A. J. Lohwater's Russian-English Dictionary of the Mathematical Sciences

1.	Present participle, having the form of an adjective; this form is listed as an adjective in the dictionary. ↑
2.	Short form, neuter, of открытый (open) indicates a predicative use with the verb "to be" omitted; note the short form, neuter, of насыщенный (saturated). ↑
3.	Note that его is not declined, even though it is linked to образ. ↑
4.	The verb являться (to be) is followed by the instrumental case. ↑
5.	отношение эквивалентности (lit. relation of equivalence) is the usual way of expressing the English phrase equivalence relation. In Russian, the noun "equivalence" cannot be used as an adjective without adding an adjectival ending. It is quite possible to form the adjective эквивалентностный (equivalence, pert. to an equivalence) and write эквивалентностное отношение (equivalence relation), but it lacks one essential ingredient: it is not the idiomatic expression for "equivalence relation" in Russian. ↑
6.	This is the present passive participle индуцируемый of индуцировать (to induce) and has a second meaning "inducible"; both meanings are given in the dictionary. ↑
7.	This phrase in parentheses may or may not be inserted, at the discretion of the translator. ↑

(3)

Если областью,¹ в которой задана² начальная температура t(x, y, z), является всё³ пространство, то⁴ решение может быть записано⁵ в замкнутой форме.

If the domain, in which the initial temperature t

(x, y, z), is prescribed, is the whole space, then the solution can be written in closed form.

Notes

1.	Instrumental case with являться (to be). ↑
2.	Short form, feminine, of заданный; a predicative use is indicated here, and the passive participle gives the passive sense to the translation. ↑
3.	всё is an adjective, and its synonyms often give a smoother translation than "all." ↑
4.	The conjunction "then" rather than the neuter of тот. ↑
5.	The same use of the passive participle as in note 2; the link-verb быть must be used here following может. Note also that может быть as a phrase has the meaning "perhaps," but that the context rules out this possibility. See also Some Special Verbs below. ↑

(4)

Для того, чтобы¹ этот процесс имел² смысл, необходимо,³ чтобы⁴ он давал единственный результат.

In order that this process have meaning, it is necessary that it give a unique result.

Notes

1.	для того, чтобы is the usual phrase "in order that," and the verb to follow is always subjunctive or conditional. ↑
2.	The conditional or subjunctive mood of иметь (to have); the particle бы has been attached to the что (that) in the clause. ↑
3.	Note the short form and the implied predicative. ↑
4.	The conditional of давать (to give) is давал бы, and the particle бы usually combines with что. ↑

(5)

Повторяя¹ это рассуждение, мы получим² тот же³ результат для функции  f (x).

By repeating this argument, we obtain the same result for the function  f (x).

Notes