of V that contains λ ( the empty string ) and is closed under the string concatenation operation.

# If V is a set of symbols or characters then V * is the set of all strings over symbols in V, including the empty string.

`` If Blue Throat has his way he'll keep us all cooped up in here for days '', he said.

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 the man on the sidewalk is surprised at this question, it has served as an exclamation.

If any one of them has any power to veto the Secretary General's decisions the nature of the organization will have changed.

If this new Bible does not increase in significance by repeated readings throughout the years, it will not survive the ages as has the King James Version.

If the case is thus determined by us to be domestic, the court has no jurisdiction.

If this capacity had not failed them, they would see that their enemy has made a disastrous miscalculation.

If the new Soviet series has followed the general pattern of previous Russian tests, the shots were roughly half fission and half fusion, meaning a fission yield of 30 to 40 megatons thus far.

If the superintendents do not receive more cooperation from Handlers, it has been suggested that licensed Judges also be qualified to judge this Class.

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.

If design head has a deep cavity, clay lid will be quite thick at this point ; ;

If you run into excess plug fouling on one truck, check to be sure that the rig has a thermostat.

If we manage to keep track of a Bombus queen after she has left her feeding place, we may discover the snug little hideout which she has fixed up for herself when she woke up from her winter sleep.

If cell Af has not previously been assigned as the information cell of a form in the text-form list, it is now assigned as the information cell of Af.

If of the founders of glottochronology Swadesh has escaped our steady plodding, and Lees has repudiated his own share in the founding, that is no reason why we should swerve.

If the transferor has substantial assets other than the claim, it seems reasonable to assume no corporation would be willing to acquire all of its properties in the dim hope of collecting a claim for refund of taxes.

If the patient can perceive figure kinesthetically when he cannot perceive it visually, then, it would seem, the sense of touch has immediate contact with the spatial aspects of things in independence of visual representations, at least in regard to two dimensions, and, as we shall see, even this much spatial awareness on the part of unaided touch is denied by the authors.

If the distant patron of the suburban branch has been frightened away from downtown by traffic problems, however, the city store can only pressure the politicians to do something about the highways or await the completion of the federal highway program.

If a statement has been assigned an address in the index word area

If she has not had such experiences, the female's normal adolescent degree of indecision will be compounded.

If the change, at first sight, seems minor, we may recall that it took the Italian painters about two hundred years to make an analogous change, and the Italian painters, by universal consent, were the most brilliant group of geniuses any art has seen.

If Barnett doesn't call a special session in 1961, it will be the first year in the last decade that the Legislature has not met in regular or special session.

If the church has followed the plan of cultivation of prospects and carried through a program of membership preparation as outlined earlier in this book, the process of assimilation and growth will be well under way.

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.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

If a is algebraic over K, then K, the set of all polynomials in a with coefficients in K, is not only a ring but a field: an algebraic extension of K which has finite degree over K. In the special case where K = Q is the field of rational numbers, Q is an example of an algebraic number field.

If μ is both inner regular and locally finite, it is called a Radon measure.

If numerical iterative methods have to be employed, the aim is to iterate until full machine accuracy is obtained ( the best that is possible with a finite word length on the computer, and within the mathematical and / or physical approximations made ).

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 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 physical reality is totally finite, then the Copenhagen interpretation might be an approximation to an information processing system below the Planck scale.

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

If black holes have a finite entropy, they should also have a finite temperature.

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

If we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: for every finite dimensional Lie algebra over F there is a simply connected Lie group G with as Lie algebra, unique up to isomorphism.

If there is a finite generator of M, then M is said to be finitely generated.

A measure μ is continuous from above: If E < sub > 1 </ sub >, E < sub > 2 </ sub >, E < sub > 3 </ sub >, … are measurable sets and E < sub > n + 1 </ sub > is a subset of E < sub > n </ sub > for all n, then the intersection of the sets E < sub > n </ sub > is measurable ; furthermore, if at least one of the E < sub > n </ sub > has finite measure, then

If has finite norm, then is infinitely close to.

If the number of solutions is finite, there are algorithms to compute the solutions.

If then we obtain a contradiction, because the sum does not exceed which is finite.

:" 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 we allow a finite automaton access to two stacks instead of just one, we obtain a more powerful device, equivalent in power to a Turing machine.

If G is a finite group and S and T are subgroups of G, then ST is a subset of G of size | ST | given by the product formula:

If A < sub > i </ sub > in R < sub > i </ sub > is an ideal for each i in I, then A = Π < sub > i in I </ sub > A < sub > i </ sub > is an ideal of R. If I is finite, then the converse is true, i. e. every ideal of R is of this form.

If ρ acts on a finite dimensional Hilbert space and has eigenvalues