Math (Несколько текстов для зачёта), страница 15

which implies that n is even, since otherwise the left side of the above equation is odd, a contradiction. Thus, T is bipartite.

Hence, there are sets A and B with A intersection cap B = *[This character cannot be converted to ASCII text], A union cup B = R, such that each edge of T is incident with one vertex in A and the other vertex in B. If both A and B were measurable, then at least one of them, say A, would have positive measure. Furthermore, for each integer k, A + 3[sup k] subset or is equal to B, which yields A intersection cup (A + 3[sup k]) = *[This character cannot be converted to ASCII text]. Since 3[sup k] right arrow 0 as k right arrow -infinity, this contradicts the following theorem, which is a standard result in measure theory. For the convenience of the reader, we include the proof from [15].

Theorem. Let M be a set of real numbers with positive Lebesgue measure. Then, there exists a delta > 0 such that for every x is an element of R, |x| < delta, M union cup (M + x) Is not equal to *[This character cannot be converted to ASCII text].

Proof. Find a closed set F and an open set G with F subset or is equal to M and F subset G such that 3lambda(G) < 4lambda(F) (where lambda is Lebesgue measure). Since G is a countable union of disjoint open intervals, there is one among them, say I, such that 3lambda(I) < 4lambda(F intersection cup I). Let delta = 1/2lambda(I) and suppose that |x| < lambda. Then, I union cup (x + I) is an interval of length less than 3/2lambda(I) which contains both F intersection cup I and x + (F intersection cup I). The last two sets cannot be disjoint, since otherwise

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which is a contradiction. Hence, *[This character cannot be converted to ASCII text] Is not equal to (F intersection cup I) union cup (x + (F intersection cup I)) subset or is equal to M intersection cup (x + M), completing the proof.

Remark. It is well known that a nonmeasurable set cannot be constructed without using the axiom of choice. Our graph T is not connected, and, in fact, each component of T has only a countable number of vertices. Thus, to define A and B, we need to make use of this axiom.

Sharkovsky's Theorem

Let f: R right arrow R be a continuous function. A point x is an element of R is called a k-periodic point off if f[sup k](x) = x and f[sup i](x) Is not equal to x for i = 1, 2, ..., k - 1. Here, f[sup n] is the nth iterate off, i.e., f[sup n] = f*[This character cannot be converted to ASCII text]f[sup n-1].

If f has a k-periodic point, is it necessary that f have an m-periodic point for some m Is not equal to k?

In 1964, Sharkovsky [13] gave a complete and amazing answer to this question with the following

Theorem. Let f: R right arrow R be a continuous function with a k-periodic point. Then, f has an m-periodic point if k precedes m in the following ordering (S) of all the natural numbers:

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This is best possible, since whenever k and m are natural numbers and m precedes k, there exists a continuous function f: R right arrow R with a k-periodic point, but no m-periodic point.

The original proof by Sharkovsky is very complicated, and, later, several mathematicians presented much simpler proofs. In some of them, graph theory was used, with the most important step being made by Straffin [14]. He defined a digraph associated with a periodic point of a function and proved the crucial result.

For this purpose, let x be a k-periodic point of a function f. Then, the distinct values [x, f(x), f[sup 2](x), ..., f[sup k-1] (x)} determine k - 1 finite intervals I[sub 1], I[sub 2], ..., I[sub k-1], labeled from left to right, after locating these numbers in their natural order on the x (and y) axis (see, for example, Fig. 2). Define a digraph G = (V, E) by V = {I[sub 1], ..., I[sub k-1]} with (I[sub i], I[sub j]) is an element of E whenever f(I[sub i]) contains or is equal to I[sub j]. For example the digraph corresponding to the 4-periodic point x = 0 of f, seen in Fig. 2, is the graph given in Fig. 3.

A closed trail in a digraph is said to be nonrepetitive if it does not consist entirely of a cycle of smaller length traced several times. For example, the digraph in Fig. 3 has nonrepetitive trails of lengths 1 and 2 only. Now, we are able to state Straffin's theorem, which turns the problem of the existence of a periodic point into a problem about the corresponding digraph.

Theorem [14]. If the digraph associated with a k-periodic point of a function f has a nonrepetitive closed trail of length m, then f has an m-periodic point.

Figure 4 shows the digraph associated with any 3-periodic point of a function. Clearly, this digraph contains a nonrepetitive closed trail of arbitrary length, showing that the existence of a 3-periodic point off implies that f has periodic points of all orders. This special case, and other results on systems with 3-periodic points, were proved in 1975 by Li and Yorke [9], when Sharkovsky's theorem was still little noticed.

The reader is referred to Straffin's one-page proof of his theorem above, which is modeled after Li and Yorke's. Straffin's proof makes essential use of two lemmas which are standard in analysis courses:

Lemma 1. Suppose I and J are closed intervals, f continuous, and J Subset f(I). Then there is a closed interval Q Subset I such that f(Q) = J.

Lemma 2. Suppose I is a closed interval, f continuous, and I Subset f(I). Then, f has a fixed point in I.

Using his theorem above, Straffin proved some parts of Sharkovsky's Theorem, and his approach subsequently allowed several authors to complete the proof (see [3, 6]). In the proof of Sharkovsky's Theorem presented in [2] graphs were used without applying Straffin's result. To give some of the flavor of the proofs in [3, 6] we sketch the proof of a partial result, showing that in the ordering S, all even integers lie after all the odd integers (see [6]).

Theorem. If a continuous function f: R right arrow R has a point of odd period 2n + 1 (n >/ = 1), then it has periodic points of all even periods.

Proof (sketch). For n = 1, the proof was given above. Now, suppose n > 1 and assume by way of induction that the theorem is true whenever f has a point of odd period 2m + 1, where 3 </ = 2m + 1 < 2n + 1. Straffin proved generally that the digraph corresponding to a periodic point of period k contains a closed trail of length k in which some vertex is repeated exactly twice. In our case, k = 2n + 1, and this closed trail can, therefore, be decomposed into two closed nonrepetitive trails, one of which has odd length, say 2m + 1 < 2n + 1. If this closed trail is of length greater than one, the assertion follows by our induction assumption and the previous theorem. If not, then Straffin proved that our digraph must contain the directed subgraph given in Fig. 5. This subgraph has a cycle of length 2, and one of length 4. For any even number t > 4, we may begin a nonrepetitive closed trail of length t at the bottom right-hand vertex, traverse the 4-cycle once, and follow this by traversing the 2-cycle exactly (t - 4)/2 times. By the previous theorem, the existence of all even periods follows.

WHAT IS ANCIENT MATHEMATICS?

Source: Mathematical Intelligencer, Summer99, Vol. 21 Issue 3, p38, 10p, 10 diagrams, 1bw Author(s): Vardi, Ilan

In my opinion, it is not only the serious accomplishments of great and good men which are worthy of being recorded, but also their amusements.
XENOPHON, SYMPOSlUM

The title of this paper is a result of comments on earlier drafts by mathematicians: "This is not mathematics, this is history!" and by historians of mathematics: "This is not history, this is mathematics!" After some reflection, I came to the conclusion that the historians were right and the mathematicians were wrong--for example, I have found little difference between reading papers of Atle Selberg (1917-, Fields Medal 1950) and Archimedes (287-212 BC) (who both lived in Syracuse!). I believe that the mathematicians I spoke to were expressing a generally held belief that reading mathematical papers that are over a hundred years old is history of mathematics, not mathematics. Thus, the reconstruction of Heegner's solution to the class-number-one problem (1952) appeared in a mathematics journal [52], while a reconstruction of the missing portions of Archimedes's The Method (250 BC) appeared in a history journal [29].

To me, reading and proving results about a mathematical paper, whether it was written in 1950 or 250 BC, is always mathematics, though the latter case might be called "ancient mathematics." At least as to Greece, this is accepted by some eminent mathematicians [30, p. 21]:

Oriental mathematics may be an interesting curiosity, but Greek mathematics is the real thing ... The Greeks, as Littlewood said to me once, are not clever schoolboys o r "scholarship candidates, " but "Fellows of another college." So Greek mathematics is "permanent, " more perhaps even than Greek literature. Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not.

I am saying that ancient Greek mathematicians were in every essential way similar to modern mathematicians. In fact, some mathematicians might find more in common with Archimedes and Euclid than with many colleagues of their departments, and even reading the original Greek--a subject traditionally taught in High School [9]--seems easier than understanding, say, the proof that every semistable elliptic curve is modular [59].

Nineteenth-century mathematicians dedicated much of their research to elementary Euclidean geometry. It is possible that some mathematicians of that era felt that the influence of the past was too great, as Felix Klein wrote [38, Vol. 2, p. 189]:

Although the Greeks worked fruitfully, not only in geometry but also in the most varied fields of mathematics, nevertheless we today have gone beyond them everywhere and certainly also in geometry.

For whatever reason, geometers recently tend to distance themselves from Euclidean geometry. For example, the book Unsolved Problems in Geometry [16], part, of a series on "unsolved problems in intuitive mathematics, " does not have a section devoted to classical Euclidean geometry, and with few exceptions, such as [10], articles on this subject are relegated to "lowbrow" publications. Yet earlier in this century, Bieberbach, Hadamard, and Lebesgue all wrote books on elementary Euclidean geometry [13] [27] [44], and excellent books and articles on ancient mathematics are still being written [31] [55]. See [17] for further analysis of these issues.

In this paper, I will give an example of ancient mathematics by using techniques that Archimedes developed in his paper The Method to derive results that he proved in his paper On Spirals. I will try to present these in a way that Archimedes might understand [57], in particular, the diagrams are intended to conform to ancient Greek standards [46]. I will also indicate how ideas in these papers can lead to some surprising results (e.g., Exercise 4 below). The paper will include such exercises as may challenge the reader to understand concepts of Archimedes as he expressed them.

(I have concentrated on the works of Archimedes because these are most similar to modern mathematical research papers, sharply focused on problems and their solution. By comparison, the works of Euclid read like a generic textbook; and so little is known about Euclid that it cannot be ruled out that he was actually a "consortium." Moreover, it seems likely that the works of Euclid are based on the efforts of earlier mathematicians [24] [39].[1])

The balance of the paper shows how a precise knowledge of ancient mathematics allows one to navigate in the sea of inaccuracies and misconceptions written about the history of mathematics. This also gives one perspective on cultural aspects of mathematics, as it forces one to understand ideas of first-rate mathematicians whose cultural background is very different from the present one. For example, it can help you read The New York Times [37]:

"Alien intelligences may be so far advanced that their math would simply be too hard for us to grasp, " [Paul] Davies said. "The calculus would have baffled Pythagoras, but with suitable tuition he would have accepted it."

Reading this paper should make it clear that Archimedes could have been Pythagoras's calculus tutor, thus refuting any implication that calculus was an unknown concept to ancient Greeks.

It is my hope that I can convince mathematicians that there are many interesting and relevant ideas to be uncovered in ancient Greek mathematics, and that it might be worthwhile to take a first-hand look, being wary of popular accounts and secondary sources, this one included!

Extending Archimedes's Method

In 1906 the Danish philologist J.L. Heiberg went to Constantinople to examine a manuscript containing mathematical writing which had been discovered seven years earlier in the monastery of the Holy Sepulchre at Jerusalem. What he found was a 10th-century palimpsest--a parchment containing works of Archimedes that, sometime between the 12th and 14th centuries, had been partially erased and overwritten by religious text. Heiberg managed to decipher the manuscript [33] and found that it included a text of The Method, a work of Archimedes previously thought lost. (The story of the transmission of Archimedean manuscripts given in [18] reads like a chapter from The Maltese Falcon. Late bulletin: Heiberg's palimpsest was sold by Christie's for $2, 000, 000--see Jeremy Grey's article in this issue.)