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Ut +UUx=ju0Ux2 +f(t),\/(t,x)^D0,(D0=(0,T0)xR), (54)
при этом (54), (2), (3) и (7) называется обратной задачей Бюргерса с фиксированным параметром, когда выполняется условие (4).
Все результаты теоремы 2 имеют место для задачи Бюргерса (54), (2), (3), когда задается условие (7).
Литература
1. Андерсон Э. Ударные волны в магнитной гидродинамике. Москва: Атомиздат, 1968. 271 с.
2. Ворожцов Е. В., Яненко Н. Н. Методы локализации особенностей при численном решении задач газовой динамики. Новосибирск: Наука, 1985. 224 с.
3. Лаврентьев М. М., Резницкая К. Г. Теорема единственности некоторых нелинейных обратных задач уравнения параболического типа // Докл. АН СССР 208. № 3, 1973. С. 531-532.
4. Омуров Т. Д. Существование, единственность и гладкость решения задачи Навье-Стокса для несжимаемой жидкости с вязкостью // КНУ им. Ж. Баласагына. Бишкек, 2014. 96 с.
5. Омуров Т. Д., Рыспаев А. О., Омуров М. Т. Обратные задачи в приложениях математической физики // КНУ им. Ж. Баласагына. Б., 2014. 192 с.
6. Полубаринова-Кочина П. Я. Теория движения грунтовых вод. Москва: Наука, 1977. 664 с.
7. Рождественский Б. Л., Яненко Н. Н. Системы квазилинейных уравнений и их приложения к газовой динамике. Москва: Наука, 1978. 688 с.
8. Соболев С. Л. Уравнения математической физики. Москва: Наука, 1966. 443 c.
9. Уизем Дж. Линейные и нелинейные волны. Москва: Мир, 1977. 622 с.
10. Friedman A. Boundary estimates for second order parabolic equations and their application. J. of Math and Mech. Vol. 7. № 5, 1958. P. 771-791.
MATHEMATICAL THEORY OF MUSIC Hang T. T.1, Thao L. D.2, Hieu L. V.3, Khoe N. H.4, Thuong T. T. M.5, Uyen V. T. P.6 Email: Hang1132@scientifictext.ru
'Hang Tran Thuy — Student,
DEPARTMENT OF IT IN THE FUEL AND ENERGY INDUSTRY, FACULTY OF LASER AND LIGHT ENGINEERING;
2Thao Le Duc — Student,
DEPARTMENT OF GEOINFORMATIONSYSTEMS, FACULTY OFINFOCOMMUNICATION TECHNOLOGIES;
3Hieu Le Van — Student; 4Khoe Nguyen Huu — Student, DEPARTMENT OF SECURE INFORMATION TECHNOLOGIES, FACULTY OF INFORMATION SECURITY AND COMPUTER TECHNOLOGIES; 5Thuong Tran Thi Mai — Student, DEPARTMENT OF COMPUTER SYSTEM DESIGN AND SECURITY, FACULTY OF INFORMATION SECURITY AND COMPUTER TECHNOLOGIES; 6Uyen Vu Thi Phuong — Student, DEPARTMENT OF IT IN THE FUEL AND ENERGY INDUSTRY, FACULTY OF LASER AND LIGHT ENGINEERING ST. PETERSBURG NATIONAL RESEARCH UNIVERSITY OF INFORMATION TECHNOLOGIES, MECHANICS AND OPTICS, SAINT PETERSBURG
Abstract: in music, it is usually forgotten a lot of mathematics. We use Western European musical system, the basis of which - two quite strict frequency and time scale. The frequencies of the scale is a geometric progression with the coefficient ',059 ... ('2 degree root of 2), and temporal organization of this sounds and pauses are in multiple relationships (often acts denominator a power of 2). The structure of the piece of music is often very simple, presenting an alternation of several "modules block" a specific length. Keywords: music, challenge, frequency, theory, harmoniousness, musical note.
МАТЕМАТИЧЕСКАЯ ТЕОРИЯ МУЗЫКИ Ханг Ч. Т.1, Тхао Л. Д.2, Хиеу Л. В.3, Кхое Н. Х.4, Тхыонг Ч. Т. М.5,
Уиен В. Т. Ф.6
'Ханг Чан Тхуи — студент, кафедра информационных технологий топливно-энергетического комплекса, факультет лазерной и световой инженерии;
2Тхао Ле Дык — студент, кафедра геоинформационных систем, факультет инфокоммуникационных технологий;
3Хиеу Ле Ван — студент;
4Кхое Нгуен Хыу — студент, кафедра безопасных информационных технологий, факультет информационной безопасности и компьютерных технологий;
5Тхыонг Чан Тхи Май — студент, кафедра проектирования и безопасности компьютерных систем, факультет информационной безопасности и компьютерных технологий;
6Уиен Ву Тхи Фыонг — студент, кафедра информационных технологий топливно-энергетического комплекса, факультет лазерной и световой инженерии, Санкт-Петербургский национальный исследовательский университет информационных технологий,
механики и оптики, г. Санкт-Петербург
Аннотация: в музыке, что обычно забывается, немало математики. Мы используем западноевропейскую нотную систему, основа которой - две вполне строгие шкалы частоты и времени. Частоты звукоряда представляют собой геометрическую прогрессию с коэффициентом 1,059... (корень 12 степени из 2), а временная организация это звуки и паузы, находящиеся в кратных отношениях (чаще всего деноминатором выступает степень 2). Структура музыкального произведения нередко оказывается очень простой, представляя собой чередование некоторых «блоков-модулей» определенной протяженности.
Ключевые слова: музыка, вызов, частота, теория, гармоничность, музыкальная нота.
УДК 51.78
I have two seemingly unrelated challenges for you. The first relates to music, and the second gives a foundational result in measure theory, which is the formal underpinning for how mathematicians define integration and probability. The second challenge, which I'll get to about halfway through this article, has to do with covering numbers with open sets, and is very counter-intuitive. Or at least, when I first saw it I was confused for a while. Foremost, I'd like to explain what's going on, but I also plan to share a surprising connection it has with music. Here's the first challenge. If I'm going to play a musical note with a given frequency, let's say 220 hertz, then I'm going to choose some number between 1 and 2, which we'll call r, and play a second musical note whose frequency is r times the frequency of the first note, 220. For some values of this ratio r, like 1.5, the two notes will sound harmonious together, but for others, like the square root of 2, they sound cacophonous. Your task it to determine whether a given ratio r will give a pleasant sound or an unpleasant one just by analyzing the number and without listening to the notes. One way to answer, especially if your name is Pythagoras, might be that two notes sound good when the ratio is a rational number, and bad when it is irrational. For instance, a ratio of 3/2 gives a musical fifth, 4/3 gives a musical fourth, of 8/5 gives a minor sixth, etc. Evidently when our brains pick up on this pattern, two notes sound nice together. However, most rational numbers actually sound pretty bad, like 211/198, or 1093/826. The issue, of course, is that these rational number are somehow more "complicated" than the other ones, our ears don't pick up on the pattern of the beats. One simple way to measure the complexity of a rational number is to consider the size of its denominator when it is written in reduced form. So we might edit our original answer to only admit fractions with low denomin ators, say less than 10 [1]. Even still, this doesn't quite capture harmoniousness, since plenty of notes sound good together even when the ratio of their frequencies is irrational, so long as it is close to a harmonious rational number. If you're curious about why this is done, the answer is, if you take a harmonious interval, like a fifth, the ratio of frequencies when played on a piano will not be a nice rational number like you expect, in this case 3/2, but will instead be some power of the 12th root of 2, in this case 2л{7/12}, which is irrational, but very close to 3/2. Similarly, a musical fourth corresponds to 2Л{5/12}, which is very close to 4/3. In fact, the reason it works so well to have 12 notes in the chromatic scale is that powers of the 12th root of 2 have a strange tendency to be within a 1% margin of error of simple rational numbers. So now you might say a ratio r will produce a harmonious pair of notes if it is sufficiently close to a rational number with a sufficiently mall denominator.
After all, maybe someone with a particularly acute musical sense would be able to hear and find pleasure in the pattern resulting from more complicated fractions like 23/21 or 35/43, as well as numbers closely approximating these fractions. This leads to an interesting question: Suppose there is a musical savant, who find pleasure in all pairs of notes whose frequencies have a rational ratio, even super complicated ratios that you and I would find cacophonous. Is it the case that she would find all ratios r between 1 and 2 harmonious, even the irrational ones? After all, for any given real number you can always find rational numbers arbitrarily close it, just as 3/2 is close to 2Л{7/12}. Well, this brings us to challenge number 2. Mathematicians like to ask riddles about covering various sets with open intervals, and the answers to these riddles have a strange tendency to become famous lemmas and theorems. My challenge to you involves covering all the rational numbers between 0 and 1 with open intervals. Once we remove the intervals from the geometry of our setup and just think of them in a list, each one responsible for only one rational number, it seems much clearer that the sum of their lengths can be less than 1, since each particular interval can be as small as you want and still cover its designated rational. In fact, the sum can be any positive number. Just choose an infinite sum with positive terms that converges to 1with powers of 2, then choose any desired value epsilon>0, like 0.5, and multiply all terms by epsilon so that we have an infinite sum converging to epsilon. Now scale the nth interval to have a length equal to the nth term in the sum. Notice, this means your intervals start getting really small, really fast, so small that you can't really see most of them, but it doesn't matter, since each one is only responsible for covering one rational. I've said it already, by I'll say it again because it's so amazing: epsilon can be whatever positive number we want, so not only can our sum be less than 1, it can be arbitrarily small! This is one of those results where even after seeing the proof, it still defies intuition [2].
Список литературы / References
1. Music and mathematics. [Electronic resource]. URL: https://en.wikipedia.org/wiki/Music_and_mathemati cs/ (date of access: 04.02.2017).
2. The magical mathematics of music. [Electronic resource]. URL: https://plus.maths.org/content/magical-mathematics-music/ (date of access: 01.05.2005).
