Exercise 5. Identify the gerund or the participle 1 in the following sentences and translate the sentences into Russian.

1) In Anderson (1) a functor R: C→ C is introduced which has the effect of localizing the homotopy groups of 1-connected space.

2) While the effect on π ₁ is not as pleasant as Hilton, Mislin and Roitberg’s localizing functor (6), R does have the advantages of being factorial on C.

3) In discussing characters of the orthogonal group, Littlewood gives six expansions of symmetric products that are basic for expressing the characters of irreducible representations of orthogonal and symplectic groups as series in those for the general symplectic group, and conversely.

4) We could continue the process of squaring some number between 4 and 5 indefinitely and never get the exact value of .

5) Some applications are given and several noteworthy types of restrictedness are introduced including ones which display compactness, connectedness and closedness as special cases of a more general concept.

6) The normal rules of probability can therefore be applied in making the choice.

7) Having a single theorem proven both for verifying Hoare’s conditions and for checking the program state at run time is one of the possibilities opened by the predicate calculus as a programming language.

8) Having fixed the sharp end of the compass we drew a curved line.

9) The approach to the problem being considered remained traditional.

10) In factoring as in all other algebraic processes the first step is always to make a careful study of the given expression.

Additional text. Read it and retell in English. You may need to write down the formulas and base your retelling on them. Pay attention to the vocabulary and correct pronunciation.