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Portuguese term or phrase: 1...que está na razao do estudo da ..  2. ..un corpo apeciável de literatura que debruça sobre o tema.. 3. ..pode ser definida por uma fuçao cracterística.. 4..conjunto de nível (set theory) 5..esperança condicional (set theory) 6. Lema 1,2, etc(describing a series of IF sentences related to set theory) 7. ..pode construirse um sistema linear de q ecuaçoes, EN TERMOS DOS paràmetros consequència...8. gradeamentos difusos . Some of these are teminology but others just won't seem to fit (I have checked dics and web), so I'm looking for ideas. Sorry to list so many, but Proz loads very slowly. 
  English translation:>> see below <<  Explanation: 1) "…which lies within the scope of research on the subject of..." (or, simply, "..."study of...")
2) "…A significant body of literature that embraces the subject…" (or, simply, "a sizable body of literature on the subject")
3) "…can be defined by a characteristic function…"
4) "level set" is the proper term. "…Level Set Methods are powerful numerical techniques which can follow the evolution of interfaces…  www.math.berkeley.edu/~sethian/level_set.html
[NOTE: J.A. Sethian (Dept. of Mathematics, Univ. of California, Berkeley, California 94720; Email: sethian@math.berkeley.edu) welcomes requests for more information about his work.]
"Numerical schemes for the HamiltonJacobi and level set equations on triangulated domains"  fano.ics.uci.edu/cites/Document
"The level set method was devised by Osher and Sethian as a simple and versatile method for computing and analyzing the motion of the interface in two or three dimensions…"  www.math.ucla.edu/~imagers/htmls/lev.html.
5) "Conditional expectation." Full definition: "If X is a random variable on a probability space (omega, F, P), the conditions expectation of X with respect to a given subsigmafield F' of F is an F'measurable random variable whose expected value over any set in F' is equal to the expected value of X over this set." (McGrawHill Dictionary of Scientific and Technical Terms (5th ed., 1994))
6) A "lemma" is a mathematical condition that is relevant to the proof of a theorem, or a subsidiary proposition introduced in proving some other proposition (i.e., a helping theorem). Hence, "lemma 1, lemma 2, etc."
7) "A linear system of equations can be constructed, in terms of [or "based on"] the consequential [?] parameters…" (Here, the word "consequènciales" may have been truncated or typo'ed, or a word or two may be missing after "paràmetros." In any event, more context would be extremely helpful.)
8. gradeamentos difusos .
8) Very hard to say without more context. Possibilities include: "multiple increments" (which seems to fit best in your general context), "indeterminate gradations," "broad gradients," etc.
Wish I could offer more help! 
 Selected response from: Heathcliff United States Local time: 08:39
 Grading comment Most of all these math terms I have come across in a previous 'English' existence, but never Lemma. Could a synonym also be Heuristic as in Heuristic 1, 2 etc? The ans was very useful, though, as usual. 4 KudoZ points were awarded for this answer 
 
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3 hrs >> see below <<
Explanation: 1) "…which lies within the scope of research on the subject of..." (or, simply, "..."study of...")
2) "…A significant body of literature that embraces the subject…" (or, simply, "a sizable body of literature on the subject")
3) "…can be defined by a characteristic function…"
4) "level set" is the proper term. "…Level Set Methods are powerful numerical techniques which can follow the evolution of interfaces…  www.math.berkeley.edu/~sethian/level_set.html
[NOTE: J.A. Sethian (Dept. of Mathematics, Univ. of California, Berkeley, California 94720; Email: sethian@math.berkeley.edu) welcomes requests for more information about his work.]
"Numerical schemes for the HamiltonJacobi and level set equations on triangulated domains"  fano.ics.uci.edu/cites/Document
"The level set method was devised by Osher and Sethian as a simple and versatile method for computing and analyzing the motion of the interface in two or three dimensions…"  www.math.ucla.edu/~imagers/htmls/lev.html.
5) "Conditional expectation." Full definition: "If X is a random variable on a probability space (omega, F, P), the conditions expectation of X with respect to a given subsigmafield F' of F is an F'measurable random variable whose expected value over any set in F' is equal to the expected value of X over this set." (McGrawHill Dictionary of Scientific and Technical Terms (5th ed., 1994))
6) A "lemma" is a mathematical condition that is relevant to the proof of a theorem, or a subsidiary proposition introduced in proving some other proposition (i.e., a helping theorem). Hence, "lemma 1, lemma 2, etc."
7) "A linear system of equations can be constructed, in terms of [or "based on"] the consequential [?] parameters…" (Here, the word "consequènciales" may have been truncated or typo'ed, or a word or two may be missing after "paràmetros." In any event, more context would be extremely helpful.)
8. gradeamentos difusos .
8) Very hard to say without more context. Possibilities include: "multiple increments" (which seems to fit best in your general context), "indeterminate gradations," "broad gradients," etc.
Wish I could offer more help!
McGrawHill Dictionary of Scientific and Technical Terms (5th ed., 1994) Reference: http://www.math.berkeley.edu/~sethian/level_set.html
 Heathcliff United States Local time: 08:39 Native speaker of: English PRO pts in pair: 231

  Grading comment Most of all these math terms I have come across in a previous 'English' existence, but never Lemma. Could a synonym also be Heuristic as in Heuristic 1, 2 etc? The ans was very useful, though, as usual. 
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1 day20 hrs re 'heuristic' (as a noun):
Explanation: Baeed on the information in the following sites (and on the absence of the term, as a noun, in the venerable McGrawHill Dictionary of Scientific and Technical Terms), I would tend to avoid "heuristic" as a synonym for the perfectly acceptable term "lemma," which is quite specific and appears to fit your context perfectly.
From the first site below: "An heuristic is a useful method for addressing problems but is not guaranteed to prove or solve, unlike an algorithm. Instead of rules, heuristics typically involve trial and error. For example, in the game of blackjack, "take a third card if you have under 14 points" is an heuristic..."
and, from PRINCIPIA CYBERNETICA WEB / Web Dictionary of Cybernetics and Systems (second site below):
HEURISTIC: (l) (adjective) Characterizing a system in which the internal parameters can be changed when necessary through feedback. (2) A heuristic idea serves as a guide for discovery. It serves as a valuable aid for empirical research but may be unproved or incapable of proof. [3] (noun) An aid to discovery; any device or procedure used to reduce problemsolving effort; a rule of thumb.
Best,
HC
Reference: http://www.hfac.uh.edu/cogsci/lang/Entries/heuristic.html pespmc1.vub.ac.be/ASC/HEURISTIC.html
 Heathcliff United States Local time: 08:39 Native speaker of: English PRO pts in pair: 231

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