1586:, which puts tight restraints on what would otherwise appear to be a large class of manifolds. This (informal) usage reflects the opinion of the mathematical community: not only should such a theorem be strong in the descriptive sense (below) but it should also be definitive in its area. A theorem, result, or condition is further called
225:
An aesthetic term referring to the ability of an idea to provide insight into mathematics, whether by unifying disparate fields, introducing a new perspective on a single field, or by providing a technique of proof which is either particularly simple, or which captures the intuition or imagination as
2215:
A rhetorical shortcut made by authors who invite the reader to verify, at a glance, the correctness of a proposed expression or deduction. If an expression can be evaluated by straightforward application of simple techniques and without recourse to extended calculation or general theory, then it can
2312:
In a context not requiring rigor, this phrase often appears as a labor-saving device when the technical details of a complete argument would outweigh the conceptual benefits. The author gives a proof in a simple enough case that the computations are reasonable, and then indicates that "in general"
111:
to a new level of abstraction...if certain mathematicians could console themselves for a time with the hope that all these complicated structures were 'abstract nonsense'...the later papers of
Grothendieck and others showed that classical problems...which had resisted efforts of several generations
947:
of a random variable is said to be finite, this implies it is a non-negative real number, possibly zero. In some contexts though, for example in "a small but finite amplitude", zero and infinitesimals are meant to be excluded. When said of the value of a variable assuming values from the extended
292:
A result is called "folklore" if it is non-obvious and non-published, yet generally known to the specialists within a field. In many scenarios, it is unclear as to who first obtained the result, though if the result is significant, it may eventually find its way into the textbooks, whereupon it
247:
The beauty of a mathematical theory is independent of the aesthetic qualities...of the theory's rigorous expositions. Some beautiful theories may never be given a presentation which matches their beauty....Instances can also be found of mediocre theories of questionable beauty which are given
2163:
can be more easily proved with additional assumptions on the objects it concerns. If the proposition as stated follows from this modified one with a simple and minimal explanation (for example, if the remaining special cases are identical but for notation), then the modified assumptions are
2194:
Rather than finding underlying principles or patterns, this is a method where one would evaluate as many cases as needed to sufficiently prove or provide convincing evidence that the thing in question is true. Sometimes this involves evaluating every possible case (where it is also known as
633:. An arbitrary choice is one which is made unrestrictedly, or alternatively, a statement holds of an arbitrary element of a set if it holds of any element of that set. Also much in general-language use among mathematicians: "Of course, this problem can be arbitrarily complicated".
489:
Although ultimately every mathematical argument must meet a high standard of precision, mathematicians use descriptive but informal statements to discuss recurring themes or concepts with unwieldy formal statements. Note that many of the terms are completely rigorous in context.
251:
Mathematicians may say that a theorem is beautiful when they really mean to say that it is enlightening. We acknowledge a theorem's beauty when we see how the theorem 'fits' in its place....We say that a proof is beautiful when such a proof finally gives away the secret of the
393:
which seem to try to resemble as little as possible the honest functions which serve some purpose....Nay more, from the logical point of view, it is these strange functions which are the most general....to-day they are invented expressly to put at fault the reasonings of our
248:
brilliant, exciting expositions.... is rich in beautiful and insightful definitions and poor in elegant proofs.... remain clumsy and dull.... vied for one another in elegance of presentation and in cleverness of proof....In retrospect, one wonders what all the fuss was about.
376:. In many occasions, these can be and often are contradictory requirements, while in other occasions, the term is more deliberately used to refer to an object artificially constructed as a counterexample to these properties. A simple example is that from the definition of a
1944:
a theorem, the use of this expression in the statement of the theorem indicates that the conditions involved may be not yet known to the speaker, and that the intent is to collect the conditions that will be found to be needed in order for the proof of the theorem to go
298:
Many of the results mentioned in this paper should be considered "folklore" in that they merely formally state ideas that are well-known to researchers in the area, but may not be obvious to beginners and to the best of my knowledge do not appear elsewhere in
1630:
meeting stronger conditions. When used in this way, the stronger notion (such as "strong antichain") is a technical term with a precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of the weaker notion (such as
409:
took on an enormous importance...as giving an incentive for the creation of new types of function whose properties departed completely from what intuitively seemed admissible. A celebrated example of such a so-called 'pathological' function...is
454:
The act of establishing a mathematical result using indisputable logic, rather than informal descriptive argument. Rigor is a cornerstone quality of mathematics, and can play an important role in preventing mathematics from degenerating into
1557: = 2.0870652... results in a sharp upper bound; the slightly smaller choice α = 2 fails to produce an upper bound, since then α = 8 < 3. In applied fields the word "tight" is often used with the same meaning.
1963:
It is often the case that two objects are shown to be equivalent in some way, and that one of them is endowed with additional structure. Using the equivalence, we may define such a structure on the second object as well, via
1804:
An obsolescent term which is used to announce to the reader an alternative method, or proof of a result. In a proof, it therefore flags a piece of reasoning that is superfluous from a logical point of view, but has some other
1156:
In a descriptive context, this phrase introduces a simple characterization of a broad class of objects, with an eye towards identifying a unifying principle. This term introduces an "elegant" description which holds for
2306:
A non-technique of proof mostly employed in lectures, where formal argument is not strictly necessary. It proceeds by omission of details or even significant ingredients, and is merely a plausibility argument.
1845:
arguments when passing from the base case to the induction step, and similarly, in the definition of sequences whose first few terms are exhibited as examples of the formula giving every term of the sequence.
3075:
2941:
in a category. Depending on authors, the term "maps" or the term "functions" may be reserved for specific kinds of functions or morphisms (e.g., function as an analytic term and map as a general term).
445:
found out in 1931, differentiable functions are colloquially speaking a rare exception among the continuous ones. Thus it can hardly be defended any-more to call non-differentiable continuous functions
469:) if it satisfies certain prevailing regularity properties, or if it conforms to mathematical intuition (even though intuition can often suggest opposite behaviors as well). In some occasions (e.g.,
941:
1896:
is algebraically closed if and only if it has no finite extensions". Often used in lists, as in "The following conditions are necessary and sufficient for a field to be algebraically closed...".
2337:
Usually applied to a claim within a larger proof when the proof of that claim can be produced routinely by any member of the audience with the necessary expertise, but is not so simple as to be
534:
having a property to mean "all except finitely many", despite the integers not admitting a measure for which this agrees with the previous usage. For example, "almost all prime numbers are
1784:
Accurately and precisely described or specified. For example, sometimes a definition relies on a choice of some object; the result of the definition must then be independent of this choice.
983:
136:(e.g., canonical map, canonical form, or canonical ordering). The same term can also be used more informally to refer to something "standard" or "classic". For example, one might say that
1174:. Grothendieck advised caution. The Platonic solids are so beautiful and so exceptional, he said, that one cannot assume such exceptional beauty will hold in more general situations.
42:: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for
2760:
2244:
A term which shortcuts around calculation the mathematician perceives to be tedious or routine, accessible to any member of the audience with the necessary expertise in the field;
1640:
In the context of limits, these terms refer to some (unspecified, even unknown) point at which a phenomenon prevails as the limit is approached. A statement such as that predicate
71:, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it is also known as
1427:
if it satisfies satisfactory continuity and differentiability properties, which are often context-dependent. These properties might include possessing a specified number of
817:
785:
721:
343:
1015:
753:
2791:
2699:
2494:
2589:
1072:
2627:
2209:
is an argument whereby a statement is not proved but instead illustrated by an example. If done well, the specific example would easily generalize to a general proof.
2188:
An informal computation omitting much rigor without sacrificing correctness. Often this computation is "proof of concept" and treats only an accessible special case.
2238:
Style of proof where claims believed by the author to be easily verifiable are labelled as 'obvious' or 'trivial', which often results in the reader being confused.
1930:): A Latin abbreviation, meaning "which was to be demonstrated", historically placed at the end of proofs, but less common currently, having been supplanted by the
1229:
1150:
set is true generically; however, it is usually not said that a property which holds merely on a dense set (which is not
Zariski open) is generic in this situation.
2319:
For proofs involving objects with multiple indices which can be solved by going to the bottom (if anyone wishes to take up the effort). Similar to diagram chasing.
2653:
2520:
1275:
1295:
if it satisfies hypotheses or properties, sometimes unspecified or even unknown, that are especially desirable in a given context. It is an informal antonym for
372:
way) if it either fails to conform to the generic behavior of such objects, fails to satisfy certain context-dependent regularity properties, or simply disobeys
2875:
2855:
2835:
2815:
2673:
2468:
2448:
1249:
885:
865:
845:
2891:
The word fundamental is used to describe a theorem with a given area of mathematics considered to be the most central theorem of that particular area (e.g.
2290:, then the proof can proceed by tracing the path of elements of various objects around the diagram as successive morphisms are applied to it. That is, one
189:
A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the
2370:, or terms that do not typically appear in more specialized glossaries. For the terms used only in some specific areas of mathematics, see glossaries in
538:". There is a more complicated meaning for integers as well, discussed in the main article. Finally, this term is sometimes used synonymously with
2411:
is a map or morphism between objects that arises naturally from the definition or the construction of the objects being mapped against each other.
1553:. This is not sharp; the gap between the functions is everywhere at least 1. Among the exponential functions of the form α, setting α =
1940:
A condition on objects in the scope of the discussion, to be specified later, that will guarantee that some stated property holds for them. When
1730:
a condition if the establishment of that condition is the only impediment to the truth of the statement. Also used when working with members of
46:
arguments or precise ideas. Much of this uses common
English words, but with a specific non-obvious meaning when used in a mathematical sense.
1590:
than another one if a proof of the second can be easily obtained from the first but not conversely. An example is the sequence of theorems:
2173:
Mathematicians have several phrases to describe proofs or proof techniques. These are often used as hints for filling in tedious details.
1513:
Often, a mathematical theorem will establish constraints on the behavior of some object; for example, a function will be shown to have an
3019:
1278:
272:
A proof or a result is called "elementary" if it only involves basic concepts and methods in the field, and is to be contrasted with
238:, saying that for example, some topics could be written about elegantly although the mathematical content is not beautiful, and some
360:) which holds independently of any choices. Though long used informally, this term has found a formal definition in category theory.
985:
the meaning is simply "not infinite". When said of a set or a mathematical object whose main component is a set, it means that the
2993:
such that the restriction of the function to the neighbourhood can be considered as a set of functions from the neighbourhood to
1902:
Proofs sometimes proceed by enumerating several conditions whose satisfaction will together imply the desired theorem; thus, one
1572:, and still others which are more complicated. Each such usage attempts to invoke the physically intuitive notion of smoothness.
1568:
is a concept which mathematics has endowed with many meanings, from simple differentiability to infinite differentiability to
1116:. A property holds "generically" on a set if the set satisfies some (context-dependent) notion of density, or perhaps if its
276:
results which require more development within or outside the field. The concept of "elementary proof" is used specifically in
3556:
3465:
3321:
3204:
899:
2918:
of an object or a space is a property or number of the object or a space that remains unchanged under some transformations.
1599:
356:
Similar to "canonical" but more specific, and which makes reference to a description (almost exclusively in the context of
3104:
on an object is an additional set of objects or data attached to the object (e.g., relation, operation, metric, topology).
3339:
Jackson, Allyn (2004), "Comme AppelĂ© du NĂ©ant — As If
Summoned from the Void: The Life of Alexandre Grothendieck",
3016:
is, roughly, a map from some space or object to another that omits some information on the object or space. For example,
2886:
226:
to why the result it proves is true. In some occasions, the term "beautiful" can also be used to the same effect, though
1955:) are equally useful in practice; one introduces a theorem stating an equivalence of more than two statements with TFAE.
3594:
3179:
2392:
is a set of ordered pairs; an element x is said to be related to another element y if and only if (x,y) are in the set.
2371:
1797:
draws repeatedly from a small pool of ideas, many of which are invoked through various lexical shorthands in practice.
3589:
3129:
2892:
1303:
ought to satisfy a certain boundedness condition "for nice test functions," or one might state that some interesting
3114:
2183:
951:
149:
There are two canonical proofs that are always used to show non-mathematicians what a mathematical proof is like:
2180:
Used to describe a geometrical proof that involves finding relationships between the various angles in a diagram.
1674:
A descriptive term referring to notation in which two objects are written one above the other; the upper one is
474:
3119:
1360:
is onto" (i. e. surjective). Not translatable (without circumlocutions) to some languages other than
English.
554:, referring to the recurrence of a phenomenon as the limit is approached. A statement such as that predicate
137:
3124:
1366:
If, for some notion of substructure, objects are substructures of themselves (that is, the relationship is
357:
3573:
17:
2704:
2153:
1885:
1591:
363:
2282:
of objects and morphisms between them, if one wishes to prove some property of the morphisms (such as
1296:
205:
is irrational is usually known to be a deep result, because it requires a considerable development of
2287:
2160:
1915:
1131:
3434:
3304:
3077:
is a projection and its restriction to a graph of a function, say, is also a projection. The terms â
790:
758:
694:
209:
before the proof can be established â even though the claim itself can be stated in terms of simple
3013:
1973:
1848:
677:
636:
1984:
and if we fix a particular isomorphism, then we may define an inner product on the other space by
1170:
Norbert A'Campo of the
University of Basel once asked Grothendieck about something related to the
1120:
satisfies some (context-dependent) notion of smallness. For example, a property which holds on a
326:
3149:
2915:
1117:
993:
419:
127:
31:
3354:
3350:
1775:
726:
3299:
3101:
2934:
2678:
2557:
2473:
2119:
2022:
1958:
1931:
1842:
1815:
1808:
1623:
1583:
1514:
1075:
390:
373:
351:
85:
3455:
2562:
2224:
by inspection is to 'notice' them, or mentally check them. 'By inspection' can play a kind of
1041:
2594:
2344:
2233:
1304:
1300:
645:; the relevant argument(s) are implicit in the context. As an example, the function log(log(
516:
508:
470:
287:
1614:
may be added to a mathematical notion to indicate a related stronger notion; for example, a
3380:
3251:
2962:
2538:
2051:
1739:
1534:
1410:
1202:
1189:
630:
535:
411:
190:
2765:
1440:
384:
which sum to Ï radians, a single straight line conforms to this definition pathologically.
8:
3291:
3078:
2632:
2499:
2496:; in other words, it is a binary relation but with the specification of the ambient sets
2279:
2196:
1878:
1832:
1747:
1731:
1320:
A function (which in mathematics is generally defined as mapping the elements of one set
1254:
551:
415:
220:
141:
133:
3384:
3255:
1525:) if it cannot be made more restrictive without failing in some cases. For example, for
1088:
Qualifies anything that is sufficiently precise to be translated straightforwardly in a
3518:
3327:
3274:
2860:
2840:
2820:
2800:
2658:
2453:
2433:
2267:
2221:
2030:
1977:
1794:
1765:
1723:
1634:
1595:
1547:
1367:
1234:
1139:
870:
850:
830:
663:
166:
108:
43:
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is satisfied by arbitrarily large values, can be expressed in more formal notation by
3552:
3545:
3510:
3461:
3420:
3408:
3403:
3361:
3317:
3279:
3239:
3200:
3175:
1743:
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if it deduces restrictive results from general hypotheses. One celebrated example is
1569:
1541:
1308:
1143:
545:
520:
500:
162:
58:
3522:
3475:
1602:, each of which is stronger than the last; another is that a sharp upper bound (see
1500:
317:
A term regarding statements. If a statement holds false, then it is said to exhibit
3530:
3502:
3398:
3388:
3331:
3309:
3269:
3259:
3235:
1909:
1615:
1504:
1147:
1103:
449:
281:
267:
198:
194:
1644:
holds for sufficiently large values, can be expressed in more formal notation by â
3490:
3479:
3365:
2794:
2427:
2389:
2273:
2253:
1952:
1889:
1735:
1560:
1414:
669:
227:
68:
64:
2415:
2245:
1171:
1113:
894:
112:
of talented mathematicians, could be solved in terms of...complicated concepts.
3506:
1918:
of an object; the object exists, and furthermore, no other such object exists.
3583:
3514:
3150:"The Elementary Proof of the Prime Number Theorem: An Historical Perspective"
3082:
2408:
1981:
1899:
need to show (NTS), required to prove (RTP), wish to show, want to show (WTS)
1627:
1496:
1452:
1195:
Most often, these refer simply to the left-hand or the right-hand side of an
1089:
442:
277:
210:
206:
3393:
3313:
2357:
to a known statement, or is a simple special case of a more general concept.
1825:
1161:" objects. Exceptions to this description may be mentioned explicitly, as "
3283:
3264:
1969:
1683:
1093:
893:
When said of the value of a variable assuming values from the non-negative
530:
of the real numbers with measure zero. One can also speak of "almost all"
504:
458:
155:
104:
3412:
2226:
2220:. It is also applied to solving equations; for example to find roots of a
1951:
Often several equivalent conditions (especially for a definition, such as
1606:
above) is a stronger result than a non-sharp one. Finally, the adjective
280:, where it usually refers to a proof that does not resort to methods from
2950:
2367:
2283:
1742:
is (categorical) isomorphism; for example, "The tensor product in a weak
986:
512:
49:
Some phrases, like "in general", appear below in more than one section.
2896:
2301:
1941:
1428:
1353:
494:
438:
368:
An object behaves pathologically (or, somewhat more broadly used, in a
35:
3296:
Proc. Tenth Annual
Structure in Complexity Theory Conference (SCT'95)
2418:
of an object is some standard or universal way to express the object.
2354:
1764:
that are integer multiples of Ï." This can also apply to limits: see
1619:
1546: = 2.7182818..., gives an upper bound on the values of the
1121:
624:
524:
437:
Note for that latter quote that as the differentiable functions are
2938:
2353:. A concept is trivial if it holds by definition, is an immediate
1819:
1707:
1699:
1695:
1196:
1135:
1036:
944:
673:
653:
becomes larger than 100"; in this context, "eventually" means "for
377:
214:
38:
of specialist and technical terms. It also has a certain amount of
2164:
introduced with this phrase and the altered proposition is proved.
1394:
990:
531:
239:
27:
Collection of commonly used phrases found in mathematical fields
3070:{\displaystyle \mathbb {R} ^{2}\to \mathbb {R} ,(x,y)\mapsto x}
2118:
for this last algebra. By transport of structure, we obtain a
1921:
1492:
527:
39:
1461:(Respectively) A convention to shorten parallel expositions. "
132:
A reference to a standard or choice-free presentation of some
1716:
1112:
but is used particularly for concepts outside the purview of
381:
242:
or proofs are beautiful but may be written about inelegantly.
2981:. It has typically the property that, for almost all points
1443:. Informally, this term is sometimes used synonymously with
345:
is compact if and only if it is bounded? This is chicanery!"
88:' — a subject then called 'general abstract nonsense'!
2366:
This section features terms used across different areas in
943:
the meaning is usually "not infinite". For example, if the
389:
Since half a century we have seen arise a crowd of bizarre
1281:
and rvalue: an RHS is primitive, and an LHS is derivative.
1277:
on the RHS. Occasionally, these are used in the sense of
1722:
An extension to mathematical discourse of the notions of
1622:
satisfying certain additional conditions, and likewise a
3435:"Some Trends in Modern Mathematics and the Fields Medal"
2879:
1431:, with the function and its derivatives exhibiting some
1035:, the intended variant is implicit. As an example, the
1841:
In the context of proofs, this phrase is often seen in
1082:
for which the value of the sequence is in the interval.
202:
3294:(1995), "A personal view of average-case complexity",
2230:
role: the answer or solution simply clicks into place.
1888:
it is necessary and sufficient that it have no finite
1756:
To assume the value 0. For example, "The function sin(
936:{\displaystyle \mathbb {R} _{\geq 0}\cup \{\infty \},}
3022:
2863:
2843:
2823:
2803:
2768:
2707:
2681:
2661:
2635:
2597:
2565:
2502:
2476:
2456:
2436:
1374:
requires the objects to be different. For example, a
1257:
1237:
1205:
1044:
996:
954:
902:
873:
853:
833:
793:
761:
729:
697:
329:
3493:(1977), "The phenomenology of mathematical beauty",
641:
In the context of limits, this is shorthand meaning
1495:(resp. triangles) have 4 sides (resp. 3 sides); or
3544:
3069:
2869:
2849:
2829:
2809:
2785:
2754:
2693:
2667:
2647:
2621:
2583:
2514:
2488:
2462:
2442:
1269:
1243:
1223:
1066:
1009:
977:
935:
879:
859:
839:
811:
779:
747:
715:
463:An object is well-behaved (in contrast with being
337:
3234:
3218:Numerous examples can be found in (Mac Lane
261:, pp.173–174, pp.181–182)
3581:
3242:(1942), "Natural Isomorphisms in Group Theory",
1078:(1/2, 3/2), because there are arbitrarily large
1024:In the context of limits, this is shorthand for
598:"can be made" arbitrarily large, corresponds to
154:—The proof that there are infinitely many
3457:Infinitesimal methods for mathematical analysis
2334:the proof is left as an exercise to the reader
3454:Pinto, J. Sousa (2004), Hoskins, R.F. (ed.),
1507:has a finite (resp. countable) open subcover.
1287:A mathematical object is colloquially called
978:{\displaystyle \mathbb {N} \cup \{\infty \},}
550:Notions which arise mostly in the context of
201:were found. On the other hand, the fact that
197:â was once thought to be a deep result until
3199:, Oxford science publications, p. 119,
1451:are not to be confused with the notion of a
969:
963:
927:
921:
484:
52:
3529:
3432:
3290:
2144:
1299:. For example, one might conjecture that a
1142:, one says that a property of points on an
305:
118:
2793:. In other words, it is a special kind of
1997:be a finite-dimensional vector space over
1447:, below. These imprecise uses of the word
3402:
3392:
3303:
3273:
3263:
3039:
3025:
2156:(WLOG, WOLOG, WALOG), we may assume (WMA)
956:
905:
441:in the space of continuous functions, as
331:
3474:
3425:Categories for the Working Mathematician
3419:
3360:
3219:
400:
193:â originally proved using techniques of
94:
84:introduced the very abstract idea of a '
3542:
3338:
2294:elements around the diagram, or does a
1180:
654:
511:to speak of. For example, "almost all
499:A shorthand term for "all except for a
177:
14:
3582:
1853:A minor variant on "if and only if"; "
1686:, the total space is often said to be
1108:This term has similar connotations as
1026:
3453:
3085:â are also synonyms for a projection.
2266:commonly reserved for jokes (puns on
2068:)....It extends to an isomorphism of
428:
3489:
3169:
3147:
1788:
1637:, suitably small, sufficiently close
480:can also be used to the same effect.
258:
3460:, Horwood Publishing, p. 246,
3194:
2887:List of theorems called fundamental
2470:is a subset of a Cartesian product
2168:
2029:....There is an isomorphism of the
1948:the following are equivalent (TFAE)
1713:, as in "bringing a term upstairs".
1503:) spaces are ones where every open
1138:) is said to hold generically. In
24:
3547:The Seventeen Provers of the World
2372:Category:Glossaries of mathematics
2286:) which can be stated in terms of
1746:is associative and unital up to a
998:
966:
924:
25:
3606:
3130:Category:Mathematical terminology
2893:Fundamental Theorem of Arithmetic
2755:{\displaystyle (a,b),(a,b')\in f}
2420:
1162:
465:
3484:, The Science Press, p. 435
3115:Glossary of areas of mathematics
2361:
2184:back-of-the-envelope calculation
2154:without (any) loss of generality
723:can be written as a composition
643:for sufficiently large arguments
3566:
1760:) vanishes for those values of
1702:is occasionally referred to as
1307:should be computable "for nice
1190:left-hand side, right-hand side
412:the one provided by Weierstrass
3478:(1913), Halsted, Bruce (ed.),
3212:
3188:
3174:. Cambridge University Press.
3163:
3141:
3120:List of mathematical constants
3061:
3058:
3046:
3035:
2989:, there is a neighbourhood of
2955:
2943:
2743:
2726:
2720:
2708:
2616:
2598:
2575:
1822:of the statement to be proved.
1526:
1455:, which is rigorously defined.
1413:word is also non-jargon for a
1158:
1055:
1045:
812:{\displaystyle h\colon B\to C}
803:
780:{\displaystyle g\colon A\to B}
771:
716:{\displaystyle f\colon A\to C}
707:
586:. The statement that quantity
323:"What do you mean a subset of
304:Russell Impagliazzo (
117:Michael Monastyrsky (
13:
1:
3533:(1991), Kandall, G.A. (ed.),
3433:Monastyrsky, Michael (2001),
3228:
3006:
1980:; if one of them is given an
1811:(BWOC), or "for, if not, ..."
77:generalized abstract nonsense
3543:Wiedijk, Freek, ed. (2006),
3125:List of mathematical symbols
3094:
2908:
2837:, there is a unique element
2401:
2327:
2241:clearly, can be easily shown
2072:to the localized algebra Sym
1814:The rhetorical prelude to a
1356:; it may even be said that "
672:referring to composition of
338:{\displaystyle \mathbb {R} }
140:is the "canonical proof" of
93:Saunders Mac Lane (
7:
3574:Encyclopedia of Mathematics
3108:
2550:
2143:Igor Shafarevich (
1031:and its relatives; as with
1010:{\displaystyle \aleph _{0}}
10:
3611:
3481:The Foundations of Science
3373:Proc. Natl. Acad. Sci. USA
3244:Proc. Natl. Acad. Sci. USA
2884:
2531:
1529:non-negative real numbers
1370:), then the qualification
748:{\displaystyle f=h\circ g}
257:Gian-Carlo Rota (
3595:Glossaries of mathematics
3222:), for example on p. 100.
2701:subject to the condition
2694:{\displaystyle A\times B}
2675:of the Cartesian product
2489:{\displaystyle A\times B}
2382:
1453:regular topological space
485:Descriptive informalities
427:J. Sousa Pinto (
399:Henri Poincaré (
273:
73:general abstract nonsense
53:Philosophy of mathematics
3590:Mathematical terminology
3537:, vol. IV, Springer
3366:"The PNAS way back then"
3135:
2584:{\displaystyle f:A\to B}
1968:. For example, any two
1932:Halmos end-of-proof mark
1916:existence and uniqueness
1877:". For example, "For a
1849:necessary and sufficient
1578:A theorem is said to be
1179:Allyn Jackson (
1067:{\displaystyle (-1)^{n}}
989:of the set is less than
232:elegance of presentation
176:Freek Wiedijk (
161:—The proof of the
142:the infinitude of primes
3507:10.1023/A:1004930722234
3394:10.1073/pnas.94.12.5983
3314:10.1109/SCT.1995.514853
2877:that corresponds to it.
2797:where given an element
2622:{\displaystyle (A,B,f)}
2522:used in the definition.
1928:Quod erat demonstrandum
1809:by way of contradiction
1793:The formal language of
1726:. A statement is true
1592:Fermat's little theorem
1405:that is different from
1386:that is different from
1324:to elements of another
32:language of mathematics
3265:10.1073/pnas.28.12.537
3170:Boyd, Stephen (2004).
3102:mathematical structure
3071:
2927:
2871:
2851:
2831:
2811:
2787:
2756:
2695:
2669:
2649:
2623:
2585:
2516:
2490:
2464:
2444:
2150:
2120:linear algebraic group
1966:transport of structure
1959:transport of structure
1906:just these statements.
1816:proof by contradiction
1690:, with the base space
1624:strongly regular graph
1271:
1245:
1225:
1186:
1146:that holds on a dense
1068:
1011:
979:
937:
881:
861:
841:
813:
781:
749:
717:
521:algebraic real numbers
434:
406:
374:mathematical intuition
339:
311:
293:ceases to be folklore.
264:
230:distinguished between
183:
124:
100:
3442:Can. Math. Soc. Notes
3072:
2872:
2852:
2832:
2812:
2788:
2757:
2696:
2670:
2650:
2624:
2591:is an ordered triple
2586:
2517:
2491:
2465:
2445:
2313:the proof is similar.
1991:
1682:. For example, in a
1517:. The constraint is
1423:A function is called
1305:topological invariant
1301:differential operator
1272:
1246:
1226:
1224:{\displaystyle x=y+1}
1168:
1074:is frequently in the
1069:
1012:
980:
938:
882:
862:
842:
814:
782:
750:
718:
414:....This function is
407:
387:
340:
296:
245:
147:
101:
82:
3298:, pp. 134â147,
3292:Impagliazzo, Russell
3089:
3020:
3001:
2963:multivalued function
2922:
2903:
2861:
2841:
2821:
2801:
2786:{\displaystyle b=b'}
2766:
2705:
2679:
2659:
2633:
2595:
2563:
2545:
2539:mathematical diagram
2526:
2500:
2474:
2454:
2434:
2396:
2377:
1886:algebraically closed
1831:An abbreviation for
1740:equivalence relation
1719:, modulo, mod out by
1671:upstairs, downstairs
1535:exponential function
1515:upper or lower bound
1255:
1235:
1203:
1042:
994:
952:
900:
871:
851:
831:
791:
759:
727:
695:
631:universal quantifier
629:A shorthand for the
327:
191:prime number theorem
3385:1997PNAS...94.5983M
3256:1942PNAS...28..537E
3197:Elementary Geometry
3172:Convex Optimization
3157:Columbia University
3079:idempotent operator
2973:is a function from
2648:{\displaystyle A,B}
2629:consisting of sets
2515:{\displaystyle A,B}
2280:commutative diagram
2263:complete intuition
2197:proof by exhaustion
2105: ⊗
2082: ⊗
2064: ⊗
1914:A statement of the
1833:logical equivalence
1748:natural isomorphism
1732:equivalence classes
1584:Donaldson's theorem
1270:{\displaystyle y+1}
507:", when there is a
134:mathematical object
3535:Algebraic Geometry
3421:Mac Lane, Saunders
3362:Mac Lane, Saunders
3240:Mac Lane, Saunders
3195:Roe, John (1993),
3148:Goldfeld, Dorian.
3067:
2977:to the subsets of
2937:between sets or a
2867:
2847:
2827:
2807:
2783:
2752:
2691:
2665:
2645:
2619:
2581:
2512:
2486:
2460:
2440:
2268:complete induction
2222:quadratic equation
2031:polynomial algebra
1934:, a square sign â.
1766:Vanish at infinity
1724:modular arithmetic
1635:sufficiently large
1600:Lagrange's theorem
1548:quadratic function
1267:
1241:
1221:
1140:algebraic geometry
1134:of countably many
1064:
1007:
975:
933:
877:
857:
837:
809:
777:
745:
713:
655:sufficiently large
335:
167:square root of two
109:algebraic geometry
3558:978-3-540-30704-4
3531:Shafarevich, Igor
3467:978-1-898563-99-0
3379:(12): 5983â5985,
3323:978-0-8186-7052-7
3236:Eilenberg, Samuel
3206:978-0-19-853456-3
2870:{\displaystyle B}
2850:{\displaystyle b}
2830:{\displaystyle A}
2810:{\displaystyle a}
2668:{\displaystyle f}
2463:{\displaystyle B}
2443:{\displaystyle A}
2096: = det(
1986:factoring through
1937:sufficiently nice
1789:Proof terminology
1744:monoidal category
1441:Hölder continuity
1352:") only if it is
1293:sufficiently nice
1244:{\displaystyle x}
1144:algebraic variety
1098:formal definition
1092:. For example. a
1027:arbitrarily large
880:{\displaystyle h}
860:{\displaystyle g}
840:{\displaystyle B}
827:any (and all) of
546:arbitrarily large
236:beauty of concept
199:elementary proofs
59:abstract nonsense
16:(Redirected from
3602:
3561:
3550:
3538:
3525:
3491:Rota, Gian-Carlo
3485:
3470:
3449:
3439:
3428:
3415:
3406:
3396:
3370:
3348:
3334:
3307:
3286:
3277:
3267:
3223:
3216:
3210:
3209:
3192:
3186:
3185:
3167:
3161:
3160:
3154:
3145:
3076:
3074:
3073:
3068:
3042:
3034:
3033:
3028:
2996:
2992:
2988:
2984:
2980:
2976:
2972:
2968:
2933:A synonym for a
2876:
2874:
2873:
2868:
2856:
2854:
2853:
2848:
2836:
2834:
2833:
2828:
2816:
2814:
2813:
2808:
2792:
2790:
2789:
2784:
2782:
2761:
2759:
2758:
2753:
2742:
2700:
2698:
2697:
2692:
2674:
2672:
2671:
2666:
2654:
2652:
2651:
2646:
2628:
2626:
2625:
2620:
2590:
2588:
2587:
2582:
2521:
2519:
2518:
2513:
2495:
2493:
2492:
2487:
2469:
2467:
2466:
2461:
2449:
2447:
2446:
2441:
2414:2. A
2407:1. A
2207:proof by example
2169:Proof techniques
2148:
2129:) isomorphic to
1988:the isomorphism.
1910:one and only one
1890:field extensions
1818:, preceding the
1774:The converse of
1734:, especially in
1616:strong antichain
1575:strong, stronger
1439:above), such as
1401:is a divisor of
1378:subset of a set
1276:
1274:
1273:
1268:
1250:
1248:
1247:
1242:
1230:
1228:
1227:
1222:
1184:
1085:formal, formally
1073:
1071:
1070:
1065:
1063:
1062:
1016:
1014:
1013:
1008:
1006:
1005:
984:
982:
981:
976:
959:
948:natural numbers
942:
940:
939:
934:
917:
916:
908:
886:
884:
883:
878:
866:
864:
863:
858:
846:
844:
843:
838:
818:
816:
815:
810:
786:
784:
783:
778:
754:
752:
751:
746:
722:
720:
719:
714:
676:. If for three
620:
581:
432:
404:
344:
342:
341:
336:
334:
309:
282:complex analysis
262:
195:complex analysis
181:
122:
98:
21:
3610:
3609:
3605:
3604:
3603:
3601:
3600:
3599:
3580:
3579:
3569:
3559:
3476:Poincare, Henri
3468:
3437:
3368:
3324:
3305:10.1.1.678.8930
3250:(12): 537â543,
3231:
3226:
3217:
3213:
3207:
3193:
3189:
3182:
3168:
3164:
3152:
3146:
3142:
3138:
3111:
3097:
3092:
3038:
3029:
3024:
3023:
3021:
3018:
3017:
3009:
3004:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2958:
2946:
2930:
2925:
2911:
2906:
2889:
2882:
2862:
2859:
2858:
2842:
2839:
2838:
2822:
2819:
2818:
2802:
2799:
2798:
2775:
2767:
2764:
2763:
2735:
2706:
2703:
2702:
2680:
2677:
2676:
2660:
2657:
2656:
2634:
2631:
2630:
2596:
2593:
2592:
2564:
2561:
2560:
2553:
2548:
2534:
2529:
2501:
2498:
2497:
2475:
2472:
2471:
2455:
2452:
2451:
2435:
2432:
2431:
2423:
2404:
2399:
2390:binary relation
2385:
2380:
2364:
2274:diagram chasing
2234:by intimidation
2171:
2149:
2142:
2137:
2113:
2104:
2091:
2077:
2059:
2049:
2020:
2009:
1953:normal subgroup
1791:
1736:category theory
1678:and the lower,
1596:Euler's theorem
1491:. For example,
1415:proper morphism
1382:is a subset of
1336:" (instead of "
1256:
1253:
1252:
1251:on the LHS and
1236:
1233:
1232:
1204:
1201:
1200:
1199:; for example,
1185:
1183:, p.1197)
1178:
1172:Platonic solids
1129:
1058:
1054:
1043:
1040:
1039:
1001:
997:
995:
992:
991:
955:
953:
950:
949:
909:
904:
903:
901:
898:
897:
872:
869:
868:
852:
849:
848:
832:
829:
828:
792:
789:
788:
760:
757:
756:
728:
725:
724:
696:
693:
692:
670:category theory
599:
594:) depending on
559:
487:
433:
426:
405:
398:
358:transformations
330:
328:
325:
324:
310:
303:
263:
256:
250:
249:
228:Gian-Carlo Rota
182:
175:
172:
123:
116:
99:
92:
69:category theory
65:tongue-in-cheek
55:
28:
23:
22:
15:
12:
11:
5:
3608:
3598:
3597:
3592:
3578:
3577:
3568:
3565:
3564:
3563:
3557:
3551:, BirkhÀuser,
3540:
3527:
3501:(2): 171â182,
3487:
3472:
3466:
3451:
3430:
3417:
3358:
3336:
3322:
3288:
3230:
3227:
3225:
3224:
3211:
3205:
3187:
3181:978-0521833783
3180:
3162:
3139:
3137:
3134:
3133:
3132:
3127:
3122:
3117:
3110:
3107:
3106:
3105:
3098:
3095:
3091:
3088:
3087:
3086:
3066:
3063:
3060:
3057:
3054:
3051:
3048:
3045:
3041:
3037:
3032:
3027:
3010:
3007:
3003:
3000:
2999:
2998:
2959:
2956:
2954:
2947:
2944:
2942:
2931:
2928:
2924:
2921:
2920:
2919:
2912:
2909:
2905:
2902:
2901:
2900:
2885:Main article:
2883:
2880:
2878:
2866:
2846:
2826:
2806:
2795:correspondence
2781:
2778:
2774:
2771:
2751:
2748:
2745:
2741:
2738:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2690:
2687:
2684:
2664:
2644:
2641:
2638:
2618:
2615:
2612:
2609:
2606:
2603:
2600:
2580:
2577:
2574:
2571:
2568:
2554:
2551:
2547:
2544:
2543:
2542:
2535:
2532:
2528:
2525:
2524:
2523:
2511:
2508:
2505:
2485:
2482:
2479:
2459:
2439:
2428:correspondence
2424:
2422:correspondence
2421:
2419:
2416:canonical form
2412:
2405:
2402:
2398:
2395:
2394:
2393:
2386:
2383:
2379:
2376:
2363:
2360:
2359:
2358:
2347:
2342:
2335:
2332:
2323:
2320:
2317:
2314:
2310:
2307:
2304:
2299:
2276:
2271:
2264:
2261:
2242:
2239:
2236:
2231:
2213:
2210:
2203:
2200:
2192:
2189:
2186:
2181:
2178:
2170:
2167:
2166:
2165:
2157:
2140:
2133:
2114:)....We write
2109:
2100:
2087:
2073:
2055:
2036:
2011:
2005:
1990:
1989:
1961:
1956:
1949:
1946:
1938:
1935:
1924:
1919:
1912:
1907:
1900:
1897:
1851:
1846:
1839:
1836:
1835:of statements.
1829:
1826:if and only if
1823:
1812:
1806:
1802:
1790:
1787:
1786:
1785:
1782:
1779:
1772:
1769:
1754:
1751:
1720:
1714:
1672:
1669:
1638:
1632:
1610:or the adverb
1576:
1573:
1563:
1558:
1511:
1508:
1484:and also that
1477:)" means that
1459:
1456:
1435:property (see
1421:
1418:
1364:
1361:
1318:
1315:
1285:
1282:
1266:
1263:
1260:
1240:
1220:
1217:
1214:
1211:
1208:
1193:
1176:
1167:
1166:
1154:
1151:
1127:
1114:measure theory
1106:
1101:
1086:
1083:
1061:
1057:
1053:
1050:
1047:
1022:
1019:
1004:
1000:
974:
971:
968:
965:
962:
958:
932:
929:
926:
923:
920:
915:
912:
907:
895:extended reals
891:
888:
876:
856:
836:
825:factor through
808:
805:
802:
799:
796:
776:
773:
770:
767:
764:
744:
741:
738:
735:
732:
712:
709:
706:
703:
700:
666:
664:factor through
661:
639:
634:
627:
622:
548:
543:
519:" because the
517:transcendental
497:
486:
483:
482:
481:
461:
456:
452:
450:rigor (rigour)
447:
424:
420:differentiable
396:
386:
385:
366:
361:
354:
348:
347:
333:
315:
301:
295:
294:
290:
285:
270:
254:
244:
243:
223:
218:
187:
173:
171:
170:
159:
151:
146:
145:
138:Euclid's proof
130:
114:
90:
81:
80:
61:
54:
51:
26:
9:
6:
4:
3:
2:
3607:
3596:
3593:
3591:
3588:
3587:
3585:
3576:
3575:
3571:
3570:
3560:
3554:
3549:
3548:
3541:
3536:
3532:
3528:
3524:
3520:
3516:
3512:
3508:
3504:
3500:
3496:
3492:
3488:
3483:
3482:
3477:
3473:
3469:
3463:
3459:
3458:
3452:
3447:
3443:
3436:
3431:
3426:
3422:
3418:
3414:
3410:
3405:
3400:
3395:
3390:
3386:
3382:
3378:
3374:
3367:
3363:
3359:
3356:
3352:
3346:
3342:
3337:
3333:
3329:
3325:
3319:
3315:
3311:
3306:
3301:
3297:
3293:
3289:
3285:
3281:
3276:
3271:
3266:
3261:
3257:
3253:
3249:
3245:
3241:
3237:
3233:
3232:
3221:
3215:
3208:
3202:
3198:
3191:
3183:
3177:
3173:
3166:
3158:
3151:
3144:
3140:
3131:
3128:
3126:
3123:
3121:
3118:
3116:
3113:
3112:
3103:
3099:
3093:
3084:
3083:forgetful map
3080:
3064:
3055:
3052:
3049:
3043:
3030:
3015:
3011:
3005:
2965:â from a set
2964:
2960:
2952:
2948:
2940:
2936:
2932:
2926:
2917:
2913:
2907:
2898:
2894:
2890:
2888:
2864:
2844:
2824:
2804:
2796:
2779:
2776:
2772:
2769:
2749:
2746:
2739:
2736:
2732:
2729:
2723:
2717:
2714:
2711:
2688:
2685:
2682:
2662:
2655:and a subset
2642:
2639:
2636:
2613:
2610:
2607:
2604:
2601:
2578:
2572:
2569:
2566:
2559:
2555:
2549:
2540:
2536:
2530:
2509:
2506:
2503:
2483:
2480:
2477:
2457:
2437:
2429:
2425:
2417:
2413:
2410:
2409:canonical map
2406:
2400:
2391:
2387:
2381:
2375:
2373:
2369:
2362:Miscellaneous
2356:
2352:
2348:
2346:
2343:
2340:
2336:
2333:
2330:
2329:
2324:
2321:
2318:
2315:
2311:
2308:
2305:
2303:
2300:
2297:
2296:diagram chase
2293:
2289:
2285:
2281:
2277:
2275:
2272:
2269:
2265:
2262:
2259:
2255:
2251:
2247:
2243:
2240:
2237:
2235:
2232:
2229:
2228:
2223:
2219:
2218:by inspection
2216:be evaluated
2214:
2212:by inspection
2211:
2208:
2204:
2201:
2198:
2193:
2190:
2187:
2185:
2182:
2179:
2177:angle chasing
2176:
2175:
2174:
2162:
2158:
2155:
2152:
2151:
2147:, p.12)
2146:
2139:
2136:
2132:
2128:
2124:
2121:
2117:
2112:
2108:
2103:
2099:
2095:
2090:
2085:
2081:
2076:
2071:
2067:
2063:
2058:
2053:
2048:
2044:
2040:
2035:
2032:
2028:
2024:
2019:
2015:
2008:
2004:
2000:
1996:
1987:
1983:
1982:inner product
1979:
1975:
1971:
1970:vector spaces
1967:
1962:
1960:
1957:
1954:
1950:
1947:
1943:
1939:
1936:
1933:
1929:
1925:
1923:
1920:
1917:
1913:
1911:
1908:
1905:
1904:needs to show
1901:
1898:
1895:
1891:
1887:
1883:
1880:
1876:
1873:if (only if)
1872:
1868:
1864:
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1856:
1852:
1850:
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1813:
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1800:
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1783:
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1773:
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1664:). See also
1663:
1659:
1655:
1651:
1647:
1643:
1639:
1636:
1633:
1631:"antichain").
1629:
1628:regular graph
1625:
1621:
1617:
1613:
1609:
1605:
1601:
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1593:
1589:
1585:
1581:
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1567:
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1487:
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1460:
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1442:
1438:
1434:
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1416:
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1404:
1400:
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1393:
1389:
1385:
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1373:
1369:
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1362:
1359:
1355:
1351:
1347:
1343:
1339:
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1331:
1328:) is called "
1327:
1323:
1319:
1316:
1313:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1283:
1280:
1264:
1261:
1258:
1238:
1218:
1215:
1212:
1209:
1206:
1198:
1194:
1191:
1188:
1187:
1182:
1175:
1173:
1164:
1160:
1155:
1152:
1149:
1145:
1141:
1137:
1133:
1126:
1123:
1119:
1115:
1111:
1107:
1105:
1102:
1099:
1095:
1091:
1090:formal system
1087:
1084:
1081:
1077:
1059:
1051:
1048:
1038:
1034:
1030:
1028:
1023:
1020:
1017:
1002:
988:
972:
960:
946:
930:
918:
913:
910:
896:
892:
889:
874:
854:
834:
826:
822:
806:
800:
797:
794:
774:
768:
765:
762:
742:
739:
736:
733:
730:
710:
704:
701:
698:
690:
686:
682:
679:
675:
671:
667:
665:
662:
659:
656:
652:
648:
644:
640:
638:
635:
632:
628:
626:
623:
619:
615:
611:
607:
603:
597:
593:
589:
585:
579:
575:
571:
567:
563:
557:
553:
549:
547:
544:
541:
537:
533:
529:
526:
522:
518:
514:
510:
506:
502:
498:
496:
493:
492:
491:
479:
476:
473:), the term "
472:
468:
467:
462:
460:
457:
453:
451:
448:
446:pathological.
444:
440:
436:
435:
430:
423:
421:
417:
413:
402:
395:
392:
383:
379:
375:
371:
367:
365:
362:
359:
355:
353:
350:
349:
346:
320:
316:
313:
312:
307:
300:
291:
289:
286:
283:
279:
278:number theory
275:
271:
269:
266:
265:
260:
253:
241:
237:
233:
229:
224:
222:
219:
216:
212:
211:number theory
208:
207:real analysis
204:
200:
196:
192:
188:
185:
184:
179:
168:
164:
163:irrationality
160:
157:
156:prime numbers
153:
152:
150:
143:
139:
135:
131:
129:
126:
125:
120:
113:
110:
107:] raised
106:
96:
89:
87:
78:
74:
70:
67:reference to
66:
62:
60:
57:
56:
50:
47:
45:
41:
37:
33:
19:
3572:
3567:Bibliography
3546:
3534:
3498:
3494:
3480:
3456:
3445:
3441:
3424:
3376:
3372:
3344:
3340:
3295:
3247:
3243:
3214:
3196:
3190:
3171:
3165:
3156:
3143:
2365:
2350:
2338:
2326:
2316:index battle
2295:
2291:
2257:
2249:
2225:
2217:
2206:
2172:
2159:Sometimes a
2134:
2130:
2126:
2122:
2115:
2110:
2106:
2101:
2097:
2093:
2088:
2083:
2079:
2074:
2069:
2065:
2061:
2056:
2046:
2042:
2038:
2033:
2026:
2017:
2013:
2006:
2002:
1998:
1994:
1992:
1985:
1972:of the same
1965:
1927:
1903:
1893:
1881:
1874:
1870:
1866:
1862:
1858:
1854:
1792:
1781:well-defined
1771:weak, weaker
1761:
1757:
1738:, where the
1727:
1710:
1703:
1691:
1687:
1684:fiber bundle
1679:
1675:
1665:
1661:
1657:
1653:
1649:
1645:
1641:
1611:
1607:
1603:
1587:
1579:
1565:
1554:
1550:
1542:
1537:
1530:
1522:
1518:
1488:
1485:
1481:
1478:
1474:
1470:
1466:
1462:
1448:
1444:
1436:
1432:
1424:
1406:
1402:
1398:
1397:of a number
1391:
1387:
1383:
1379:
1375:
1371:
1357:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1311:
1297:pathological
1292:
1288:
1169:
1163:pathological
1148:Zariski open
1132:intersection
1124:
1109:
1097:
1094:formal proof
1079:
1032:
1025:
824:
820:
688:
684:
680:
657:
650:
646:
642:
617:
613:
609:
605:
601:
595:
591:
587:
583:
582:. See also
577:
573:
569:
565:
561:
555:
539:
513:real numbers
505:measure zero
488:
477:
466:Pathological
464:
459:well-behaved
408:
388:
369:
364:pathological
322:
318:
297:
246:
235:
231:
180:, p.2)
148:
105:Grothendieck
102:
83:
76:
72:
48:
29:
3341:AMS Notices
2957:multivalued
2951:mathematics
2945:mathematics
2881:fundamental
2430:from a set
2368:mathematics
2349:Similar to
2284:injectivity
2191:brute force
2161:proposition
1942:working out
1708:denominator
1570:analyticity
1521:(sometimes
1429:derivatives
987:cardinality
823:is said to
394:fathers....
370:degenerated
252:theorem....
34:has a vast
18:Deep result
3584:Categories
3427:, Springer
3229:References
3014:projection
3008:projection
2897:Arithmetic
2309:in general
2302:handwaving
2202:by example
1978:isomorphic
1863:sufficient
1838:in general
1711:downstairs
1692:downstairs
1680:downstairs
1666:eventually
1566:Smoothness
1411:overloaded
1354:surjective
1192:(LHS, RHS)
1153:in general
1118:complement
1110:almost all
1033:eventually
1021:frequently
668:A term in
651:eventually
637:eventually
584:frequently
495:almost all
455:fallacies.
416:continuous
268:elementary
36:vocabulary
3515:0039-7857
3448:(2 and 3)
3300:CiteSeerX
3096:structure
3062:↦
3036:→
2969:to a set
2916:invariant
2910:invariant
2747:∈
2686:×
2576:→
2481:×
2450:to a set
2403:canonical
2355:corollary
2322:obviously
2050:onto the
2001:....Let (
1974:dimension
1892:" means "
1869:" means "
1859:necessary
1843:induction
1805:interest.
1700:numerator
1648: : â
1620:antichain
1527:arbitrary
1368:reflexive
1159:arbitrary
1136:open sets
1049:−
1029:arguments
999:ℵ
967:∞
961:∪
925:∞
919:∪
911:≥
804:→
798::
772:→
766::
740:∘
708:→
702::
674:morphisms
625:arbitrary
604: : â
564: : â
525:countable
391:functions
319:chicanery
314:chicanery
128:canonical
3523:44064821
3495:Synthese
3423:(1998),
3364:(1997),
3284:16588584
3109:See also
2939:morphism
2935:function
2780:′
2762:implies
2740:′
2558:function
2552:function
2288:elements
2278:Given a
2141:â
2092:, where
1945:through.
1820:negation
1706:and the
1704:upstairs
1696:fraction
1694:. In a
1688:upstairs
1676:upstairs
1656: :
1612:strongly
1588:stronger
1540:, where
1501:Lindelöf
1420:regular
1409:. This
1390:, and a
1197:equation
1177:â
1165:" cases.
1076:interval
1037:sequence
945:variance
608: :
572: :
542:, below.
532:integers
471:analysis
425:â
418:but not
397:â
378:triangle
302:â
288:folklore
255:â
240:theorems
215:geometry
174:â
115:â
91:â
86:category
44:rigorous
3413:9177152
3381:Bibcode
3349:(Parts
3347:(9, 10)
3332:2154064
3275:1078535
3252:Bibcode
3081:â and â
2533:diagram
2351:clearly
2345:trivial
2339:obvious
2328:clearly
2258:Ă©vident
2250:obvious
2246:Laplace
2227:gestalt
2052:algebra
1523:optimal
1499:(resp.
1497:compact
1493:squares
1473:(resp.
1465:(resp.
1449:regular
1425:regular
1395:divisor
1104:generic
819:, then
678:objects
540:generic
523:form a
509:measure
380:having
352:natural
221:elegant
165:of the
3555:
3521:
3513:
3464:
3411:
3401:
3330:
3320:
3302:
3282:
3272:
3203:
3178:
2384:binary
2292:chases
2254:French
1922:Q.E.D.
1884:to be
1865:) for
1801:aliter
1776:strong
1753:vanish
1698:, the
1618:is an
1608:strong
1580:strong
1561:smooth
1533:, the
1445:smooth
1392:proper
1376:proper
1372:proper
1363:proper
1344:" or "
1309:spaces
1279:lvalue
890:finite
867:, and
691:a map
687:, and
552:limits
528:subset
475:smooth
443:Banach
439:meagre
382:angles
299:print.
40:jargon
3519:S2CID
3438:(PDF)
3404:33670
3369:(PDF)
3328:S2CID
3153:(PDF)
3136:Notes
2248:used
2023:basis
2021:be a
1879:field
1828:(iff)
1795:proof
1728:up to
1717:up to
1626:is a
1604:sharp
1519:sharp
1510:sharp
1505:cover
1458:resp.
1348:into
1332:onto
1122:dense
755:with
103:[
3553:ISBN
3511:ISSN
3462:ISBN
3409:PMID
3353:and
3318:ISBN
3280:PMID
3220:1998
3201:ISBN
3176:ISBN
2949:See
2895:for
2537:See
2325:See
2145:1991
2025:for
1993:Let
1976:are
1437:nice
1433:nice
1317:onto
1289:nice
1284:nice
1231:has
1181:2004
1096:, a
787:and
616:) â„
515:are
429:2004
401:1913
306:1995
274:deep
259:1977
234:and
213:and
186:deep
178:2006
119:2001
95:1997
30:The
3503:doi
3499:111
3399:PMC
3389:doi
3310:doi
3270:PMC
3260:doi
2985:of
2961:A "
2929:map
2914:An
2857:of
2817:of
2054:Sym
2037:1â€
2012:1â€
1857:is
1469:)
1340:to
1291:or
649:))
536:odd
503:of
501:set
75:or
3586::
3517:,
3509:,
3497:,
3446:33
3444:,
3440:,
3407:,
3397:,
3387:,
3377:94
3375:,
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3357:).
3355:II
3345:51
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3326:,
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3308:,
3278:,
3268:,
3258:,
3248:28
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3238:;
3155:.
3100:A
3012:A
2899:).
2556:A
2426:A
2388:A
2374:.
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2256::
2205:A
2199:).
2131:GL
2123:GL
2045:â€
2041:,
2016:â€
1750:."
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63:A
3562:.
3539:.
3526:.
3505::
3486:.
3471:.
3450:.
3429:.
3416:.
3391::
3383::
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3335:.
3312::
3287:.
3262::
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3184:.
3159:.
3090:S
3065:x
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3056:y
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3031:2
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2997:.
2995:B
2991:x
2987:B
2983:x
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2967:A
2953:.
2923:M
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2865:B
2845:b
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2805:a
2777:b
2773:=
2770:b
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2730:a
2727:(
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2721:)
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2683:A
2663:f
2643:B
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2637:A
2617:)
2614:f
2611:,
2608:B
2605:,
2602:A
2599:(
2579:B
2573:A
2570::
2567:f
2546:F
2541:.
2527:D
2510:B
2507:,
2504:A
2484:B
2478:A
2458:B
2438:A
2397:C
2378:B
2341:.
2331:.
2298:.
2252:(
2138:.
2135:n
2127:V
2125:(
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2086:)
2084:V
2080:V
2078:(
2075:k
2070:k
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2062:V
2060:(
2057:k
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2034:k
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2018:n
2014:i
2010:)
2007:i
2003:e
1999:k
1995:V
1926:(
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1875:B
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1867:B
1861:(
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1778:.
1768:.
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1471:X
1467:B
1463:A
1417:.
1407:n
1403:n
1399:n
1388:S
1384:S
1380:S
1358:f
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1342:B
1338:A
1334:B
1330:A
1326:B
1322:A
1312:X
1265:1
1262:+
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1219:1
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1213:y
1210:=
1207:x
1157:"
1130:(
1128:ÎŽ
1125:G
1100:.
1080:n
1060:n
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1052:1
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1018:.
1003:0
973:,
970:}
964:{
957:N
931:,
928:}
922:{
914:0
906:R
887:.
875:h
855:g
835:B
821:f
807:C
801:B
795:h
775:B
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763:g
743:g
737:h
734:=
731:f
711:C
705:A
699:f
689:C
685:B
681:A
658:x
647:x
621:.
618:y
614:x
612:(
610:f
606:x
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600:â
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332:R
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284:.
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203:Ï
169:.
158:.
144:.
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97:)
79:.
20:)
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