A. J. Lohwater's Russian–English Dictionary of the Mathematical Sciences. — Second Edition, Revised and Expanded. Edited by R. P. Boas. — American Mathematical Society. Providence, Rhode Island.  [Если R. P. Boas в своём предисловии говорит, что этот словарь адресован, в первую очередь, англоязычному читателю, которому требуется по роду деятельности читать русские статьи и книги по математике, то я, готовя этот файл, делал всё в расчёте на русскоговорящую аудиторию. Поэтому, несмотря на оставленный дайджест по грамматике русского языка, я выкинул из этого словаря все ударения, произношения и тому подобное (в конце концов, эта книга рассчитана на математиков, а не лингвистов). — E.G.A.]

Если обнаружите опечатки, которые ускользнули от моего взгляда, то сделайте доброе дело — сообщите мне об этом (просто выделите double click'ом слово с ошибкой в Internet Explorer, перекиньте его через буфер обмена в тело письма и отправьте по адресу ega-math@narod.ru) и я оперативно внесу исправления.

This dictionary is intended for use primarily by speakers of English who need to understand Russian mathematical writing and/or to translate such writing. As far as possible, the "definitions" are translations rather than explanations, and reflect current usage in mathematical writing. The vocabulary has been extensively enlarged and brought up to date, although it retains some obsolete terms that may be needed by users who have to consult older literature. Users may encounter material that has not been composed in the most approved or correct style. Therefore, some words that are frowned on by Soviet purists have been retained, or added, because they occur in writing by Soviet authors whose native language is not Russian, and whose knowledge of Russian is less than perfect.

The dictionary must not be inverted: that is, read backwards as an English-Russian dictionary. We have attempted to give idiomatic translations, but a Russian word is not necessarily a correct or exact equivalent of the English word that is used to translate it in a particular context. Just because краевая задача can be correctly translated as "boundary value problem" does not mean that either, or any part, of the Russian words means "value."

Many users of the dictionary have asked for the addition of stress markings on Russian words, and these have now been supplied. [В этом файле ударения оставлены только в исключительных случаях, например, в фамилиях некоторых математиков. Я и сам обычно произношу "Кронекер" и "Якоби" не так, как надо. :) — E.G.A.] We have added many new entries and amplified old entries; and we have eliminated as many as possible of the mistakes of the first edition. However, some inconsistencies remain. A few spurious words have been added as a trap for plagiarists.

The grammar section has been rewritten by Alana I. Thorpe of the Brown University Slavic Department. We have added some notes on Russian mathematical notation. An appendix contains complete paradigms of a large number of selected words, to facilitate the identification of irregular inflected forms. The grammar is intended only for reference. Users who need instruction in Russian grammar should consult publications that are devoted to that topic. S. H. Gould's A Manual for Translators of  Mathematical Russian, published by the American Mathematical Society, contains many useful suggestions that are beyond the scope of the Dictionary.

We have assumed that users will know enough Russian to be able to recognize inflected forms of words. We have followed the common practice of listing nouns in the nominative singular, adjectives in the masculine nominative singular, and verbs in the imperfective infinitive, with the perfective (if there is one) in parentheses.

Theoretically, the Russian spelling of foreign names is supposed to represent the pronunciation used in the original language. Thus, Эйлер = Euler, Надь = Nagy, Гатó = Gâteaux. Nevertheless, it is often difficult to determine the correct spelling of the original name, especially when it is French or Hungarian. We have included a selection of Russian versions of mathematicians' names.

Adjectives formed from proper names are not capitalized in Russian. If a word seems strange, it very well may be such an adjective.

Some entries may seem unnecessary, because they mean just what they look like. There are, however, enough Russian words that resemble English words, but mean something entirely different, that we preferred to include too many rather than too few.

We thank the hundreds of mathematicians, from many countries, who have made the revision of the Dictionary possible by contributing suggestions for corrections and additions.

The Society intends to update the Dictionary from time to time. Consequently, users are encouraged to submit additions and corrections to Russian-English Dictionary, c/o American Mathematical Society, P.O.Box 6248, Providence, RI 02940-6248.

abbrev. abbreviation

acc. accusative case

adj. adjective

adv. adverb

cf. compare

colloq. colloquial

compar. comparative

conj. conjunction

dat. dative case

f. feminine

gen. genitive case

imperf. imperfective

instr. instrumental case

loc. locative case

m. masculine

n. neuter

nom. nominative case

num. numeral

ord. num. ordinal numeral

part. participle

perf. perfective

pl. plural

p.n. proper name, personal name

pred. predicate

prep. preposition

pron. pronoun

sing. singular

v. verb

Like any inflected Indo-European language, the Russian language is studied by means of an analysis of the declension of nouns and adjectives and the conjugation of verbs, together with their use in conjunction with prepositions, connectives, etc. The first obstacle which the beginner encounters, however, is the problem of the alphabet. Once the alphabet has been learned, the study of Russian becomes relatively simple, certainly no more difficult than the study of German. The Russian, or Cyrillic, alphabet is an interesting combination of Roman, Greek, and Hebrew characters; following the Bolshevik Revolution, the orthography was revised and modernized, and, in the alphabet given in the following table, the older characters are listed for the convenience of those readers who must make occasional references to the pre-Revolution literature. Since the italicized alphabet is used in places of such critical interest as the statements of theorems, etc., many phrases and sentences are duplicated in italics in the grammatical sketch so that the reader will become accustomed to reading italicized printing.

No universal agreement has been reached on the transliteration of the Russian alphabet, and the most commonly used systems of transliteration are given in the following table. System I is that used by the Library of Congress and is the most commonly used system for nontechnical literature in the United States. System II is the system used in Mathematical Reviews from 1980 onward. System III was used in Mathematical Reviews from 1962 to 1979. System IV is a German system which is frequently encountered in mathematical literature.

Russian is an inflected language; its nouns and adjectives are declined into six cases depending upon their function in a sentence. Also, nouns are grouped into three genders, masculine (m.), feminine ( f .), and neuter (n.), and the gender of a given noun may generally be determined by its ending.

1. Nominative case. A noun in the nominative case denotes the subject or the predicate nominative.
   1. Subject.

      Мы знаем. → We know.

      Студент знает. → The student knows.

   2. Predicate nominative.

      Они студенты. → They are students.

   3. The nominative form is the one listed in the dictionaries.
       
2. Genitive case. A noun appears in the genitive case when expressing any of the following:
   1. Possession.

      работа профессора → the professor's work, the work of the professor

      У профессора работа. → The professor has work.

   2. "Of" clauses.

      решение уравнения → the solution of the equation

   3. Object of negation.

      Число X не превосходит этой величины. → The number X does not exceed the magnitude.

   4. Complement of certain verbs.

      Мы достигнем результата. → We shall achieve the result.

   5. Following numbers (except the number one and any compound with the number one):
      1. Genitive singular following 2, 3, and 4, as well as any compounds in which the final component is 2, 3, or 4 (e.g., 22, 33, 54): три (3) книги, "three books"
      2. Genitive plural following 5 through 20, as well as any compounds in which the final component is 5, 6, etc.: пять (5) книг, "five books"
   6. Following words denoting quantity.

      Сколько теорем? → How many theorems?

      много теорем → a lot of theorems
       

3. Dative case. A noun appears in the dative ease in the following environments:
   1. Indirect object.

      Студент читает доклад профессору. → The student gives (reads) a paper to the professor.

   2. Impersonal expressions (sentences in which there is no grammatical subject):

      Ивану легко читать по-русски. → It's easy for John to read Russian.

      Студенту надо заниматься. → The student must study. It is necessary for the student to study.
       

4. Accusative case. A noun in the accusative case denotes the direct object: рассмотрим сумму, "let us consider the sum." In the accusative singular of masculine animate nouns, the accusative coincides in form with the genitive singular, while inanimate nouns exhibit the usual accusative endings: он читал доклад (inanimate masculine object доклад), "he read the paper"; он видел студента (animate masculine object студент), "he saw the student." Feminine nouns have only one form of the accusative in the singular. However, in the plural, all animate nouns, regardless of gender, have the form of the accusative which coincides with the genitive plural, while the accusative for inanimate nouns coincides with the nominative plural.
    
5. Instrumental case. A noun in the instrumental case denotes the following:
   1. Means by which something is done.

      выражать формулой → to express by a formula

      Он пишет мелом. → He writes with chalk.

   2. Agent in a passive construction.

      Статья написана Иваном. → The article is written by John.

   3. Manner of action.

      говорить тихим голосом → to speak in a quiet voice
       

6. Locative case. A noun occurs in the locative case after the following prepositions:

   в: в городе → in the city

   на: на собрании → at the meeting

   о: о проблеме → about the problem

   In addition to the above uses, each preposition requires its object to be in a particular case. As described above under the Locative case, the three prepositions в, на, and о require the locative case. The preposition между requires the instrumental case, while the preposition около requires the genitive case. The preposition and its object, which may include one or more adjectives as well as the noun or pronoun, are adjacent to each other in a given sentence. See the table of prepositions below.

   Also, certain verbs require their objects to be in a particular case. For instance, the verb называться (to be called, to be named) generally requires its object to be in the instrumental case. However, there has been a tendency for this verb to require the nominative case, as well.
    

7. Word order. Word order in Russian is flexible because each noun and its modifiers are in a particular case which expresses a specific meaning. In English, the word order subject-verb-object is the standard, while in Russian this order is not mandatory. Because the subject of a sentence is in the nominative case, the subject may appear at the end of the sentence; the reader can always identify the subject by its case. Similarly, the direct object is in the accusative case, while the indirect object appears in the dative case; these objects may appear anywhere in the sentence because the reader can always identify them by their cases. However, the word order subject-verb-object appears frequently. There are constraints, of course, in the ordering of components within a sentence. For instance, a preposition is not separated from its object, nor is a noun separated from its adjective in a noun phrase. Thus, noun phrases and prepositional phrases are generally placed as units within a sentence.

   Mathematical symbols are of course not inflected, so that, strictly speaking, Russian cannot say "the number of edges of Q" but has to say число граней конуса Q ("the number of edges of the cone Q"). The extra word is not needed in English, and consequently a translation will read better if the extra конуса is omitted. However, some Russian authors now disregard the rule, apparently when "the syntax is clear from the context," and write число граней Q to mean "the number of edges of Q," where the translator has to insert the "of" to prevent the reader from thinking that the edges are what is denoted by Q.

   A further very common ambiguity arises when an author wants to describe an object and give it a name. For example, if we have a sequence of increasing continuous functions and want to name it S, we would be likely to write "a sequence S of increasing continuous functions." Russian prefers to put the descriptive phrase first: последовательность возрастающих непрерывных функций S. A reader who is not aware of this construction is quite likely to translate the phrase as "a sequence of increasing continuous functions S," and give the impression that the functions are called S, whereas the author's intention was to call the sequence itself S.
    

In general, masculine nouns end in a consonant with or without a soft sign, feminine nouns in the vowels -а, -я, and -ь, and neuter nouns in the vowels -о, -е. However, a more complete listing is given below. Note that the nouns presented here are in the nominative case, which is the case in which they are typically presented in the dictionary. The endings for each of the other cases are presented later.

1. Masculine nouns may end in a consonant, with or without a -ь, -й.
   1. Consonant.

      интеграл, ранг, выигрыш

   2. Consonant plus -ь.

      делитель

   3. -й: случай, край
       
2. Feminine nouns may end in -а, -я, -ия, -ь.
   1. -а: сторона, задача
   2. -я: потеря
   3. -ия: функция
   4. -ь: часть, приводимость, вещь
       
3. Neuter nouns may end in -о, -е, -ие, -мя.
   1. -о: место, множество
   2. -e: поле, море
   3. -ие: отображение
   4. -мя: время, имя
       
4. The plural endings for masculine, feminine, and neuter nouns in the nominative are as follows:
   1. Masculine and feminine plural: -ы, -и, -а

      интегралы, случаи, профессора, края (m.)

      стороны, задачи (f.)

   2. Neuter plural: -а, -я

      множества, поля, отображения
       

The endings given above in part B are those of the nominative case. Each of the other five cases exhibits its own set of endings for singular and plural. For nouns derived from adjectives, refer to the section on the adjective for a complete list of endings.

1. Genitive case.
   1. Masculine nouns have the following endings:
      1. Singular: -а, -я

         случай (nom. sing.), случая (gen. sing.)

         интеграл (nom. sing.), интеграла (gen. sing.)

      2. Plural: -ов, -ев, -ей

         ранг (nom. sing.), рангов (gen. pl.)

         случай (nom. sing.), случаев (gen. pl.)

         делитель (nom. sing.), делителей (gen. pl.)

   2. Feminine nouns have the following endings:
      1. Singular: -ы, -и

         перестановка (nom. sing.), перестановки (gen. sing.)

         сторона (nom. sing.), стороны (gen. sing.)

         задача (nom. sing.), задачи (gen. sing.)

         потеря (nom. sing.), потери (gen. sing.)

      2. Plural: -Ø, -ь, -и, -ей ¹

         сторона (nom. sing.), сторон_ (gen. pl.)

         задача (nom. sing.), задач_ (gen. pl.)

         перестановка (nom. sing.), перестановок_ (gen. pl.) ²

         потеря (nom. sing.), потерь (gen. pl.)

         часть (nom. sing.), частей (gen. pl.)

         функция (nom. sing.), функций (gen. pl.)

   3. Neuter nouns have the following endings:
      1. Singular: -а, -я

         место (nom. sing.), места (gen. sing.)

         отображение (nom. sing.), отображения (gen. sing.)

      2. Plural: -Ø, -и, -ей

         место (nom. sing.), мест_ (gen. pl.)

         отображение (nom. sing.), отображений (gen. pl.)

         поле (nom. sing.), полей (gen. pl.)
          

2. Dative case.
   1. Masculine and neuter nouns have the following endings in the singular: -у, -ю.
      1. Examples of masculine nouns:

         интеграл (nom. sing.), интегралу (dat. sing.)

         случай (nom. sing.), случаю (dat. sing.)

      2. Examples of neuter nouns:

         место (nom. sing.), месту (dat. sing.)

         отображение (nom. sing.), отображению (dat. sing.)

   2. Feminine nouns have the following endings in the singular: -е, -и.

      сторона (nom. sing.), стороне (dat. sing.)

      потеря (nom. sing.), потере (dat. sing.)

      функция (nom. sing.), функции (dat. sing.)

      часть (nom. sing.), части (dat. sing.)

   3. The plural endings in the dative case are identical for masculine, feminine, and neuter nouns: -ам, -ям.
      1. Examples of masculine nouns:

         интеграл (nom. sing.), интегралам (dat. pl.)

         случай (nom. sing.), случаям (dat. pl.)

      2. Examples of neuter nouns:

         место (nom. sing.), местам (dat. pl.)

         отображение (nom. sing.), отображениям (dat. pl.)

      3. Examples of feminine nouns:

         сторона (nom. sing.), сторонам (dat. pl.)

         потеря (nom. sing.), потерям (dat. pl.)

         функция (nom. sing.), функциям (dat. pl.)
          

3. Accusative case.
   1. Masculine nouns have the following endings:
      1. Singular: -Ø, -а ³

         интеграл (nom. sing.), интеграл_ (acc. sing.)

         профессор (nom. sing.), профессора (acc. sing.)

         ранг (nom. sing.), ранг_ (acc. sing.)

      2. Plural: -ы, -и, -ов (and other genitive plural endings)

         интеграл (nom. sing.), интегралы (acc. pl.)

         студент (nom. sing.), студентов (acc. pl.)

   2. Feminine nouns have the following endings:
      1. Singular: -у, -ю, -ь

         сторона (nom. sing.), сторону (acc. sing.)

         потеря (nom. sing.), потерю (acc. sing.)

         часть (nom. sing.), часть (acc. sing.)

      2. Plural: -ы, -и ⁴

         сторона (nom. sing.), стороны (acc. pl.)

         перестановка (nom. sing.), перестановки (acc. pl.)

   3. Neuter nouns have the following endings:
      1. Singular: -о, -е

         место (nom. sing.), место (acc. sing.)

         поле (nom. sing.), поле (acc. sing.)

      2. Plural: -а, -я

         ме́сто (nom. sing.), места́ (acc. pl.)

         по́ле (nom. sing.), поля́ (acc. pl.)
         Note the normal stress shift.
          

4. Instrumental case.
   1. Masculine and neuter nouns have the following endings in the singular: -ом, -ем.
      1. Examples of masculine nouns:

         интеграл (nom. sing.), интегралом (instr. sing.)

         делитель (nom. sing.), делителем (instr. sing.)

      2. Examples of neuter nouns:

         место (nom. sing.), местом (instr. sing.)

         поле (nom. sing.), полем (instr. sing.)

   2. Feminine nouns have the following endings in the singular: -ой, -ей, -ью.

      сторона (nom. sing.), стороной (instr. sing.)

      функция (nom. sing.), функцией (instr. sing.)

      часть (nom. sing.), частью (instr. sing.)

   3. The plural endings in the dative case are identical for masculine, neuter, and feminine nouns: -ами, -ями.
      1. Examples of masculine nouns:

         интеграл (nom. sing.), интегралами (instr. pl.)

         делитель (nom. sing.), делителями (instr. pl.)

      2. Examples of neuter nouns:

         место (nom. sing.), местами (instr. pl.)

         поле (nom. sing.), полями (instr. pl.)

      3. Examples of feminine nouns:

         сторона (nom. sing.), сторонами (instr. pl.)

         функция (nom. sing.), функциями (instr. pl.)
          

5. Locative case.
   1. Masculine nouns have the following ending in the singular: -е.

      интеграл (nom. sing.), интеграле (loc. sing.)

      делитель (nom. sing.), делителе (loc. sing.)

   2. Neuter nouns have the following endings in the singular: -е, -и.

      место (nom. sing.), месте (loc. sing.)

      отображение (nom. sing.), отображении (loc. sing.)

   3. Feminine nouns have the following endings in the singular: -е, -и.

      сторона (nom. sing.), стороне (loc. sing.)

      часть (nom. sing.), части (loc. sing.)

   4. The plural endings in the locative case are identical for masculine, feminine, and neuter nouns: -ах, -ях.
      1. Examples of masculine nouns:

         интеграл (nom. sing.), интегралах (loc. pl.)

         делитель (nom. sing.), делителях (loc. pl.)

      2. Examples of feminine nouns:

         сторона (nom. sing.), сторонах (loc. pl.)

         потеря (nom. sing.), потерях (loc. pl.)

      3. Examples of neuter nouns:

         место (nom. sing.), местах (loc. pl.)

         отображение (nom. sing.), отображениях (loc. pl.)
          

There are several sets of nouns which exhibit stem changes throughout their declensions. Among these are the р-stem nouns, с-stem nouns, and н-stem nouns. Some examples are given below in partial declension, while more complete declensions are given in the Appendix.

1. Two important feminine nouns are found in the р-stem declension: мать (mother) and дочь (daughter).

   мать (nom. sing.), матери (gen. sing.)

   дочь (nom. sing.), дочери (gen. sing.)

2. Among the с-stem nouns is the neuter noun небо (sky).

   небо (nom. sing.), небеса (nom. pl.)

3. Two important neuter nouns are found in the н-stem declension: имя (name) and время (time)

   имя (nom. sing.), имени (gen. sing.)

   время (nom. sing.), времени (gen. sing.)
    

Proper names in Russian are declined. Last names may end in an adjectival ending, which decline as adjectives, or they may end in -ов (-ев) or -ин. The latter forms exhibit a mixed declension. The masculine singular is declined like a noun, except for the instrumental, which is declined like an adjective. The feminine singular is declined like a pronoun. In the plural, the nominative is declined like a short-form adjective, while the remaining forms are declined like long-form adjectives.

Russian names ending in -ых, -их, -аго, and -ово are invariable. Last names ending in -а/-я follow the feminine noun pattern, regardless of the gender of the person to whom the name refers. Sample paradigms of declined names are given in the Appendix.

The adjective in Russian may occur in both a long form and a short form. It also expresses the positive, comparative, and superlative degrees. Moreover, the long-form adjective is declined; it agrees with the corresponding noun in gender, number, and case.

Adjectives can be used in the attributive position, as in новая книга, "the new book," or the predicate position, as in книга новая, "the book is new." ¹ Thus, an adjective used attributively occurs next to the noun it modifies without a linking verb, while an adjective used predicatively is separated from the noun it modifies by a linking verb. In the latter case, the predicate adjective refers back to the subject of the clause, which is in fact the noun the adjective modifies. The long form of an adjective is used in either an attributive or predicative position.

The long-form adjective is declined into all six cases, and both types of adjective express the three genders in the singular, while there is no gender distinction in the plural. The difference in form between the two types of adjective is in the form of the ending; the short form exhibits a simple ending, while the long form exhibits a compound ending. ² In the above example, the short-form adjective ending is -а (simple), and the long-form ending is -ая (compound).

Adjectives may in some instances stand alone and act as nouns, although they still decline as long-form adjectives: кривая, "a curve," is a feminine noun; подынтегральный, "an integrand," is a masculine noun.

1. The long form of the adjective. The adjectives сложный, "complicated," and дуговой ³, "arc," are declined into all the cases in singular and plural, and in the three genders.

   сложный вопрос → a complicated question (m.)

   сложная задача → a complicated problem (f.)

   сложное решение → a complicated solution (n.)

   1. сложный; stem: сложн-

      	Singular						Plural
      	m.		f.		n.
      nom.	сложный		сложная		сложное		сложные
      gen.	сложного		сложной		сложного		сложных
      dat.	сложному		сложной		сложному		сложным
      acc.	сложный		сложную		сложное		сложные
      instr.	сложным		сложной		сложным		сложными
      loc.	сложном		сложной		сложном		сложных

      
       
   2. дуговой; stem: дугов-

      	Singular						Plural
      	m.		f.		n.
      nom.	дуговой		дуговая		дуговое		дуговые
      gen.	дугового		дуговой		дугового		дуговых
      dat.	дуговому		дуговой		дуговому		дуговым
      acc.	дуговой		дуговую		дуговой		дуговые
      instr.	дуговым		дуговой		дуговым		дуговыми
      loc.	дуговом		дуговой		дуговом		дуговых

      
       
2. The short form of the adjective. The adjectives правый and голодный appear below in their short forms. The simple endings are added to the adjectival stem, which is identical to the stem of the long form.
   1. Правый; stem: прав-

      прав (m. sing.), права (f. sing.)

      право (n. sing.), правы (pl.)

   2. Голодный; stem: голодн-

      голоден (m. sing.), голодна (f. sing.)

      голодно (n. sing.), голодны (pl.)

      Note that in the adjective голодный, the masculine singular short form exhibits what appears to be an alternate stem. This occurs in a number of adjectives; an -е or an -о appears between the two final consonants. Two other adjectives which exhibit this alternation are редкий, "infrequent," and верный, "correct," whose stems are редк- and верн-. The corresponding short forms are редок and верен.
       

3. A complete reference chart of adjectival endings appears in the Appendix.
    

The Russian adjective occurs not only in the positive degree, whose forms have been given in the previous section, but also in the comparative and superlative degrees.

1. The comparative degree of the adjective may be expressed in two ways.
   1. Suffixation. The suffix -ее (and -е in certain adjectives) is added to the adjectival stem. This form is invariable. For the adjective сложный, "complex," the comparative form is сложнее, "more complex," while the comparative of большой, "large," is больше, "larger."
   2. Use of the words более, "more," and менее, "less." These words are used in conjunction with the positive adjective in either its long or short form, depending upon its use in the sentence. Более and менее are not inflected, but the long-form adjective used is declined.

      более ранняя теорема Белинского → an earlier theorem of Belinskii

      в более узком смысле → in a narrower sense
       

2. The comparative conjunction (English "than") in Russian is the word чем. However, a comparison may be expressed without the word чем by declining the second item into the genitive case.

   Неравенство (11) слабее чем неравенство (12). → The inequality (11) is weaker than the inequality (12).
    

3. The superlative degree may be expressed in several ways, some used more frequently than others.
   1. Use of the word самый followed by the adjective. Both the adjective and the word самый, which is also an adjective, are declined,

      самый сложный → most complex

      самой интересной теоремы → of the most interesting theorem

   2. Suffixation. The suffix -ейш, -айш is added to the adjectival stem, and the adjectival endings are added to the resultant form.

      новый, новейший → new, newest

      важный, важнейший → important, most important

      высокий, высочайший → high, highest ⁴

      While a common translation of this form is "the most complex theorem," the use of this form generally indicates a high, although not superlative, degree of the adjective, such as "a most complex theorem."

   3. Use of the prefix наи-. This occurs infrequently.

      наисильнейший → strongest

   4. Use of the word наиболее instead of самый. Although far less common in usage than самый, наиболее may precede the adjective and is indeclinable.

      наиболее интересный → most interesting

   5. Use of the word всех in the predicate. The comparative form (that form of the adjective ending in -ее) plus the genitive plural of весь (all) may be used to express the superlative degree of an adjective.

      сложнее всех → (the) most complex one (of all)
       

Adjectives may be formed from proper names by adding the following suffixes: -овский, -овый, or -овой. For instance, the adjective corresponding to the name Эйнштейн is either Эйнштейновский or Эйнштейновый (Einsteinian). However, as adjectives these words may also appear in a short form; the adjective Евклидовый (Euclidean) may be rendered as Евклидов. Such adjectives are usually listed in the dictionary in the more modern short form. Unlike other short-form adjectives, this type does decline and is used nonpredicatively. The declension is given in the Appendix.

The various pronouns in Russian, like the nouns and adjectives, are declined. The use of each type of pronoun is described below, while complete paradigms of the pronouns are found in the Appendix.

The Russian personal pronouns correspond to the English "I," "you," "he," "she," "it," etc. However, unlike English, and like French, Russian expresses the difference between the "formal you" and the "informal you." That is, the second-person singular form ты is used in familiar circumstances, while вы, the second-person plural form is used in more polite and formal address. (However, the informal style is uncommon in mathematical writing.) Вы is also used to express plurality when more than one person is addressed in conversation.

1. The following table lists the personal pronouns in the nominative case.

   		Singular		Plural
   1st person		я (I)		мы (we)
   2nd person		ты (you)		вы (you)
   3rd person		он (he, it) она (she, it) оно (it)		они (they)

   
    
2. The third-person personal pronouns, in the singular and plural, prefix an н- when they are objects of a prepositional phrase.

   его: m. acc. sing., 3rd-person personal pronoun without a preposition
   Профессор видел его. → The professor saw him.

   ним: m. instr. sing., 3rd-person personal pronoun as object of preposition с
   Профессор говорил с ним. → The professor spoke with him.
    

Presentation of the possessive pronoun-adjectives is best divided into two categories: first- and second-person pronouns and third-person pronouns.

1. First- and second-person possessive pronouns. The pronouns of the first- and second-person agree with the noun possessed in gender, number, and case. Also, in Russian, there is no distinction (as in English) between "my X" and "mine," "your X" and "yours," etc.
   1. The pronouns in the nominative.

      1st person singular (my, mine): мой (m.), мой (f.), моё (n.), мой (pl.)

      2nd person singular (your, yours): твой (m.), твой (f.), твоё (n.), твои (pl.)

      1st person plural (our, ours): наш (m.), наша (f.), наше (n.), наши (pl.)

      2nd person plural (your, yours): ваш (m.), ваша (f.), ваше (n.), ваши (pl.)

   2. Examples of the possessive pronouns.

      моё решение → my solution

      Это решение моё. → This solution is mine.

      наша задача → our problem

      Эта задача наша. → This problem is ours.

2. Third-person possessive pronouns. These pronouns often cause some confusion; they are identical in form to the genitive of the third-person personal pronouns.

   Unlike the first- and second-person possessive pronouns, the third-person pronouns do not agree in gender, number, or case with the noun possessed, nor do they ever change form. They are: его (his), её (her, hers), их (their, theirs).

   Её профессор читал доклад. → Her professor read the paper.

   Он говорил с её профессором. → He spoke with her professor.

   The second sentence above indicates that unlike the third-person personal pronouns, these possessive pronouns never prefix н- when they are part of the object of a preposition.
    

3. The pronoun свой declines like мой, твой. This pronoun-adjective helps to avoid certain ambiguities often found in English: e.g., "He read his paper" ("his" may refer to the subject of the clause, "he," or to another male previously indicated).

   Он читал его доклад. → He read (gave) his (someone else's) paper.

   Он читал свой доклад. → He read (gave) his (own) paper.

   The pronoun свой and the subject to which it refers must be in the same clause or sentence.
    

This category includes the pronouns этот (this), тот (that), and такой (such). The full declension of these pronouns is given in the Appendix. Demonstrative pronouns agree in gender, number, and case with the nouns they modify.

Тот and такой may be followed by же to express identity and similarity:

из того же равенства следует → from the same equation follows

такая же группа → a similar group

The interrogative pronouns include кто (who), что (what), чей (whose), какой (which, what kind), and который (which, who). Declensions are given in the Appendix.

The relative pronoun requires some explanation. Который is a relative pronoun, and as such it links two clauses together:

Проблема, которую мы ставили, были сложная. → The problem, which we posed, was complex.

Note that the relative pronoun not only agrees in gender and number with the noun it replaces, but it also appears in the case required by its function within the relative clause. In this sentence, the pronoun is feminine and singular as it refers to проблема, and is in the accusative case because it is the direct object of the verb ставили.

The pronouns никто (nobody), ничто (nothing), никакой (no, no sort of), and ничей (nobody's) are declined like the corresponding positive interrogative pronouns. Note, however, that when used in conjunction with a preposition, the preposition is inserted between the ни- and the appropriate form of the pronoun.

ничто: ни о чём → about nothing

Also, double negatives are the rule in Russian phrases containing negative pronouns.

Никто не ответил на вопрос профессора. → No one answered the professor's question.

The category of definite pronouns includes the following: каждый (each, every, everybody), всякий (any, anybody, all sorts of), сам (myself, yourself, himself, etc.), and весь (all, whole, everybody, everything). The full declensions appear in the Appendix.

Каждый and всякий act attributively as well as standing alone as subjects:

У каждого студента решение. → Each student has a solution.

У каждого решение. → Everyone has a solution.

Сам is used to intensify or emphasize. In the following sentence, сам is used to emphasize the subject pronoun я:

Я сам не знаю, как решить эту задачу. → I myself don't know how to solve this problem.

One of the distinguishing characteristics of the Russian language is the notion of verbal aspect. There are two aspects in Russian: the imperfective aspect and the perfective aspect. In its simplest definition, the imperfective aspect indicates that an action is in progress, that it has not necessarily been, nor will be, completed, or is continuous or recurring. Conversely, the perfective aspect indicates that an action has been, or will be, completed, or that there has been, or will be, a definite result.

In languages that do not have special grammatical forms to express verbal aspect, the distinctions described above are rendered by the many verbal tenses, such as the perfect and pluperfect. Consequently, since Russian does exhibit verbal aspect, its inventory of tenses is minimal. In fact, Russian has only three tenses: past, present, and future. Furthermore, perfective verbs have no present tense, since they describe a completed action, and thus are used only in the past and future tenses.

Most Russian verbs have two infinitives — an imperfective infinitive and a perfective infinitive. For instance, the verb "to read" in Russian has the imperfective infinitive читать and the perfective infinitive прочитать. The imperfective verb читать appears in all three tenses, while the perfective form прочитать appears only in the past and future tenses:

читать (imperfective infinitive)

Я читаю книгу. → I read, I am reading, I do read the book

Я читал (читала, читало, читали) ¹ книгу. → I read, I was reading, I did read the book (but did not finish reading it)

Я буду читать книгу. → I will read, I will be reading the book (but will not finish reading it)

прочитать (perfective infinitive)

Я прочитаю книгу. → I will read the book (and will finish reading it)

Я прочитал (прочитала, прочитало, прочитали) ¹ книгу. → I read, I have read, I did read the book (and did finish it)

Notice that the imperfective future tense is compound, while the perfective future is simple. The first component of the compound future construction, буду of буду читать, is the future tense of the verb "to be" and is used to form all imperfective future tenses. The following sentence illustrates the imperfective future tense:

Мы будем называть функцию  f (z) непрерывной, если ... . → We will call a function  f (z) continuous if ... .

Although a great number of imperfective and perfective verb pairs may be derived from each other by affixation (adding or removing a prefix, suffix, or infix), not all pairs are easily recognized as pairs by their form. In the dictionary, the imperfective verb is listed with its definition, with the perfective form given in parentheses. The perfective form may be listed with or without a definition, but is usually referenced to its imperfective. When no perfective form is indicated, this is not necessarily an oversight: some verbs do not have a perfective aspect, some have only the perfective, and many, especially verbs adopted from other languages, have the same form in both aspects. (These special situations are not usually mentioned in the dictionary, since nonexistent forms do not have to be translated.) The definition may be repeated for the perfective form when the imperfective and perfective are widely separated in the dictionary. The perfective forms of reflexive or passive verbs are not usually mentioned if the corresponding active forms are given.

The verb "to be" (быть) is very important in Russian, as in all languages. Following is the conjugation of быть:

present tense 2
он, она, оно есть		he, she, it is		они суть		they are
future tense
я буду		I will be		мы будем		we will be
ты будешь		you will be		вы будете		you will be
он, она, оно будет		he, she, it will be		они будут		they will be
past tense был, была, было, были
я был, была		I was		мы были		we were
ты был, была		you were		вы были		you were
он был она была оно было		he, it was she, it was it was		они были		they were

1. It has already been illustrated that the present tense of быть is usually omitted in Russian sentences. Review the following sentence:

   Неравенство (11) слабее чем неравенство (12). → The inequality (11) is weaker than the inequality (12).

   Consider also the following sentence:

   Предложение P эквивалентно Q. → Proposition P is equivalent to Q.

   The present tense of быть, "to be," is omitted in both sentences. However, often a dash (—) is used in place of the omitted verb, as in the following example:

   Если E — некоторое множество точек комплексной плоскости, ... → If E is a set of points of the complex plane, ...

   Occasionally, the forms есть and суть, from быть, are used:

   Граница этой области есть окружность K_r, где r — постоянная. → The frontier of this domain is the circle K_r, where r is a constant.

   Матрицы суть спиновые матрицы Паули. → The matrices are the Pauli spin matrices.
   
   

2. Unlike the present tense, the future and past tenses of быть are never omitted. Furthermore, when быть is used in the future or the past, the predicate is often in the instrumental case:

   Если α алгебраическая величина относительно Σ, a Σ — алгебраическое поле относительно Δ, то α будет алгебраическим относительно Δ. → If α is algebraic with respect to Σ, and Σ is algebraic with respect to Δ, then α will be algebraic with respect to Δ.
   
   

3. The verb являться. The verb являться (to appear, to present oneself) is often used to replace the present tense of "to be" in sentences which would be confusing if lacking a verb. In such instances, the verb is translated as the appropriate form of "to be." The third-person singular and plural forms are, respectively: является, являются. The predicate of these verb forms is always in the instrumental case. The following sentences illustrate this construction:

   Такая функция существует и является единственной. → Such a function exists and is unique.

   Единственными изоморфизмами топологической группы T в себя являются тождественное отображение и симметрия X→–X. → The only isomorphisms of the topological group T into itself are the identity map and the symmetry X→–X.

   Вероятность P будет являться абстрактной копией эмпирической частоты. → The probability P will be an abstract counterpart of the empirical frequency ratio.

Russian verbs are classified as members of either the first or second conjugation, according to their nonpast personal endings. It is possible, however, for a verb to exhibit a mixed conjugation; that is, some of its endings are those of the first conjugation, while the rest are from the second. The following chart shows the endings of the two conjugations:

The verb читать is of the first conjugation: я читаю, ты читаешь, он читает, мы читаем, вы читаете, они читают. The verb говорить is of the second conjugation: я говорю, ты говоришь, он говорит, мы говорим, вы говорите, они говорят. The forms in parentheses, first person singular and third person plural, are common alternate endings and occur when dictated by certain spelling conventions.

A verb typically consists of several components when fully conjugated, although there are variations; perfective verbs, for instance, have no present tense, and thus no present participles. Among the inventory of components are the infinitive, the present, future, and past tenses (the indicative mood), the imperative mood, the conditional mood, the present participles, the past participles, and the gerund, or adverbial participle.

1. Sample verb conjugations. The following are sample conjugations. The Appendix includes a more comprehensive set of paradigms, as well as a list of common irregular verbs and their conjugations. The paradigms in the Appendix follow the given format, but are somewhat condensed and abbreviated where possible (for instance, only the masculine form of a participle is given).
   
   

   делать (imperfective), "to do"
   present tense
   я делаю		I do, I am doing		мы делаем		we do
   ты делаешь		you do		вы делаете		you do
   он делает		he, it does		они делают		they do
   она делает		she, it does
   оно делает		it does
   future tense
   я буду делать		I will do		мы будем делать		we will do
   ты будешь делать		you will do		вы будете делать		you will do
   он будет делать		he, it will do		они будут делать		they will do
   она будет делать		she, it will do
   оно будет делать		it will do
   past tense   делал, делала, делало, делали
   я делал		I did, I was doing		мы делали		we did, we were doing
   ты делал(а)		you did, you were doing		вы делали		you did, you were doing
   он делал		he, it did, he, it was doing		они делали		they did, they were doing
   она делала		she, it did, she, it was doing
   оно делало		it did, it was doing
   imperative делай (singular); делайте (plural) do! make!
   participles
   active passive  present   делающий   doing, making   делаемый   being made, being done  past   делавший   (who was) doing, (who was) making деланный   made
   adverbial participle
   делая		doing, making, if we make
   сделать (perfective), "to do"
   no present tense
   future tense
   я сделаю		I will do		мы сделаем		we will do
   ты сделаешь		you will do		вы сделаете		you will do
   он сделает		he, it will do		они сделают		they will do
   она сделает		she, it will do
   оно сделает		it will do
   past tense   сделал, сделала, сделало, сделали
   я сделал		I did		мы сделали		we did
   ты сделал(ла)		you did		вы сделали		you did
   он сделал		he, it did		они сделали		they did
   она сделала		she, it did
   оно сделало		it did
   imperative сделай (singular); сделайте (plural) do! make!
   participles
   active passive  past   сделавший   having done, having made сделанный   done, made having been done
   adverbial participle
   сделав		having done, having made

   
    
2. The infinitive. The infinitive does not indicate person or tense — it describes the action. The infinitive is translated as "to X"; the imperfective infinitive делать is translated as "to do," as is the perfective infinitive сделать. The infinitive of a verb consists of the infinitive stem plus the infinitival ending. In Russian, there are three infinitive types: those ending in -ть, those ending in -ти,, and those ending in -чь. The ending -ть is the most common and includes such verbs as делать (to do), сделать (to do), сесть (to sit), говорить (to speak), and заниматься (to study). ⁴ The ending -ти is found in some infinitives, such as: идти (to go), расти (to grow), and найти (to find). The third ending, -чь, occurs in the following verbs: жечь (to burn), течь (to flow), and мочь (to be able to).

   It is the infinitive of the verb which is listed in the dictionary. Therefore, it is important for the reader of Russian to convert to the infinitive before looking in the dictionary. While it would be impossible here to give a formal and complete description of the relation between the infinitive types and all the conjugations, without a lengthy linguistic explanation, the following generalizations should be helpful. The reader should remove the conjugated ending (i.e., the present and future tense endings, which are collectively called nonpast endings, and the past tense ending) to find a stem. If working with a past tense, whose stem is equivalent to the infinitive stem, the infinitive is easily determined by dropping the past tense suffix and adding the infinitive ending. Generally, if the stem ends in a vowel, the -ть suffix will apply (дела/ют, дела/ть).

   However, when working from a verb with two different stems, one in the infinitive and one in the nonpast, sometimes a regular consonant mutation is present: писать, пишу, пишешь, etc. The infinitive-past tense stem is писа-, and the present tense stem is пиш-. This is a regular alternation. The following is a list of common consonant alternations:

   
   

   щ   from   ск, ст	ш   from   с, к
   ж   from   з, г, д	ч   from   к, т
   жд   from   д	мл, пл, вл, бл, фл   from   м, п, в, б, ф

   
   

   If the stem ends in a -ч, -к, -г, and the verb is first conjugation, the -чь suffix will apply (мог/ут, мо/чь; тек/ут, течь). ⁵ These verbs exhibit stem consonant alternations; к alternates with ч, г with ж, with the first-person singular and the third-person plural being the same: течь (to flow), теку, течёшь, течёт, течём, течёте, текут. The past tense of these verbs lacks the suffix -л- in the masculine form: тёк, текла, текло, текли.

   If the stem ends in some other consonant and the verb is first conjugation, the -ти suffix applies (ид/ут, ид/ти; раст/ут, рас/ти). A warning is necessary here, however; these generalizations are not comprehensive linguistic rules, nor are they without exception. The reader should therefore keep in mind the list of consonant alternations; these often occur and subsequently mask the relation between the infinitive and its conjugated forms.

   When an alternation occurs in a first-conjugation verb, except those in -чь, the alternation occurs in all the persons, while in second-conjugation verbs, the alternation occurs only in the first-person singular; the remaining forms retain the consonant of the infinitive.

   Also, in the Appendix are several sample paradigms which will assist the reader in determining infinitives. The Appendix also lists several of the most common irregular verbs and their conjugations.

   
   
3. The present, future, and past tenses (the indicative mood). These forms have already been introduced at the beginning of this section. To review, the imperfective verbs have all three tenses, while perfective verbs have only past and present tenses. The future tense of imperfective verbs is compound; the future tense of perfective verbs is simple. ⁶ The compound future tense is formed with the appropriate form (person and number) of the future tense of "to be" plus the infinitive of the main verb. The past-tense suffix is -л, to which is added the endings -Ø, -а, -о, -и (masculine, feminine, neuter, and plural). To determine the past-tense stem, which is identical to the infinitive stem, remove the suffix -л and the endings.

   Reflexive verbs end in the particle -ся (-сь), and are conjugated as are other verbs. The particle -ся is added to the end of the verb; -ся is added if the verb form ends in a consonant, and the alternate form -сь is added if the verb form ends in a vowel.

   заниматься,   "to study"
   present		я занимаюсь ты занимаешься он занимается		мы занимаемся вы занимаетесь они занимаются
   past		занимался, занималась, занималось, занимались

   
    
4. The imperative mood. The imperative form of a verb is also known as the "command": "Solve the problem, read your paper." The imperative, as in English, has the understood subject "you." In Russian, since there are two "you's," there are two imperative forms, one for ты and one for вы: Делай работу! Делайте работу! "Do the work!" singular and plural; Расти! Растите! "Grow!" singular and plural. The singular endings are -и or -й after a vowel; the plural endings are the same but with an additional -те. Reflexive verbs, those ending in the particle -ся retain this particle at the end of the imperative: занимайтесь from заниматься, "study."
   
   
5. The conditional mood. The conditional is formed by following the past tense form of the verb with the particle бы.

   Он читал бы. → He should have read. He would have read.

   
   
6. Participles. Participles are verb forms which act as adjectives. They retain the quality of transitivity if the verb is transitive, intransitivity if the verb is intransitive. A participle also retains the aspect of the verb from which it is formed.
   1. Participles may be present or past tense:

      студент, читающий доклад → a student who read/is reading a paper

      студент, читавший доклад → a student who read/was reading/had been reading a paper

      Note the adjectival endings on the participles. Participles agree in gender, number, and case with the nouns they modify.

   2. Participles are also active or passive. An active participle describes the noun which is the agent of the action, whereas a passive participle describes the noun which is the object of the action.

      студент, прочитавший доклад → the student who had read the paper

      статья, прочитанная студентом → the article read by the student

      Note that the agent of the passive participle is in the instrumental case. Passive participles are formed only from transitive verbs (those which take direct objects); reflexive verbs, which are intransitive, thus do not form passive participles.

   3. Formation of the participle.
      1. Present active participles
         Add the suffix plus adjectival endings to the present-tense stem.
         • 1st-conjugation verbs add -ущ-, -ющ-:
           читать – чита/ют – чита/ющ/ий, читающий
         • 2nd-conjugation verbs add -ащ-, -ящ-:
           говорить – говор/ят – говор/ящ/ий, говорящий
      2. Past active participles
         Add the suffix plus adjectival endings to the past-tense stem.
         • If the stem ends in a vowel, add -вш-:
           читать – чита/л – чита/вш/ий, читавший
         • If the stem ends in a consonant, add -ш-:
           нес/ти – нёс – нёс/ш/ий, нёсший
      3. Present passive participles
         Add the suffix plus adjectival endings to the stem of the present tense.
         • 1 st-conjugation verbs add -ем-:
           читать – чита/ют – чита/ем/ый, читаемый
           рисовать – рису/ют – рису/ем/ый, рисуемый
         • 2nd-conjugation verbs add -им-:
           любить – люб/ят – люб/им/ый, любимый
      4. Past passive participles
         Add the suffix plus adjectival endings to past-tense stem.
         • -нн-, or -т- if stem ends in a vowel:
           прочитать – прочита/л – прочита/нн/ый, прочитанный
           взять – взя/л – взя/т/ый, взятый
         • -енн- if stem ends in a consonant or -и-:
           принести – принёс – принес/ённ/ый, принесённый
           изучить – изучи/л – изуч/енн/ый, изученный ⁷
      5. Participles from reflexive verbs
         Participles ending in the reflexive particle always take the -ся form.
      6. Short-form participles
         Because participles are adjectival in form, they may occur as short-form participles. However, only passive participles have both the long and short forms; active participles occur only in the long form. Short-form participles are used as the predicate of a passive construction. They consist of stem plus suffix plus short-form adjective endings. Like short-form adjectives, they are only used as predicate adjectives, and so they do not decline. The suffixes -н-, -ен- (-ён-), and -т- are added to the stem:

         прочитать – прочитан

         решить – решён

         занять – занят

   
   
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.
   1. 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.

   2. 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.

   3. 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.
   1. 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.

   2. 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.

   3. 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.

   4. 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.

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:

The perfective of the verb "to go by one's own means" is пойти:

Где профессор? Он пошёл в лабораторию. → Where is the professor? He has gone to the lab.

The conjugation of the verb пойти is as follows:

Other common verbs of motion include the following:

Furthermore, these verbs of going and carrying may be prefixed by "directional prefixes." For example, the verb приносить/принести means "to bring"; the verb уносить/унести means "to carry away," or "to take away." The prefix при- means "up to," while у- means "away." These prefixed compounds have only two forms: imperfective and perfective.

The difference in the sense of the imperfective and perfective can be emphasized by exhibiting both aspects of the same verb in one sentence. (It is recommended that each word in the sentence be looked up in the dictionary; the superscripts refer to the comments immediately following the translation.)

Прежде чем¹ доказывать,² что все собственные значения матрицы A удовлетворяют³ уравнению (4), мы докажем,⁴ что если h(λ) — общий наибольший делитель⁵ элементов присоединённой к A – λI матрицы,⁶ то⁷ φ(λ) = h(λ)ψ(λ).

Before we go on to prove (or, Before we prove) that all the eigenvalues of the matrix A satisfy equation (4), we shall prove that if h(λ) is the highest common divisor of the elements of the matrix (which is) adjoint to A – λI, then φ(λ) = h(λ)ψ(λ).

Для каждой строго возрастающей¹ последовательности σ индексов множество A_σ открыто² в L_n+1,p+1 и насыщено² по Δ_n,p, поэтому его³ канонический образ C_σ в P_n,p, является открытым множеством,⁴ гомеоморфным фактормножеству множества A_σ по отношению эквивалентности⁵ θ_σ, индуцируемому⁶ в A_σ отношением Δ_n,p.

For every strictly increasing sequence of indices σ, the set A_σ is open in L_n+1,p+1 and saturated for Δ_n,p; hence its canonical image C_σ in P_n,p is an open set (which is)⁷ homeomorphic to the quotient of the set A_σ by the equivalence relation θ_σ induced in A_σ by the relation Δ_n,p.

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.

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.

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).

Распределение, задаваемое¹ функцией плотности² s_n(x) или функцией распределения² S_n(x) известно под названием распределения Стьюдента или t-распределения; оно было впервые использовано³ в одной важной статистической проблеме В. Госсетом, писавшим⁴ под псевдонимом «Стьюдент» (Student).⁵

The distribution defined by the frequency function s_n(x) or the distribution function S_n(x) is known under the name of Student's distribution or the t-distribution; it was first used in an important statistical problem by W. Gosset, writing under the pseudonym of "Student."

Примеры § 1 показывают, что ряд может быть сходящимся,¹ не будучи² абсолютно сходящимся.¹

The examples of § 1 show that a series can be convergent without being absolutely convergent.

Examples of a number of auxiliary verbs and forms, such as must, can, etc., will be given in this section. Please refer to the section concerning modals in The Verb for further discussion.

1. Если скоростью частицы нельзя пренебречь по сравнению со скоростью света c или энергией её нельзя пренебречь по сравнению с энергией покоящейся массы, то необходимо пользоваться формулой λ_D = ... → If the velocity of the particle is not negligible compared to the velocity of light, c, or if the energy is not negligible compared to the rest mass energy, then it is necessary to use the formula λ_D = ... (Literally, If one must not, or cannot, neglect the velocity of the particle in comparison to the velocity of light, c, or if one must not neglect the energy, etc.)

The English words "must," "ought to," and "should" may also be rendered by следует followed by the infinitive; the past is следовало бы (should have, ought to have); for example

2. Следует быть острожным при использовании этой формулы. → One must be careful in using this formula.
3. Вам следовало бы исключить сначала A, затем B. → You should have eliminated A first, and then B.

Note. Следует may also have its usual meaning "follows," for example

4. Из (2) следует, что ... → It follows from (2) that ...
5. Самодуальному тензору G^ρσ можно сопоставить симметрический спинор второго ранга g_rs. → With a self-dual tensor G^ρσ one can associate a symmetric spinor g_rs of rank two. (Or: It is possible to associate with a self-dual tensor G^ρσ a symmetric spinor g_rs of rank two.)
6. Для заданного ε можно найти такое δ, что¹ |I₁| < ε/3 для всех значений n; зафиксировав² это значение δ, можно найти такое n₀ = n₀(δ), что³ |I₂| < ε/3 и |I₃| < ε/3 при n > n₀. → For given ε, we can find δ such that |I₁| < ε/3 for all values of n; having fixed δ, we can find n₀ = n₀(δ) such that |I₂| < ε/3 and |I₃| < ε/3 for n > n₀.  
   Notes

   1.	Literally, "such а δ that"; this is the usual Russian construction. ↑
   2.	Adverbial participle; note that фиксировать is both perfective and imperfective. ↑
   3.	Same comment as in note 1. ↑

7. Может показаться, что такое явление трудно объяснить. → This phenomenon may seem difficult to explain.
8. Если бы мы имели в своём распоряжении большее число точных наблюдений, ... → If we could have at our disposal a large number of precise observations, ...
9. Если мы проинтегрируем функцию 1/(1 + x²) от 0 до ∞, мы сможем определить значение π. → If we integrate the function 1/(1 + x²) from 0 to ∞, we shall be able to determine the value of π.
10. Ответ на этот вопрос может дать ключ ко всей проблеме. → The answer to this question may give the key to the whole problem.

Unlike the noun and the adjective in Russian, the form of the adverb is invariable.

1. Most adverbs are formed from corresponding adjectives by replacing the adjectival ending (-ый, -ий, -ой) with the adverbial suffix (-о, -е):

   хороший – хорошо → good – well

   плохой – плохо → bad – badly

   интересный – интересно → interesting – interestingly

   могучий – могуче → powerful – powerfully

   Note that these forms may resemble the neuter short-form adjective and should not be confused.

   
   
2. Adjectives ending in -ский form corresponding adverbs in -и, with or without the prefix по-:

   английский – по-английски → English – in English

   практический – практически → practical – practically

   теоретический – теоретически → theoretical – theoretically

3. Adverbs may also be formed by use of case endings, resulting in an invariable form:
   1. по- plus the dative singular of the corresponding adjective:

      настоящий – по-настоящему → real – really

   2. The instrumental singular of a noun:

      полностью → fully

4. Some adverbs are not formed from corresponding adjectives by adverbial suffixation. These include the following:

   здесь → here

   там → there

   сюда → to here

   туда → to there

   отсюда → from here

   оттуда → from there

   очень → very

5. The adverb may express degree:
   1. The adverbial comparative degree is equivalent in form to that of the adjective. The suffix -ее is used:

      интересно → interestingly (adv.)

      интереснее → more interesting (adj.), more interestingly (adv.)

      Он говорит интересно. → He speaks interestingly.

      Она говорит интереснее, чем он. → She speaks more interestingly than he does.

   2. The superlative degree is the comparative degree plus всех (of all):

      Он решил задачу раньше всех. → He solved the problem earlier than all the rest.

   3. As in adjectival comparative constructions, the comparative conjunction than is expressed by using the word чем:

      Они решили задачу раньше, чем мы. → They solved the problem earlier than we (did).

1. Adverbs may modify verbs, adjectives, and other adverbs:

   Он правильно решил задачу. → He solved the problem correctly.

   Профессор читал очень интересный доклад. → The professor read a very interesting paper.

   Студент очень быстро ответил на вопрос профессора. → The student answered the professor's question very quickly.

2. Adverbs may be used in the predicate in impersonal constructions (sentences which lack a grammatical subject):
   1. These constructions may express a state or condition:
      1. State of environment

         Здесь холодно. → It is cold here.

         В кабинете тепло и светло. → It is warm and light in the study.

         Эту задачу трудно решить. → It is difficult to solve this problem.

      2. State of necessity or possibility

         Нужно правильно решить задачу. → It is necessary to solve the problem correctly.

         Здесь нельзя курить. → Smoking is not allowed here.

   2. When a logical subject is required, a noun or pronoun in the dative case may be used:

      Ему холодно. → He is cold.

      Эту задачу студенту трудно решить. → It is difficult for the student to solve this problem.

      Note that the verb in such constructions is in the infinitive.

   3. In the past tense, these constructions use the neuter past form of to be (было), while in the future tense the singular form of to be is used (будет). This occurs because there is no grammatical subject in the sentences; thus the verb has nothing with which to agree.

      Трудно было решить задачу. → It was difficult to solve the problem.

      Профессору будет легко выполнить это задание. → It will be easy for the professor to carry out this task.

The meanings of many of the common English prepositions are already embodied in the case endings of the Russian nouns and pronouns; the most obvious instance is, of course, the implicit rendering of the preposition "of" by the genitive case, the preposition "by" with the instrumental, for example, применённый автором (applied by the author), etc. The table below gives some of the more commonly used Russian prepositions, including adverbial participles, and the corresponding cases together with some typical phrases.

In Russian, as in English, numerals may be divided into two categories, cardinal (one, two, three, etc.) and ordinal (first, second, third, etc.). The following table lists the cardinal and ordinal numbers in Russian; the cardinal numbers appear first, and the ordinals follow the semicolon:

Both types of numeral are declined. Ordinal numbers are adjectival in form, and thus simply follow the adjectival declension. In compound ordinals, however, only the final component is in the ordinal (adjectival) form:

The only ordinal number which requires additional explanation is the adjective "third," or третий (m.), третья (f.), третье (n.). Notice that the feminine and neuter adjectival endings are not the expected compound endings -ая (-яя), -ое (-ее). This is so in the nominative and accusative, both singular and plural. The complete declension of this ordinal appears in the Appendix.

Declensions of the cardinal numbers are found in the Appendix. Note that only the numbers "one" and "two" distinguish gender; the number "one" has the following forms which follow the declension of этот: один (m.), одна (f.), одно (n.), and одни (pl.); the number "two" exhibits the form два for masculine and neuter nouns and the form две for feminine nouns.

1. Use of case with numbers. Numbers may occur in any of the cases, depending upon their function in the sentence.
   1. The number "one" is essentially an adjective, and as such agrees in gender, number, and case with its noun. The plural form, одни, is used when the following noun occurs only in the plural:

      Если одни часы колеблются с периодом T, ... → If one clock vibrates with period T, ...

      When the number "one" is part of a compound number, both the number "one" and the noun remain in the singular:¹

      Он написал пятьдесят одну статью. → He wrote fifty-one articles.

      The word один may also express the meanings "certain," "alone," and "a":

      Одни математики согласились с Лобачевским. → Certain mathematicians agree with Lobachevsky.

      одна теорема Лобачевского → a theorem of Lobachevsky

   2. When the numbers два, две, три, and четыре (and compounds ending in them) are used in the nominative and inanimate accusative (which looks like the nominative), the accompanying noun appears in the genitive singular:

      два примера: (gen. sing. of пример) → two examples

      разбить на 2 интеграла (gen. sing. of интеграл) → to split into two integrals

      Пусть i, j, k — три вектора единичной длины, ... → Let i, j, k be three vectors of unit length, ...

      If the noun following the number is preceded by an adjective, it may be in either the genitive plural or the nominative plural (although the noun is genitive singular in both instances). Generally, the genitive plural is used with masculine and neuter nouns, while the nominative plural is used with feminine nouns:

      три сложных примера → three complex examples (masculine noun)

      три сложные книги → three complex books (feminine noun)

      Вводя три единичных вектора i₁, i₂, i₃, ... → If we introduce the three unit vectors i₁, i₂, i₃, ...

      If the item following the number is a substantivized adjective, that is, an adjective acting as a noun, the word is treated as an adjective under the above conditions:

      две переменных → two variables

      две кривые → two curves

   3. When the numbers пять (five) and up through девятнадцать (nineteen), and any compound numbers ending in five, six, seven, eight, nine, or zero are in the nominative or inanimate accusative cases, both the noun and its adjective appear in the genitive plural:

      тридцать шесть русских книг → thirty-six Russian books

   4. When a number is used in a case other than the nominative or inanimate accusative, the number and the noun and adjectives following that number are in the plural of that particular case.² In other words, the number no longer determines the case of the adjective and noun following it; rather, it agrees in case with the following noun. In compound numbers, all components decline:

      из двадцати новых книг → from twenty new books

      из двадцати двух новых книг → from twenty-two new books

      Если угол между двумя зеркалами равен α, ... → If the angle between two mirrors is equal to α, ...

      V является функцией 12 величин. → V is a function of twelve variables.

   5. Nouns following the numbers тысяча, миллион, and миллиард are usually in the genitive plural, regardless of the case in which the number is being used. These words are declined as nouns.
   
   
2. Indefinite numerals. When the actual number is unspecified, the noun following the number is usually in the genitive plural:

   Пусть уравнение L = 0 имеет n различных корней. → Let the equation L = 0 have n distinct roots.

   Similarly, the expressions of quantity много (much, many), мало (little, few), несколько (several, some, a few) are indefinite numerals. When the noun following them can be counted, it appears in the genitive plural; when the noun can be measured but not counted, it is in the genitive singular:

   Приведём несколько примеров. → We cite a few examples.

   Имеется много различных вариантов. → Many different variants are possible.

   мало света, много воды → a little light, a lot of water

   The above are sentences in which the indefinite numerals are in the nominative or accusative case. These expressions may themselves occur in the other cases as well; the nouns following them occur also in that particular case, and generally in the plural:

   много новых книг → many new books (nominative plus genitive plural)

   из многих новых книг → from many new books (all in genitive plural)

   со многими новыми книгами → with many new books (all in instrumental plural)

tg = tan

ctg = cotangent

sh = sinh

ch = cosh

Ареа- before a hyperbolic function means the inverse of the function

i = 1, n stands for i = 1, ..., n.