Examples of English Collocations in Mathematical Texts
Collected by Mati Pentus
2016
Errors are marked with *.
\sk1 Math jargon
\sk2 <==
_ follows from Lemma 1 of [3]
_ now follows from _
_ since [commaguidelines.html]
, since
, for
(for ...)
, as
_ in view of
_ in accordance with
_ because [commaguidelines.html]
This is evident from
_ by virtue of [Springer]
_ due to _, which has already been proved
\sk2 ==>
Then _
Suppose that _; then
* Suppose that _, then [Springer]
therefore
, and, therefore,
, and therefore
* , and therefore, [Springer]
(and, therefore, _)
* (and therefore _) [Springer]
Therefore,
Therefore a+b=c, where a=2. [CMS14, 13.21]
Hence it is apparent that abc=xyz. [CMS14, 13.25]
Hence,
it follows that
it now follows that
? thus
Thus, [Springer]
* So, [Springer]
* So _ [Springer]
Consequently,
consequently
, and, consequently, [Springer]
the polynomial f and, consequently, the polynomial g, is
By _,
by Definition 3,
Since _, by Lemma 1 _
* Since _, by Lemma 1, [Springer]
But by assumption
by construction
in view of _,
since _, it follows that _
since _, we see that _
* since _, then _
as _, we have _
* as _, then _
According to _,
Then, according to _, we have
, whence
whence
This topology is the greatest and, hence, coincides with the Mackey topoplogy.
and, hence, M is a coatom
* and hence, M is a coatom
; hence, [Springer]
, forcing
guarantees that
this implies
this proves
this amounts
this entails
this provides
Accordingly,
But then,
using this, we get
, which yields
, so that _
This is only true for _
by the suppositions on _,
From Lemma 1 it follows that
From Lemma 1 we obtain
It is immediate, from Lemma 1, that
Use of Lemma 1 yields
An application of _ to _ yields
The application of Lemma 1 yields
Applying _, we obtain
* Applying _ we
* Applying _, this gives [Springer]
Applying Lemma 3, we obtain
* By applying Lemma 3
Using (2.1), we get
Using this, we obtain the following result.
* Using this, the following result is obtained. [Springer]
Using (2.1), we find that
* Using (2.1), it is found that [Springer]
Combining (2.1) and (2.2),
If we combine this with Lemma 1,
Now if we recall (2.1),
we arrive at the conclusion that
we come to the conclusion that
we draw the conclusion that
we reach the conclusion that
_ is a consequence of _
* a consequence from
, which completes the verification of the conditions of Lemma 3
Now by the preceding paragraph
it follows that m+n>5
* it follows m+n>5 [Sosinsky, p. 36]
Owing to the choice of _ we have
* Due to the choice of _ we have [Springer]
Owing to this fact,
Then, owing to the definition of _, there exists
* due to the definition of [Springer]
, and so [Springer]
On account of Lemma 1, [Springer]
By the above, we have
It follows from the above that
As a result, we obtain
, according to the definition,
* , according to definition,
Because of _,
Since, for a Lie algebra, the Jacobian vanishes,
* Since for a Lie algebra the Jacobian vanishes,
\sk2 GET
see that
have
obtain
get
show
establish
prove
derive
deduce
conclude
check
verify
note that
* notice that [Sosinsky: OK. Springer: incorrect.]
*? remark that
We remark that [Springer]
* Remark that [Springer]
observe that
know
point out
holds
we immediately get
we have m+n>5
* we have that m+n>5 [Sosinsky, p. 36]
we obtain m+n>5
We obtain that f(i)>n for all i.
* We obtain f(i)>n for all i. [Springer]
we see that m+n>5
The following theorem holds.
* theorem takes place
\sk2 <--
_ if
_ provided
, provided
, provided that
_ whenever _
_ when
_ in the case where
_ unless
, under the assumption that
* in the assumption
* under assumption that
\sk2 -->
if _, then
whenver _, we have _
under the assumption/conditions of _, we have _
_ only if
_ only when
_ is a sufficient condition for _ be _
in the presence of _, the hypothesis _ implies the conclusion _
For invertible _, the adjoint _ also is invertible.
, from _, it follows that _
\sk2 <-->
iff
if and only if [Springer]
_ for those and only those _ for which _
for _ to be _ it is necessary and sufficient to have _
a necessary and sufficient condition for _ be _ is that _ be _
Then the following conditions are equivalent:
\sk2 NOT, AND, OR
it cannot be that _
it is contradictory to assume that
such that
and
but
But, in fact,
the object of this work is not only to find _ but also to use _
not only is it finite, but it even consists of
Assume that kn.
_, although _ [Springer]
_ although _ [commaguidelines.html]
Although _, _ [commaguidelines.html]
, which is always _,
, which of course are _,
, which is impossible
* , what is impossible
in addition to
or
if either _ or _, then _
and, in addition,
* and, additionally,
; otherwise
x=1 if a=b and x=0 otherwise
* else
* in the other case [Springer]
all the more
neither _ nor _
, where
_ is both _ and _
must also have
This is impossible.
it is impossible to express _ as _
cannot
* can not [Springer 2005/05]
Let _ and let _. [Springer 2009]
Let _, and let _. [Springer 2010]
Let M be a ring, and Z be a semigroup. [Springer 2012]
we show that if _, then
* we show that, if _, then [Springer]
one, two, or three points
* one, two or three [Springer]
are not equal to 13, 17, or 19
if this is not the case
\sk2 FOR ALL, THERE EXISTS
each _ may be written as _
* any _ may be written as _
for all _ such that _, we have _
for any _ such that _, it follows that _
for any _ and any _, we have _
_ for all x
_ for any x \in M
?* _ for each x \in M
for all integers n>3
for every integer n>3
* for every integers n>3
for every
for any _, there exists a _
* for any _ there exists a _ [Springer]
, for any _, there exists
there is a _
there exists a unique _ such that
* there exists the unique
G contains a unique maximal element
there exists some
there exists only one
There exists a set X for which f(X)=0.
there exist no such
there does not exist
for an arbitrary semiring S
* for arbitrary semiring
for an arbitrary n>3
each of its submodules
each of its subgroups is _
* every its subgroup is _
* each its subgroup is _
all of our results
* all our results [Springer]
all of its elements
? all its elements
all of whose coordinates
* whose all coordinates
a graph each of whose components is a tree
* a graph whose each component is a tree
each element of which
* whose each element
any of its
* any its
for all sufficiently large n,
the coefficients _ are not all zero
Let, for every _, [Springer]
For an arbitrary _,
for any three points
* for every three points [Trzeciak]
for any integer n, however large,
the space C is a superalgebra if, for all integers n, there are
some (maybe all) of
none of the above-listed symmetries
all cones in M, except maybe the smallest one,
some such ring
\sk2 LET
Let _ be _
Suppose that m+n>5.
* Suppose m+n>5.
Suppose that m+n>5 is false.
Suppose that M is a finite module.
* Suppose M is a finite module. [Springer]
Assume that
* Assume _ has the form _ [Springer]
Assuming that _, we obtain
is assumed to
In addition, suppose that
Furthermore, assume that
Suppose _ satisfies the assumption/assumptions of Lemma 1
Suppose _ satisfies the conclusion of Lemma 1
Let _ be an arbitrary _
can be assumed to be convergent
* can be assumed convergent
* can be supposed to be convergent
Let _ be _, and let _ be _.
Let _ be _, and _ be _. [Springer]
* Let _ be _ and _ be _. [Springer]
Under the assumptions of Lemma 1,
* In the assumptions of Lemma 1,
Under the above assumptions,
In what follows, M is assumed to be
Let us assume that
* Let us consider that
\sk2 CONSIDER
consider _
Considering _, we obtain
Turning now to _, we see that
we begin by considering
we consider
we are dealing with [Springer]
we often deal with
we regard
we treat
Conversely, [Springer]
For the converse,
For the other direction, suppose that
we shall take _ to be some fixed
Let us now consider
To be definite
For the sake of being definite
consider a _ whose _ _
has already been considered
the case considered
* the considered case [Springer]
\sk2 TAKE
put
take
choose
pick
We take for Q the _
take any _ such that _
take an arbitrary n>3
consider an arbitrary n>3
For _, take the _ constructed previously
to each _ assign _
_ corresponds to _
let _ be given by _
let the function _ take each _ to _
let _ be the function that takes each _ to _
the function f defined on _ by f(x)=_
We can apply Lemma 1, taking _
* We can apply Lemma 1 taking _ [Springer]
Let us be given _
\sk2 INTENTION
Let us prove that
Let us prove property (2.1).
Let us show that
We now prove that
We must
* We have to
We also show how
We claim that _. Indeed,
* Really
We want to
it will be proved that
It will appear later that
We seek
, which will be discussed in more detail in _,
the next three theorems
The purpose of _ is to prove
* The purpose of _ is proving
\sk2 PREPARE
We will make use of
In preparation for
To do this we _
Below, we will use abundantly some lemas on _ from [3].
\sk2 IT REMAINS TO
it remains to check that
now we must only prove that
To _,
? To _ we _
_ in order to
it suffices to
Then we only need to
For this goal
To compute _ it is sufficient to _
What remains is to find
* It is left to find [Springer]
\sk2 END OF PROOF
is thereby proved
We have thereby proved that
* We thereby proved that
This completes the proof of Lemma 1.
This concludes the proof.
This is what was to be proved.
To conclude the proof, it remains to note that
The result is
This contradiction proves the theorem.
This contradicts Lemma 1. The theorem is proved.
* contradicts to
This contradicts n being odd.
But this contradicts (3).
This leads to a contradiction with Lemma 1.
Now we arrive at a contradiction with
* Now we come to a contradiction with
Hence we get a contradiction.
We have a contradiction with
We have obtained a contradiction with
* We obtained a contradiction with [Springer]
and we reach a contradiction.
* and we reach the contradiction.
, a contradiction.
* , contradiction. [Springer]
* . A contradiction. [Springer]
, in contradiction to
* , in contradiction with the condition of [Springer]
This is the value we are seeking.
* This is the value we are looking for.
This graph enjoys the required property.
, as required.
, as desired.
, which proves the theorem.
By _, the desired result follows.
, which contradicts the assumption
* what contradicts
\sk2 PROOF
Proof. (Sketch)
Here is a sketch of the proof.
an outline of the proof
We shall leave the proof to the next section.
The proof is found in [3].
This lemma was proved by _ (see [3]).
This was proved by _ in [3].
The proof is given in Section 2.
the `if' part
the `only if' part
The proof is by reductio ad absurdum.
Assume the converse.
Suppose to the contrary that
Suppose that, on the contrary,
Assume the contrary, i.e.,
Suppose the contrary, i.e.,
The proof is in 3 steps. Step 1:
\sk2 INDUCTION
The proof is by induction on/over _
by transfinite induction
induction on the complexity of
induction on the structure of formulas
induction on derivations
induction on the length of a proof
induction on ordinal numbers
we proceed by induction
straighforward induction on _
by the induction hypothesis,
by the inductive assumption,
by the induction assumption,
a mechanical inductive argument shows that
The induction step must be proved for every t.
the inductive step
the induction base
Using induction,
* Using the induction,
Since, by the induction hypothesis,
\sk2 CASES
Let us consider 2 cases: (a) _; (b) _. [Springer]
The following n cases arise:
We consider 3 cases depending on _
We distinguish 3 cases.
There are n possible cases to consider
the case where
the case that
* the case when
the situation where
as a special case
Case 1:
In this case,
In the case n=2,
subcases
In our case,
In either case,
If, in this situation, we take
\sk2 CONSTRUCT
construct
is constructed from
exhibit a proper device
show that this construction works
prove that this reduction works
For f, take the function _ constructed previously.
Suppose that _ have already been defined
\sk2 FOR SIMPLICITY
For convenience,
For the convenience of the reader,
For the reader's convenience [Springer]
For the sake of convenience,
* of reader convenience [Springer]
to facilitate
Without loss of generality, we can assume that
Without loss of generality it can be assumed that
We may assume without loss of generality that
There is no loss of generality in assuming
we assume for simplicity that
* we suppose for simplicity that [Springer]
Assume, for the moment, that
For simplicity, suppose that
For simplicity we shall discuss the case _
the general case differs from this special case in notation only.
leaving the reader an easy transfer to the dual variant
Although conceptually simple, this would complicate notation and
would give no new insight.
, which does not affect the argument
Sometimes it turns out to be more convenient to use
In order not to make the formulas too cumbersome
\sk2 BELOW
Below,
* Below _ [Springer]
Further on,
In the sequel,
* In the sequel _ [Springer]
In what follows,
* In all what follows
Here and in what follows
\sk2 NEXT
First, we need
First, we consider
* At first, [Springer]
Secondly,
First of all,
* First of all _ [Springer]
Next _
In addition,
Further,
Moreover,
* Moreover _
* Besides,
Also, n runs through S.
* Also n runs through S. [Springer]
Now _
Now let us show that
* Let now [Springer]
* Now, let [Springer]
Let us now show that
* Let us show now that
Finally, we note that [Springer]
* At last, [Springer]
Finally(,)
Let us observe that
? Let us remark that
, in addition,
, in turn,
As to _, consider _
For _, note that _
We may now _
It then becomes possible to express
In the following lemmas,
we need several lemmas
The following lemmas are needed for the proof of Theorem 3.
For the proof of the main result of this section, we need three lemmas.
* For the proof of the main result of this section, we need three lemmas. [Springer]
Concluding this section,
Then we apply
This can be done as follows.
Now we can prove Theorem 3.
Let us first prove
For our purposes, we also need
\sk2 SIMILAR
Similarly,
Similarly to _,
in the same way
* by the same way
in exactly the same way
, as above,
In a similar way,
In an analogous way,
in a similar manner
* in the similar manner
Continuing in the same way, we see that
Likewise,
* Likewise _ [Springer]
Again
The opposite implication is similar.
In a manner analogous to that of
The same situation holds for
we only indicate necessary modifications
Dually,
is treated in a dual way
The proof of _ is dual.
is treated symmetrically
the mirror image
Exploiting the left-right symmetry,
As in the algebraic case,
* Like in the algebraic case,
As before,
can still be defined as before
a similar problem
a similar formula
a similar description
By analogy with Theorem 1,
follows from considerations analogous to the ones above in the proof of
Further, reasoning as in the proof of Lemma 1,
The proof of this theorem is not fundamentally different from the proof of
By the same reasoning,
can be shown in the same way as in
* may be shown [Springer]
\sk2 BUT
However,
* However _ [Springer]
Thus, Z is, however, a ring.
; however there is no
Both X and Y are countable, but neither is finite.
Thus, Z is countable but not finite.
the interval concentric with Z but of twice its length
This class contains three elements but does not satisfy Lemma 3.
* contains three elements, but does not _ [Springer]
, whereas
Nevertheless,
* Nevertheless _ [Springer]
* Though [Springer]
nevertheless
while _ _
, unlike
On the one hand,
On the other hand,
nonetheless
on the contrary
Surprisingly,
Curiously,
despite
* in spite of
contrary to
in contrast to
* in contrast with [Springer]
\sk2 THAT IS
, that is,
That is,
, i.e.,
(i.e.,
I.e.
(that is, _)
this means that
* it means that
in other words,
Namely,
In fact,
Actually,
In short,
More precisely,
In particular,
, which means, in particular, that
* , which means in particular that
Specifically,
_ asserts that _
_ states that _
, which means that
In this way,
* In this way _
* This way,
\sk2 EVIDENT
Evidently,
? Evidently
Obviously,
Clearly
Clearly,
Of course,
Observe that
Note
Apparently
It is clear that
It is obvious that
It is evident that
It is easily proved that
it can be proved that (British English, American English)
it can be proven that (American English)
It is easily seen that
It turns out that
is trivially true
is readily seen
for trivial reasons
, as is easily checked.
as is easy to check
* as it is easy to [Springer]
It is routine to
One easily proves _
The verification that _ is routine.
For n=1, there is nothing to prove
For the case AB=CD, there is nothing to prove
There is nothing to prove if
We leave the proof to the reader.
The proof is omitted.
The proof is trivial.
We omit an easy proof.
This lemma can be proved by standard methods of _
This lemma can be proved by direct calculations.
a direct calculation shows that
the obvious embedding $ A \subset B $
* the evident embedding $ A \subset B $
the obvious equality
* the evident equality [Springer]
obviously holds
folkloric exercise [Springer]
we can easily verify
\sk2 WE NEED SOME TERMINOLOGY
Now we shall give the following definition.
Now we introduce the following concept.
This suggests the following definition.
we need some notation.
We need some terminology.
Let us introduce the following notation.
* notations
Let us establish some notational conventions.
Let us fix some notation we use in this paper.
We introduce the following definitions and notions.
To prove this lemma, we need the concept of _
Before we give the proof of _, we need the concept _
the notion of differential equation
We also fix the natural homomorphism _
\sk2 USUAL/UNUSUAL
As usual,
* As usually,
As customary
we standardly extend _ to
is extended in the natural way to concern
in the usual way
Contrary perhaps to normal usage,
We use the usual abbreviations _ for _; _ for _
We follow the customary approach by using the definition
under the usual matrix multiplication
\sk2 USUALLY
More generally,
* More generally _ [Springer]
In general,
In the general case,
* In general case, [Springer]
for the most part
\sk2 DEFINE
denote by _ any _
denote by _ the _ that _
a _ is a pair _, where _
we say that _ if [Lambek, Sipser]
we say that _ iff (?)
we shall say that _
a _ is called a group if the following conditions hold:
a _ is said to be a group if _
a _ is a group if _
a group is a _ such that _
Suppose _. Then this _ is called _
Suppose _. Then any _ such that _ is called _
and is denoted by _
If, _ then we say that _ is _ and write _
By definition, put Z = _
* By the definition,
Now let us put, by definition,
define the _ of two _ as _
For any _, _
* For any _ _ [Springer]
For _,
* For _ let [Springer]
? Here _
Here, [Springer]
Hereafter, [Springer]
? Hereafter _
We now agree to
We agree to identify
If _
take _ to _
we take _ to _
one takes _ to _
_ is taken to be _
_ generates _
_ will _
_ is intended to _
let _ _
by _ we _
define _ to _
put _ to _
_ is defined to _
mean
be
denote
stand for
by _ we understand _ [Springer]
* under _ we understand _
by _ we mean _ [Springer]
by a _ is meant a
we refer to _ as
let us refer to _ as
_ is referred to as
we write ABC for ((AB)C)
we write ((AB)C) as ABC
we use _ as _
any _ is called _
, which we shall call _
setting _ we get
, or, if there is no danger of confusion,
* misunderstanding [Springer]
_ consists of _, subject to the following conditions:
somewhat wider sense
we say that _ or _, according as _ or _
_ is defined inductively by the rules:
we call a _ _ if
* we term _ by _
Define a _ by the prescription _ if _
the elements of _ are called _
_ is _ whose _ is determined by the following condition:
_ are named for John Smith
_ is a _ along with a _
_ is as follows.
in the following way:
if it satisfies one of the following equivalent conditions:
if, given a binary operation + defined in V,
the following four properties hold for all $ a , b \in V $:
_ is described as being linearly independent
Given _, one can define _. These are _.
We give three definitions:
a binary operation labeled +
\sk2 NOTATION
We adopt the convention
notation
Here
denotes
, or simply
abbreviate _ to _
is introduced as an abbreviation defined as
is an abbreviation for _
We write
, written _,
, written with the symbol _,
, written with the _ symbol,
will be symbolized
, which we denote by _,
is designated by the _ symbol
the letter _ (with or without indices) is reserved for _
_ is understood according to the equality _
_ is to be read as
We abbreviate _ as
By convention, we take
it is clear from the context
without introducing ambiguity
For _, by a _ we understand any _
* For _, under a _ we understand any _
by _, we shall understand
We let \nabla be the _ and we call \nabla the _
We will write _ to indicate that _
_ inherits a natural _ from _
notational collision
Throughout this paper
throughout the calculations
In the notation of Definition 3 above, we have _
* In notation of
For the sake of brevity, [Springer]
For brevity, we write [Springer]
\sk2 RANGE
capitals
capital letters, X, Y, etc.,
Latin letters
capital Greek letters
We use capitals for
we shall usually employ capitals as variables ranging over all
will be represented by capitals and will be called
range over
ranged over by
The metasymbols _ range over
meta-variable ranging over
In stating the axiom schemata and inference rules,
A, B, C are any _, p is any _
\sk2 REPLACE
_ arises from _ by _
results from
becomes _ if
is obtained from _ by replacing _ by _ at some occurrence of _ in
? to replace _ by _ in _
?? to replace _ with _
can be replaced by
* be replaced with
to come from _ by interchanging _ with
result of substituting _ for _ in _
_ can be renamed without affecting _
a substitution of _ into the polynomial
_ changes sign under replacement of _ by _ [Springer]
\sk2 CLAIM
assertion
clause
the condition _=_
the following condition: [Springer]
* the condition: [Springer]
the following property:
statement
fact
observation
claim
proposition
lemma
a corollary of
* a corollary to [Google]
consequence
conjecture
digression
remark
Concluding remarks
desired conclusion
see the Introduction [Springer]
* see the introduction [Springer]
Preliminaries
the fact that
the converse statement
The converse is not always true.
principle
equality
equivalence
draw the conclusion that
* make the conclusion that
Using assertion (5) of item (d),
item I(2)
* item I (2) [Springer]
\sk2 FOR EXAMPLE
For example,
for example(,)
for instance,
, say
, e.g.,
\sk2 INTUITIVELY
intuitively _ is to be thought of as
In intuitive language, the theorem says that
\sk2 MODIFICATION
modification
the resulting system
one has to adjoin to the system _ _
\sk2 ADJECTIVES
usual
ordinary
analogous to
more general
above-sketched
above-mentioned
given
indicated
mutually exclusive
auxiliary
earlier-mentioned
suitable
concerning
replaceable
interchangeable
substitutible for each other in
the following
distinct from
three (not necessarily distinct, possibly empty) words
applicable
essentially stronger
the required subalgebra
the _ under consideration
classical
* classic
the smallest index such that
the underlying structures
twofold
periodic words
cyclic words
the above-considered bimodule M
\sk2 COINCIDE
to coincide
is the same as
two points with the same values
* two points with same values [Springer]
are precisely those
\sk2 SYMBOLS
between parenthese
a close parenthesis
quasi-quotes \ulcorner \urcorner
we routinely drop the parentheses from
nonterminal (American English)
\sk1 Front and back matter
\sk2 GRANTS
This research was partially supported by RFBR grant 12-245.
This work was partially supported by grant ABC-12-345.
* The work
* is partially supported
* by the grant ABC-12-345
by a "Universitety Rossii" grant
partially supported by the Russian Foundation for Basic Research
and by the Russian Ministry for Education
the grant of the President of the Russian Federation No. ABC-12.345
\sk2 ACKNOWLEDGEMENTS
Acknowledgements [Elsevier, Oxford: OK. Springer: incorrect.]
Acknowledgment(s?) [IEEE]
Acknowledgments [Springer]
Professor
Dr.
The author is grateful to
The author expresses his gratitude to
* expresses gratitude to
The second author wishes to express his gratitude to
I am deeply grateful to
most grateful
I would like to thank
I also thank
for constant attention to this work (sc. adv.)
for the help throughout the work
for his support and advice
for his guidance
for useful discussions
for helpful discussions
for discussing matters treated here
for the supportive environment
The presentation of _ benefited greatly from suggestions of
These suggestions are gratefully acknowledged.
for a number of comments on a draft of this paper
for finding an error in our first draft
for making several suggestions to improve the exposition
_, as well as for _,
for his encouragement to extend our original work
for his valuable help
for providing information about
for their hospitality
, particularly Prof. J. Smith
for helpful guidance and assistance
for useful comments and for _
for very valuable comments on a preliminary version of this work
for many helpful and stimulating discussions while writing this paper
during the preparation of this paper
for several very constructive suggestions
a number of
on the subject of
also
his supervisor, _,
my thesis advisor, _,
my advisor, Prof. J. Smith, [Springer]
* professor J. Smith [Springer]
my scientific advisor, Prof. J. Smith, [Springer]
Profs. A. B. Ivanov and J. Smith [Springer]
the research reported in this paper
this research
was initiated during the author's visiting stay
was finished during
the stay has been granted by
The idea of the paper originated during the author's stay at the university of
I
The author
The second author
would like to
wish to
thank
express my gratitude to
am deeply grateful to
most
very
helpful
useful
valuable
stimulating
constructive
a number of
several
many
discussions
conversations
suggestions
comments on
guidance
assistance
encouragement to
for the statement of the prblem
* the problem statement [Springer]
their hospitality
discussing
finding an error in
our first draft
the subject of
a preliminary version of
a draft of
this work
this paper
present paper
matters treated here
while writing
during the preparation of
to improve the exposition
to extend our original work
who shared their insights and experiences so willingly
indebted to
very helpful in formulating these ideas
thoughtful suggestions and improvements
much of the progress made in these studies
Published with the kind permission of the journal's editorial board
and authors.
\sk2 OTHER BACK MATTER
Conclusions
Appendix
Appendices
\sk2 PREAMBLE
Dedicated to Professor _ on the occasion of his _th birthday
This issue is dedicated to
\sk2 INTRODUCTION
In this paper,
* In this article, [Springer]
* In the paper, [Springer]
previous work involves _
we consider certain problems related to _
the problem on the isomorphism of
* the problem about isomorphism of [Springer]
a question on the isomorphism of
* a question about the isomorphism of
questions on the completeness of _ are considered
some facts on
* some facts about
problems on the classification of
results on
* results about
In _, appropriate _ are/were constructed
the aim of this paper is to prove the following
this generalizes results of Ivanov
* results by Ivanov
a slight generalization of _
this strengthens a theorem of _
using methods of _, we show that
as related to
technically involved
elaborated
logically guided
essentially incorrect
partial
further
the so-called local case
general
resulting
familiar from
Recently [3],
starting from
aspects
problems
survey
framework
behaviour
an early proof
positive results
main goal
partial results towards
technique
a proof technique
ultimate solution of
detailed verification
pattern
have been discussed
design
investigated into
appear
can be adapted
expect
have been obtained
attempt a general solution
refine the methods of
these efforts fail in
provide
is modified
handle
transforming
reducing
explain the basic terminology and notation
achieve the same effect
is motivated
motivating idea
Section 3 is devoted to the proof of Theorem 3.
The present paper is devoted to the study of this problem.
a theorem on
it was pointed out that
close connection between
In this paper, we are mainly concerned with
is concerned with
* this paper concerns
the major result of
remain open
this is an open problem at present
Whether P includes Q is an open question.
The existence of _ is still an open question.
It is an open question whether
* It is open question
This question has not yet been solved.
In Stanley's work [3] the following problem has been raised: Is there _?
in the works of Janet [3] and Thomas [4]
* in papers by Janet [Springer]
provide a quick review
we give a brief review of some recent results
* a small review
considerations
syntactic methods
syntactic results
semantic results
develop a theory
in group theory
* in the group theory
from matrix theory
from category theory
in model theory
in string theory
the NBG theory
extend
elaborate
advance
introduced in
presents
present a method
determine
approach
analyze
analysis
proof covers
give
include
explain
handle
treat
capture
use
employ
adopt
accompany
go on
move on
play a key role
summary
outline
expository discussions
studied from _ perspective
(for details, see [3]) [Springer]
see [3] and the references therein [Springer]
* see [3] and references therein [Springer]
(cf. [3])
* (compare with [3]) [Springer]
(see the paper [3])
in his paper [3], _ proved that
_ was considered by _ in his book [3]
are also discussed in detail in [3]
stronger than Lemma 1
the following condition is weaker than (2.1)
the article/section is devoted to describing
* devoted to the describing
can be adapted for the present purpose
can be adapted for the latter purpose
for purposes of this exposition
has attracted the attention of many algebraists [Springer]
confine attention to
the question is not addressed here
Ivanov prposed calling them _
reasoning
* reasonings
Currently,
complete description
complete answer
* full answer
further studies are necessary
In the second section, we
* In the second section we
In this paper, we
* In this paper we
In [3], A. Ivanov introduced
We shall briefly discuss an extension
This paper contains some preliminary results.
Section 1 includes preliminary definitions.
to include in it
for free algebras this was noted by
Our approach is equivalent to Ivanov's on a certain class of
The function obtained the name of Dehn function.
At present
* Now there are results
can be done at present by
* can be done today by
To date, it is not known whether
At present, it is not known if
the theory of _ has been developed on the basis of the theory of
the propositions also give some intuition on how to
the last few years
the classical results on
* the classical results about [Springer]
the 1930s and 1940s [CMS14, 8.40]
* the 1900s [CMS14, 8.40]
* the 1910s [CMS14, 8.40]
in the 1990s [Springer]
at the beginning of the 1990s
in the early 1990s
the twentieth century [CMS14, 8.40]
during the eighties and nineties [CMS14, 8.40]
in the last few decades
in recent decades
attempts at using standard bases have shown that
* attempts of using
he was the first to introduce
* he was the first who introduced
was first formulated by Smith in 1999
* was for the first time formulated [Springer]
fundamental difference
* principal difference
we focus our attention on
the following result of Ivanov
* the following result by Ivanov
two approaches toward the definition of a topological prime radical
* two approaches for the definition of
we are considering _ with respect to _
A detailed account of _ can be found in [3].
The reader is assumed to be familiar with
* The reader is supposed to be familiar with [Springer]
The reader can refer to
* The readers can refer to [Springer]
* A reader
in the algebraic language
in the topological language
by using the
* by the usage of the [Springer]
the greater part of this material
the idea of generalizing the equations to the case of
* the idea to generalize
Throughout the paper all rings are associative unless the contrary is explicitly stated.
Actually this is not so.
, which also has theoretical value
We start by recalling
\sk2 REVIEW
clear
impressive
thoroughly
relatively large
helpful
harmonious
enjoy a growing popularity among
broad range
systematic account
The author's central claim is that
The author's strategy is to
one of the primary threads of the paper is
\sk1 Grammar
\sk2 CAPITALS
non-Euclidean
(in title) Product-Free Calculi and Non-Archimedean Fields
(in title) Maps That Are Not Invertible in the Syntax
(in title) The Case where the Rank Is Countable
(in title) $c$-Latgroups
the lemma
* the Lemma [Springer]
Noetherianity
$\pi$-Monic polynomials exist.
* $\pi$-monic polynomials exist. [Springer]
the Diamond lemma
* the Diamond Lemma [Springer]
\sk2 HYPHENS
zero divisor
nonzero-divisor
the grammar is context-free [Partee, 2005]
* the grammar is context free [Partee, 2005]
this well-known theorem
this theorem is well known
As is well known in the theory of _,
_ is left-associative
invertibility-preserving operators
the just-defined topology
self-injective
* selfinjective [Springer]
pointwise convergence
* point-wise [Springer]
quasi-identity
? quasiregular
? quasi-regular
? quasiorder
? quasi-order
\sk2 COMMA
the minimal subalgebras, i.e., the atoms of the lattice A(X),
\sk2 ARTICLES
the Lagrange theorem
Lagrange's theorem
* the Lagrange's theorem
the famous Lagrange's theorem
this theorem of Lagrange
the area theorem
the density theorem
{\L}os's theorem
* {\L}os' theorem [Springer]
Taurinus's results
Taurinus' results
Gauss' theorem
Zorn's lemma
the Zorn lemma [Springer]
the Hahn--Banach theorem
Hahn and Banach's theorem
* Hahn--Banach's theorem
the Minkowski functional
* Minkowski's functional
the Thomas division
Peano arithmetic
Peano's successor function
of the second order
the notion of equality on the set of _ [Lambek]
the notion of commutativity [Springer]
* the notion of the commutativity [Springer]
the notion of regular expression [Sipser]
the notion of almost orthogonality [Springer]
the notion of differential equation
the definition of regular expressions [Sipser]
the definition of a uniformly recurrent pseudoword [Springer]
By the definition of a partition, [Springer]
* the definition of partition [Springer]
* By definition of a partition, [Springer]
the concept of a vector space [Wikipedia]
the concept of a correct universal algebra [Springer]
the concept of involutive monomial division was proposed [Springer]
the almost orthogonality condition
the condition of homogeneity of [Springer]
a criterion for completeness is proved
a criterion for the completeness of _ is proved
is a maximal set of _
a finiteness condition
prove the continuity of _ on
Taking into account the irreducibility of M, [Springer]
the isomorphism A \cong B
the isomorphism of
the connection between computability and completeness
the correctness of this definition
due to the arbitraryness of
* due to arbitraryness of [Springer]
in view of the faithfulness of
* in view of faithfulness of [Springer]
the compactness of
the equivalence of
the continuity of
by the symmetry of
* by symmetry of
In view of the density of
This proves the injectivity of
implies the existence of
the conditions of existence of
* the conditions of the existence of [Springer]
the problem of existence of
* the problem of the existence of [Springer]
the Lambek calculus
a group of order n
* a group of the order n
an element of degree n
The group G has order n.
an element with weight 3
* an element with the weight 3
all subgroups of finite index
in the case of problem (2.1) with weight function w(x)=1
* with the weight function w(x)=1 [Springer]
every subset of size n
an algebra with generators _
the ordered monoid of ... with order induced by
* the ordered monoid of ... with the order induced by
an algebra of arbitrary rank
in such a way that
proved such a statement for
such a finitely generated ideal exists
* such finitely generated ideal exists
for such a set
* for a such set
every such category
* every such a category
condition (3) of Theorem 2
* the condition (3) [Springer]
equation (2.4) [CMS14, Fig. 13.2]
has property (2.1) [Springer]
has the important property (2.1) [Springer]
in case (2.1)
of the form (2.1)
the least of the natural numbers k with the property _
the pair property
the first and second summands
with the help of
* with help of
Proceedings of the International Algebraic Conference on the Occasion of the 90th Birthday of A. G. Kurosh
on the Occasion of _ and _
the Max-Planck-Institut f\"ur Mathematik
a minimal generating set
methods of differential analysis
* methods of the differential analysis
in homogenization theory
in quantum group theory
in the theory of quantum groups
the theory of topological groups
in the weak and strong sense
to present a mathematical approach
the third and fourth lines
* the third and the fourth lines
the first and second semicycles
the coincidence of their algebraic geometries
with the usual ring operations
* with usual ring operations
the case of a finite group
the case of finite groups
implemented in the C++ language
from right to left
* from the right to the left
two of the n_1,...n_k
* two of n_1,...n_k
with the help of a computer
* with the help of computer
has the natural structure of a Poisson algebra
* has the natural structure of Poisson algebra
has the structure of a group
* has a structure of a group [Springer]
in view of the properties of
* in view of properties of [Springer]
In _, J. Smith studied the properties of
the example of a pseudouniversal class
in the Z-grading
a similar argument
to prove a similar result for rings
Note that r does not depend on the choice of the element b.
* does not depend on a choice of
This number depends on the choice of the basis.
of the form m/n
* of form m/n
at least one of the numbers x_1,...,x_n
with none of the x_1,...,x_n equal to zero
* with none of x_1,...,x_n equal to zero [Springer]
Observe that P is a singleton, say {p}.
the number of elements of this set
* the number of the elements of this set
the number of good cubes
* the number of the good cubes [Springer]
, which contradicts common sense
* contradicts the common sense [Springer]
equations in dimensionless form
* equations in the dimensionless form [Springer]
the set of parameters for which m+n>5
with the use of (2.1)
* with use of
this problem has a positive solution if
in general position
is an open set in the Zariski topology
play an important role in the study of
_ equals double the number of _
* the doubled number [Springer]
has nonzero intersection with
the system of linear homogeneous equations _=0
the system of the equations _=0 [Springer]
generated by rotation by $120^\circ$
* by the rotation by [Springer]
this is equivalent to multiplication by
both cases
* both the cases
both its uses
both of these matrices [Springer]
one of the weak points
every nonempty subset has a least element
there exists a maximal subset Z satisfying _
there is a least ordinal satisfying _
* there is the least ordinal
there exists a unique
* there exists the unique
there exists a least upper bound
used for the construction of the
a method for constructing the
in the preparation of the
in preparing the
for the computation of the
* for computation of the
modern combinatorial algebra
* the modern combinatorial algebra [Springer]
two terms, one of which lies in M and the other in N
These modules have a structure similar to
These modules have a structure that is similar to
* These modules have the structure that is similar to [Springer]
Z has nonzero intersection with every nonzero submodule.
the so-called Morita theorems
* The main results are so-called Morita theorems. [Springer]
the Internet
* Internet [Springer]
one of the promising ideas
one of the possible solutions
one of the problems considered
one of the natural generalizations of [Springer]
one of the important applications of Theorem 3
in explicit form
* in the explicit form [Springer]
We see that the function f, which belongs to Y, can be represented as...
We see that a function f that belongs to Y can be represented as...
The role of _ is reduced to a minimum. [Springer]
the set of differential 2-forms for [Springer]
* the set of the differential 2-forms for
of a more general character
We call the attention of the reader to the fact that
in general position
contains n as a summand
has finite length
neither a left nor a right zero divisor
\section{Main Results}
* \section{The Main Results} [Springer]
in the ordinary sense
in the sense of Definition 3
by the assumption of the proposition
in increasing order
with a finite number of
has a finite number of
at most a countable set of [Springer]
the first or second row
The group has trivial intersection with
in normal form
\sk2 THAT/WHICH
there exists _ that
* there exists _ which [Springer]
the smallest set that
* the smallest set which
the smallest such M for which
denotes the _ that
* denotes the _ which [Springer]
Suppose G is a group that
* Suppose G is a group which [Springer]
We construct a group that satisfies
* We construct a group which satisfies
Abelian groups that are _ are considered in [3].
* Abelian groups which are _ [Springer]
contains a _ that
we obtain a _ that
a diagram D that contains three squares
the diagram D, which contains three squares,
an identity s=t that holds in any Leibniz algebra
the identity x(yz)=xyz-xzy, which holds in any Leibniz algebra,
\sk2 POSITION OF ADVERBS
we can also consider
we also consider
may also be
* may be also
must also be taken into account
* must be also taken
we also have
The map f is also right-continuous.
there also exists
* there exists also [Springer]
can always be represented as
* can be always represented as [Springer]
* can be also directly proved [Springer]
will also be denoted
this set is also finite
* also this set is finite [Springer]
we again get
but are also monotone
* but also are monotone [Springer]
Consider also [Partee, 2005]
Also be sure [Partee, 2005]
* Be also sure [Partee, 2005]
Also set
* Set also [Springer]
Note also that
Also note that
Again, set
* Set again [Springer]
Let us also consider
Let us note here that
* Let us here note that [Springer]
It now follows similarly to _ that
* It follows now
Now suppose that we have already
Suppose now that we have already
Now consider the formula
Consider now the formula
Let us now consider
Now let us consider
Let us now prove that
Now let us prove that
Let us now formulate the main statement [Springer]
* Let us formulate now the main statement [Springer]
We now show that
Now we show that
The proof repeats word-for-word the considerations from
* word-for-word repeats [Springer]
the second property means exactly that _
we will return later
* we will later return
*
\sk1 Areas of mathematics
\sk2 GRAPHS
directed graph
digraph
oriented graph
undirected graphs without loops or multiple edges
* nonoriented graphs
associated undirected graph
vertex (pl. vertices)
directed edge
undirected edge
endpoints of an edge
initial vertex of an edge
terminal vertex of an edge
reduced path
reduced form of a path
simple reduction
circuit
loop
planar graph
the number of edges eminating from a vertex v is called the degree of v
the valence of v
_ is drawn by connecting circles representing _ together by an arrow if _
the arrow originating at _ and ? arrowhead pointing to _
the apex of the cone (pl. apices)
cone with apex _
connected components
connected by a path
* connected with a path [ams.org]
\sk2 CATEGORIES
category
object
morphism
identity
domain
codomain
full subcategory
ordered class
_ precedes _
_ follows _
_ is left adjoint to _
\sk2 STRINGS
a finite sequence (possibly of length 0) of these symbols
a string (or word) over an alphabet
the juxtaposition of
For any finite set _ of symbols, we denote by _ the set of all finite
strings of symbols from _.
the empty string _
the empty word of length 0 will be written _
If _ is a subset of _, we say that _ is a language over the alphabet
_.
nonempty strings over _
length of a string
the rightmost
* the right-most [Springer]
obtained by juxtaposing
the reversal of a string
the language of properly nested parentheses [Sipser]
all words ending in c
* all words ending by c
the lexicographic order
in lexicographic order
\sk2 INFERENCE RULES
the calculus admits the following axioms and inference rules:
these calculi
contraction rule
exchange rule
weakening rule
Sequents have the form
right-sided sequents
axiom schemes
to drop an axiom
rules of inference
derived rules of inference
conservative extension of
repeated application of _ rules
premise
premiss
conclusion
the final step
the rule can be eliminated from
to drop the rules
deducible from
lies faithfully embedded in
the introduction rule for / on the right
the result of adjoining _ as an axiom to _
\sk2 SETS
set theory
set
family
a family _ of _ indexed by _
collection
disjoint union
Cartesian product
a proper superset of
stable subset
a countably infinite set
a countable set
countably many
denumerably many
(denumerably infinite)
cardinality of a set
exactly one
unique
non-void
nonvoid (American English)
non-empty
nonempty
disjoint parts
proper inclusion
persistent
, which may be empty,
a binary relation on the set _
a limit ordinal
a successor ordinal
? limiting ordinal
go in ascending order
* stand in ascending order
the set of natural numbers
the reverse inclusion
the reverse inequality
All _ have the same cardinality.
Y contains X and is contained in Z [Wikipedia]
the intervals are specified by their end points
the axiom of choice
* the choice axiom
a (possibly empty) word
* (possibly, _) [Springer]
total order = linear order
well-order = well-ordering = well-ordered set
symmetric difference
* symmetrical difference
\sk2 FUNCTIONS
single-valued binary relation
valuation
onto injection
identity
a one-to-one mapping
* an one-to-one mapping
are in one-to-one correspondence with
there is a one-to-one correspondence between _ and _
1-1 function _ from _ into _
to preserve order
to preserve complementation
preserves the orientation of a _
to map _ to _
the restriction of f to _
the extension of f to X
* the extension of f on X
_ is majorized by _
the map of R to R such that _ (total function)
the map from R to R such that _ (partial function)
a function whose domain is _
the inverse function
the inverse image of _ under the map _
the sequence tends to $ A $ as $ n \to \infty $
the sequence tends to infinity
Let n tend to infinity.
is monotonically increasing in x
a monotonically decreasing sequence
\sk2 FIRST-ORDER
function symbol
predicate symbol
Each function and relation symbol of _ comes with a fixed arity.
individual variables
inconsistent
contingent
arithmetical
an occurrence of a subformula
* a placed subformula
concrete occurrence
primitive term constant
logical constant
propositional letter
functional symbols having arity r
\sk2 MODELS
formulas
* formulae [Springer]
a formula is valid in a model
valid
the intended semantics
\sk2 ALGORITHMS
given _, find _
to find _, multiply _ by _
the algorithm of computation of involutive bases
the involutive bases computation algorithm
an algorithm for constructing
recursively enumerable
deterministic one-tape Turing machine (DTM)
a two-way infinite sequence of tape squares
a transition function
an instantaneous description of a Turing machine is given by a quadruple _
the smallest contiguous portion of the tape that includes _
a DTM program recognizes
an algorithm solves
a polynomial translation from a language _ to a language _
satisfiability
truth assignment
a clause is satisfied by a truth assignment
effectively decidable
the decidability of the word problem
the membership in G is decidable
decision procedure
the process offers an effective test whether
is a consequence of _ with respect to the production _
the inverse of the production _
the semi-Thue production associated with
the normal production associated with
antinormal production
A combinatorial system _ consists of a single nonempty word
called the axiom of _ and a finite set of productions.
a word on a combinatorial system
is a combinatorial system all of whose productions are
a proof in a combinatorial system
a step of the proof
a theorem of a combinatorial system
configuration _ yields configuration _ in one step
after at most _ steps
after some number, possibly zero, of steps
By the decision problem for a combinatorial system, we mean the problem of
determining, of a given word, whether or not it is a theorem of the system.
We say that the decision problem for _ is recursively solvable or unsolvable,
according as _ is or is not a recursive set.
the problem (1): given _, to decide whether _
whose alphabet is that of _ with some additional symbol
we form _ by adjoining to _ the productions _
computational complexity
the complexity of the derivability/satisfiability problem for _
elementary operations are performed in constant time
Provability of multiplicative formulas is NP-complete.
The decidability of this fragment is an open problem.
an encoding of Turing machines by constant-only formulas
this encoding is reminiscent of the standard proof of the PSPACE-hardness of
this encoding is sound
this encoding is faithful
at each step
* on each step
\sk2 LOGIC
accessibility relation
deducible
relevant logic
fusion
embedding of _ into
validity
an instance of
simultaneous substitution
application of modus ponens
designated truth-values
is closed under the rules
a logic has the Craig interpolation property
a first-order language
* first order language
second-order logic
recursion-theoretic
* recursion-theoretical
model-theoretic
proof-theoretic
set-theoretic
noncommutative linear logic
to determine whether a given formula is a tautology
* to define whether
The list of axioms is complete.
* The list of axioms is finished.
the power of the continuum
elementarily equivalent to
* elementary equivalent to
Linear negation is a defined concept, not a basic connective.
One may define negation by recursion on the structure of formulas.
\sk2 ALGEBRA
alphabet
defining relations
Abelian group
a simple group
unit
a noninvertible element of a ring = a non-unit of a ring
* irreversible element
a ring with identity [Collins]
a commutative ring with identity [Kurakin]
a commutative ring with unit [Joyner, Kreminski, Turisco, Springer]
algebra with unit element [Kovalenko]
an associative ring with unity [Lambek, Springer]
* an associative ring with a unity [Springer]
not equal to the identity [Springer]
unity element
a root of unity
other diagonal elements are equal to unity
* other diagonal elements are equal to the unit [Springer]
idempotent
complemented lattice
proper ideal
two-sided ideal
to operate on _
to operate on the left on _
multiplying on the left by _
the identity map
* the identical map
the inverse of
the inverse value of
kernel
image
the preimage of
short exact sequence
commutative diagram
semigroup
neutral element
irreducible
closure under
the reflexive, transitive closure
the left-cancellation law
to distribute over
the distributive law
distributivity
Multiplication is both-side distributive over addition.
to commute with
For notational convenience, it is usually assumed that
_ associates to the left
we associate multiplication to the left
a left-normed Lie monomial
* monom
_ takes precedence over _
_ has higher precedence than _
conjunction has precedence over disjunction
products bind more tightly than sums
the order of strength of _ is the natural one: _ is performed before _
Brauer groups
infinite-dimensional
a finite-dimensional vector space over C
H is finite dimensional
a polynomial in x
a polynomial of degree n over a ring R
a polynomial in three variables
a polynomial with all real distinct roots x_1