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 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 his scholarship and formal musicianship were not all they might have been, Mercer demonstrated at an early age that he was gifted with a remarkable ear for rhythm and dialect.

If a distinction is to be made between divination and fortune-telling, divination has a formal or ritual and often social character, usually in a religious context, as seen in traditional African medicine ; while fortune-telling is a more everyday practice for personal purposes.

If civil commitment proceedings follow, the evaluation is presented in a formal court hearing where testimony and other evidence may also be submitted.

If an appropriate form of written text is made available before formal schooling begins, reading should be learned inductively, emerge naturally, and with no significant negative consequences.

If a manager decides that his client wrestler should tap out, but cannot convince the wrestler himself to do so, he may throw in the towel ( literally taking a gym towel and hurling it into the referee's line of sight ); this is the same as a submission, as the manager is, in kayfabe, considered the wrestlers agent, and therefore, authorized to make formal decisions ( such as forfeiting a match ) on the client's behalf.

If the software is safety-critical and can have negative impact on human life ( e. g., nuclear power systems, medical equipment ), more formal requirements documentation is often required.

If the male is an Abdi Dalem royal servant, courtier or particularly " peko-peko " ( taken directly from Japanese to mean obsequious ) or even a highly formal individual, he will retreat face and head downcast, never show his side or back to his superior, and retreat backwards in the following posture: left-arm crossed against the chest, right-arm hanging and walking stooped.

More precisely, if A is a finite set of generators for G then the word problem is the membership problem for the formal language of all words in A and a formal set of inverses that map to the identity under the natural map from the free monoid with involution on A to the group G. If B is another finite generating set for G, then the word problem over the generating set B is equivalent to the word problem over the generating set A.

If one spoke in the lingua or sermo Latinus one merely spoke Latin, but if one spoke latine or latinius (" more Latinish ") one spoke good Latin, and formal Latin had latinitas, the quality of good Latin, about it.

If we denote by exp ( X ) the formal power series

If the formal power series g with g ( 0 ) =

If I is an index set and X < sub > I </ sub > is the set of indeterminates X < sub > i </ sub > for i ∈ I, then a monomial X < sup > α </ sup > is any finite product of elements of X < sub > I </ sub > ( repetitions allowed ); a formal power series in X < sub > I </ sub > with coefficients in a ring R is determined by any mapping from the set of monomials X < sup > α </ sup > to a corresponding coefficient c < sub > α </ sub >, and is denoted.

If she accepts, she is tonsured in a formal service during which she is given the outer robe ( Exorassa ) and veil ( Epanokamelavkion ) to wear, and ( because she is now dead to the world ) receives a new name.

If Regency era is being used to describe the transition between Georgian and Victorian eras, the focus is on the pre-Victorian period from 1811, when the formal Regency began, through 1837 when Queen Victoria succeeded William IV.

* If the formal specification is in an operational semantics, the observed behavior of the concrete system can be compared with the behavior of the specification ( which itself should be executable or simulateable ).

* If the formal specification is in an axiomatic semantics, the preconditions and postconditions of the specification may become assertions in the executable code.

If implemented on a global scale, the inclusion of women in the formal workforce ( like a fortune 500 company ) can increase the economic output of a nation.

( If you're working in a formal system that doesn't allow you to introduce new symbols after proving theorems, then you will have to use relation symbols to get around this, as in the next section.

If either party takes an appeal, the lower court produces a copy with a unique seal to authenticate the formal record.

If it's considered formal communications as they occur in traditional military organizations, messages have a " one-way " directional characteristic.

If the pathologic examination of the resected specimen shows incomplete resection or deep invasion by tumor, the patient would need a formal stomach resection.

If he had stipulated that the name was a formal name, governed by rules of nomenclature, then his spelling would have been an orthogonal mutation and one would simply correct the spelling in subsequent publications ( e. g. Straminopiles ).

If five or more MPs challenge the Speaker's opinion, a formal division follows.