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