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

7. The adverbial participle, or the gerund. The adverbial participle exhibits properties of both the verb and the adverb. Verbs ending in the reflexive particle exhibit the particle in the gerund, and the gerund exhibits the aspect of the verb from which it is derived. The form of the gerund, as with all adverbs, is invariable. It acts as a modifier by describing the environment in which the action of the predicate verb occurs. The adverbial participle may be past or present; past participles are formed from perfective verbs and refer to actions occurring prior to the action of the main verb, while present participles are formed from imperfective verbs and refer to actions occurring simultaneously with the action of the main verb.

a. Past adverbial participles. The suffix -в- (and less frequently -вши-, -ши-) is added to the stem of the past tense.

прочитать – прочита/л – прочита/в, прочитав → having read

сказать – сказа/л – сказа/в, сказав → having said

написать – написа/л – написа/в, написав → having written

принести – принёс – принёс/ши, принёсши → having brought

Прочитав доклад, он ответил на вопросы профессора. → Having read the paper, he answered the professor's questions.

b. Present adverbial participles. The suffix -я (-а after ж, ч, ш, щ) is added to the present-tense stem of the imperfective verb.

читать – чита/ют – чита/я, читая → reading

Отвечая на вопрос профессора, он сделал ошибку. → In answering the professor's question, he made a mistake.

c. Gerunds from verbs ending in the reflexive particle -ся. Gerunds may be formed from verbs ending in -ся, although the ending always appears as -сь.

8. Auxiliaries: должен, нужно, нельзя, можно, мочь. These verbs, adverbs, and adjectives are translated as auxiliaries.

a. Должен, "must, ought to." This form is used as a short adjective and is usually followed by an infinitive.

должен (m.), должна (f.), должно (n.), должны (pl.)

Должны быть удовлетворены граничные условия. → The boundary conditions must be satisfied.

b. Нужно, надо, "must, it is necessary," нельзя, "must not, it is impossible, cannot." These forms are invariable and take the logical subject if there is one, in the dative case, and are followed by the infinitive.

Чтобы представить уравнение в номографической форме, нужно сначала найти зквивалентный детерминант. → То represent an equation nomographically (in nomographic form), we must first find an equivalent determinant.

Теорему Абеля нельзя распространить на пути, касательные единичной окружности. → It is not possible to extend Abel's theorem to paths which are tangent to the unit circle.

c. Можно, "may, can, it is possible." This form is invariable and is followed by the infinitive.

Используя интеграл Фурье, можно сконструировать сферические или цилиндрические волны из плоских волн. → By using the Fourier integral, it is possible to construct spherical or cylindrical waves out of plane waves.

d. Мочь, "to be able to." This verb is conjugated as follows:

present tense
я могу ты можешь он может		мы можем вы можете они могут
past tense   мог, могла, могло, могли
participles
present  past		active могущий могший

Видоизменения центральной предельной теоремы могут всё ещё показать, что распределение приближённо нормально. → Modifications of the central limit theorem may still show that the distribution is approximately normal.

C. The Verbs of Motion

The verbs of motion, or more specifically, of going and carrying, exhibit a further distinction within the imperfective category. While the majority of verbs have only one imperfective form to describe two separate types of action (i.e., action in progress and habitual action), the verbs of motion have two imperfective forms which distinguish these separate meanings. The determinate imperfective describes an action (going or carrying) in progress and in a specific direction. The indeterminate imperfective describes habitual actions (going or carrying) and actions in which no specific direction is involved.

For instance, the two imperfective forms of "to go by one's own means" are the indeterminate ходить and the determinate идти:

Профессор ходит (ходил) в лабораторию каждый день. → The professor goes (went) to the laboratory every day.

Профессор идёт (шёл, когда я его видела) в лабораторию. → The professor is going (was going, when I saw him) to the laboratory.

The conjugations of these two verbs are as follows: