Who founded fractal geometry?
1973, when B.B.Mandelbrot gave a lecture at the French Academy, he first put forward the idea of fractal dimension and fractal geometry. The word fractal was coined by Mandelbrot, which means irregularity and fragmentation. Fractal geometry takes irregular geometry as the research object. Because irregularities are common in nature, fractal geometry is also called geometry that describes nature. After the establishment of fractal geometry, it quickly attracted the attention of many disciplines, because it has important value not only in theory but also in practice.
What are the characteristics of fractal geometry compared with traditional geometry?
(1) On the whole, fractal geometry is irregular everywhere. For example, the shapes of coastlines and mountains and rivers are extremely irregular from a distance.
⑵ On different scales, the regularity of graphics is the same. From a close distance, the local shapes of the above coastlines and mountains and rivers are similar to the whole, and they are self-similar from the whole to the part. Of course, there are some fractal geometric figures, which are not completely self-similar. Some are used to describe general random phenomena, while others are used to describe chaotic and nonlinear systems.
What is fractal dimension?
In Euclidean space, people are used to treating space as three dimensions, plane or sphere as two dimensions and straight line or curve as one dimension. It can also be generalized that a point is zero-dimensional, or a high-dimensional space can be introduced, but usually people are used to the dimension of integers. Fractal theory regards dimension as a fraction, which is an important concept that physicists need to introduce when studying chaotic attractors and other theories. In order to quantitatively describe the "irregular" degree of objective things, in 19 19, mathematicians introduced the concept of dimension from the perspective of measure, and extended the dimension from integer to fraction, thus breaking through the boundary that the dimension of general topological sets is integer.
The concept of fractal dimension can be established from two aspects: on the one hand, we draw a line segment, a square and a cube, and their side lengths are all 1. Divide their side lengths into two halves. At this time, the linearity of the original image is reduced to 1/2, and the original image is divided into several similar graphics. Its line segments, squares and cubes are divided into two similar subgraphs 1, two 2's and two 3's, respectively, where the exponents 1, 2 and 3 are exactly equal to the corresponding empirical dimensions of the graphs. Generally speaking, if a graph is composed of similar B graphs and the original graph is simplified to 1/a, there are:
a^D=b，D=logb/loga
If the relationship is established, the index d is called similarity dimension, and d can be an integer or a fraction. On the other hand, when we draw a straight line, if we measure it with 0-dimensional points, the result is infinite, because the straight line contains infinite points; If we measure with a plane, the result is 0, because there is no plane on the straight line. So, what scale will get a finite value? It seems that only by measuring a small line segment with the same dimension can a finite value be obtained, where the dimension of the straight line is 1 (greater than 0 and less than 2). Similarly, if we draw a Koch curve, its whole is folded by an infinitely long line. Obviously, if we use a small straight line segment, the result will be infinite, while if we use a plane segment, the result will be 0 (the plane is not included in this curve). Only when we find a ruler with the same dimension as Koch curve, will it get a finite value, and this dimension is obviously greater than/kloc-0. In fact, the dimension of Koch curve is 1.5438+08.
According to Professor Mandelbrot himself, the word fractal came to him on a quiet night in the summer of 1975, when he was thinking hard and accidentally looked through his son's Latin dictionary. The word comes from the Latin adjective fractus, and the corresponding Latin verb is frangere ("breaking" and "generating random fragments"). In addition, it has the same root as the English words "fragment", "fraction" and "fragment". Before the mid-1970s, Mandelbrot always used the English word fractal to express his fractal thoughts. Therefore, it is irregular, fractured and fractal to start with Latin words and end with English words. Mandelbrot wants to use this word to describe a large class of complex and random geometric objects that traditional Euclidean geometry cannot describe in nature. For example, winding coastline, rugged mountains, rugged road sections, fickle clouds, winding rivers, criss-crossing blood vessels, dazzling stars and so on. They are characterized by extremely irregular or extremely unsmooth. Intuitively and roughly, these objects are fractal.
Definition of fractal
Mandelbrot once gave two definitions of fractal:
(1) meets the following conditions
dim(A)gt； Dim (a)
The set a of is called a fractal set. Where dim(A) is Hausdoff dimension (or fractal dimension) of set A and Dim(A) is its topological dimension. Generally speaking, Dim(A) is not an integer, but a fraction.
(2) A shape whose part is similar to the whole in some form is called fractal.
However, after the test of theory and application, it is found that these two definitions are difficult to contain such rich contents as fractal. In fact, so far, we can't give an exact definition of what fractal is. Just as there is no strict and clear definition of "life" in biology, people usually list a series of characteristics of biology to explain it. The definition of fractal can also be treated in the same way.
(i) The fractal set has any small scale details, or it has a fine structure.
(2) Fractal set can't be described by traditional geometric language. It is neither the locus of points satisfying certain conditions nor the solution set of some simple equations.
(iii) Fractal sets have some form of self-similarity, which may be approximate self-similarity or statistical self-similarity.
(iv) Generally, the "fractal dimension" of a fractal set is strictly greater than its corresponding topological dimension.
(v) In most interesting cases, fractal sets are defined by a very simple method and can be generated by iteration of transformation.
Mythological vocabulary: chaos/confusion
New Oriental Education Online
Greek and Roman mythology had a great influence on English literature, especially during the Renaissance. Stories and words permeate all aspects of English. To master these words, you'd better know their origins. In this issue, Mr. Li Chuanwei, who is familiar with western culture, is invited to write relevant articles to help us understand myths, learn western culture and memorize vocabulary.
The real meaning of chaos (noun) is "chaos and disorder"; When C is capitalized, it means "chaos"-the legendary vague scene before the formation of the universe. The adjective chaos is chaotic, meaning "chaotic, messy; Chaos. " Commonly used in the following structures: in chaos; A mess; Create chaos; Throw sth into chaos
Chaos is the first kind of "power" at the beginning of the formation of the universe, and it is the Greek name for the vast and dark space of the universe. In Greek, chaos is the mother of heaven and earth, which means "crack". Hesiod, the author of Greek divination, regarded chaos as the mother of heaven and earth, probably because he thought that "something was created out of nothing"; However, since "nothing" can be "born out of nothing", it can be seen that chaos cannot be a space without existence, but may be a messy space composed of intangible messy substances.
This chaotic concept is reminiscent of the creation story in the Bible. Genesis in the King James Bible wrote: "The earth is empty and chaotic; The earth is unformed and empty, and darkness covers the surface of the abyss. But the earth here does not refer to Gaia, nor is it created by chaos-it was created by God. In Greek mythology, chaos is a great originality; In the Bible, chaos must be tamed and shaped by the creator God. The creation myths of other civilizations, such as those of Australian aborigines, have similar concepts of chaos.
The Greeks are not particularly attached to chaos, but it is the masterpiece of Zeus, the supreme god, aiming at establishing a strong order on the basis of disorderly primitive and things. However, in the Christian tradition, the concept of chaos is frightening. The opposite of chaos is the universe, because the latter refers to the universe regarded as a harmonious system. Cosmic adjectives are cosmic. Cosmo, as a root word, has strong word-building ability. Such as: cosmogony origin of the universe; Astrochemistry; Cosmology cosmology; Cosmic philosophy.
Cleverly remembering the word chaos can be remembered not only according to the source, but also by the split method: chaos = chaos (spelled "noisy" in Chinese pinyin) +s (spelled "dead" in Chinese pinyin) → "noisy "→ chaos.
Stagnant growth may damage London' s own long-term $ TERM and short-term $ TERM. Despite the poor transportation system, London succeeded; If it is to thrive in the future, it needs a better one. But the danger is that when the pressure disappears, everyone forgets the need for new investment; Then, next time, the chaos will be more serious. Slow economic growth will do harm to London in the short and long term. Despite the poor transportation system, London has achieved economic success. If London is to prosper in the future, its transportation system must be improved. But there is a danger in doing so: after the pressure is eliminated, people will forget that they need new investment to improve the transportation system. In this way, the next time a similar situation occurs, it will be even more chaotic. ) (2003 1 month 15, economist)
What is an optical soliton?
Soliton, also known as solitary wave, is a special type of ultrashort pulse, or a pulse-like traveling wave whose shape, amplitude and velocity remain unchanged during propagation. Some people define solitary waves as being able to keep their amplitude, shape and velocity unchanged after encountering other similar solitary waves.
The word soliton was first put forward in physical fluid mechanics. 1834, American scientist john scott russell observed a phenomenon: in a narrow river, a ship was pulled forward quickly. When the ship stopped suddenly, an isolated water wave formed at the bow quickly left the bow and advanced at the speed of 14 ~ 15 km per hour, but the shape of the wave remained unchanged and disappeared after 2 ~ 3 km. He called this wave a solitary wave.
Later, in 1895, Cavite and others further studied it, and people had a clearer understanding of solitons, and acoustic solitons, electric solitons and optical solitons were discovered successively. From the point of view of physics, soliton is a special product of nonlinear effect of matter. Mathematically, it is a stable and energy-limited non-dispersive solution of some nonlinear partial differential equations. In other words, it can always keep its waveform and speed unchanged. Solitary waves can keep the same shape and velocity after colliding with each other, just like particles, so people call them solitons for short.
Soliton has been widely used in plasma physics, high-energy electromagnetism, fluid mechanics and nonlinear optics because of its special properties.
From 65438 to 0973, the viewpoint of solitary wave was introduced into optical fiber transmission. When the frequency shifts, due to the balance between the nonlinear change of refractive index and the group dispersion effect, the optical pulse will form a basic soliton and propagate stably in the abnormal dispersion region. As a result, a new electromagnetic theory-optical soliton theory appears gradually, which leads communication to a new field of nonlinear optical fiber soliton transmission system. Soliton is a kind of optical pulse that can propagate in optical fiber and keep its shape, amplitude and speed unchanged for a long time. Using the characteristics of optical soliton, optical communication with ultra-long distance and ultra-large capacity can be realized.
Optical soliton communication
In optical fiber communication, the main reasons for limiting transmission distance and transmission capacity are "loss" and "dispersion". "Loss" makes the energy of optical signal weaken continuously when transmitting; And "dispersion" means that the optical pulse is gradually broadened during transmission. The so-called optical pulse is actually an electromagnetic wave set composed of a series of light wave oscillations with different frequencies. The dispersion of optical fiber makes light waves with different frequencies propagate at different speeds, which makes the optical pulses that start at the same time travel at different speeds due to different frequencies, and the time to reach the end point is also different, resulting in pulse broadening and signal distortion. Now with the development of optical fiber manufacturing technology, the loss of optical fiber has been reduced to the level close to the theoretical limit, and the dispersion problem has become the main problem to realize ultra-long distance and ultra-large capacity optical fiber communication.
The dispersion of optical fiber broadens the pulse of optical signal, and there is also a nonlinear characteristic in optical fiber, which will make the pulse of optical signal produce compression effect. The nonlinear characteristics of optical fiber change the frequency when the light intensity changes, thus changing the propagation speed. This change of optical fiber makes the trailing edge frequency of optical pulse higher and the propagation speed faster; The frequency of the leading edge becomes lower and the propagation speed becomes slower. This causes the trailing edge of the pulse to move faster than the leading edge, so that the pulse is compressed and narrowed.
If there is a way to make the broadening and narrowing effects of optical pulses cancel each other out, optical pulses will form optical solitons like isolated particles, which can remain unchanged in optical fiber transmission and realize ultra-long distance and ultra-large capacity communication.
Optical soliton communication is an all-optical nonlinear communication scheme. Its basic principle is that the nonlinear (self-phase modulation) effect of optical fiber refractive index leads to the balance between optical pulse compression and optical pulse broadening caused by group velocity dispersion. Under certain conditions (the anomalous dispersion region of optical fiber and the pulsed light power density are large enough), optical solitons can travel a long distance in optical fiber without deformation. It completely gets rid of the limitation of optical fiber dispersion on transmission rate and communication capacity, the transmission capacity is 1~2 orders of magnitude higher than that of the best communication system today, and the relay distance can reach hundreds of km. It is considered as one of the most promising next generation transmission modes.
From the analysis of optical soliton transmission theory, optical soliton is an ideal optical pulse, because it is very narrow and the pulse width is in the order of picosecond (ps, that is, s). In this way, the interval between adjacent optical pulses can be made very small, so that pulse overlap and interference will not occur. Using optical soliton to communicate, its transmission capacity is extremely large, which can be said to be almost infinite. The transmission rate will probably be as high as megabits per second. Such a high speed will mean that all books in the Library of Congress, the largest library in the world, can be delivered in 100 second. This shows how powerful the optical soliton communication is.
Main technical contents
In recent years, optical soliton communication has made a breakthrough. The application of optical fiber amplifier is very beneficial to the amplification and transmission of soliton, which advances the dream of soliton communication to the practical development stage. In the process of optical soliton transmission in optical fiber, the following problems need to be solved: the influence of fiber loss on optical soliton transmission, the interaction between optical solitons, the influence of higher-order dispersion effect on optical soliton transmission, and the birefringence phenomenon in single-mode fiber. The technologies involved mainly include:
Optical fiber technology suitable for optical soliton transmission. An important task of studying optical soliton communication system is to evaluate the evolution of optical soliton transmission along optical fiber. The effective distance of optical soliton transmission under specific optical fiber parameters is studied, and the relay distance of energy supplement is determined. This research not only provides data for the design of optical soliton communication system, but also usually leads to the emergence of new optical fibers.
Optical soliton source technology. Optical soliton source is the key to realize ultra-high-speed optical soliton communication. According to theoretical analysis, only when the output optical pulse is strictly hyperbolic secant and the amplitude meets certain conditions, the optical soliton can be stably transmitted in the optical fiber. At present, there are many kinds of optical soliton sources studied and developed, including Raman soliton laser, parametric soliton laser, erbium-doped fiber soliton laser, gain-switched semiconductor soliton laser and mode-locked semiconductor soliton laser. At present, most optical soliton communication test systems use gain-switched DFB semiconductor lasers or mode-locked semiconductor lasers with small volume and high repetition rate as optical soliton sources. Their output optical pulses are Gaussian, and their power is small, but after being amplified by fiber amplifier, the peak power enough to form optical soliton transmission can be obtained. Theoretical and experimental results show that optical soliton transmission does not need strict waveform. When Gaussian optical pulse propagates in dispersive fiber, the central part of optical pulse can gradually evolve into hyperbolic secant shape due to the interaction of nonlinear self-phase modulation and dispersion effect.
Optical soliton amplification technology. All-optical soliton amplifier can directly amplify the optical signal, avoiding the optical/electrical and electrical/optical conversion modes in the current optical communication system. It can be used as both a preamplifier of optical transceiver and an all-optical repeater, and it is an extremely important device in optical soliton communication system. In fact, optical solitons inevitably have losses in the process of optical fiber propagation. The loss of optical fiber only reduces the pulse amplitude of soliton, but does not change the shape of soliton. Therefore, compensation for these losses has become one of the key technologies of optical soliton transmission. At present, there are two ways to compensate soliton energy. One is to use distributed optical amplifiers, that is, stimulated Raman dispersion amplifiers or distributed erbium-doped fiber amplifiers. The other is the lumped optical amplifier method, which uses erbium-doped fiber amplifier or semiconductor laser amplifier. Optical amplifier using stimulated Raman scattering effect is a typical distributed optical amplifier. Its advantage is that the optical fiber itself becomes an amplification medium. However, the gain coefficient of stimulated Raman scattering in optical fiber is very small, which means that high-power laser is needed as the pump source of stimulated Raman scattering in optical fiber. In addition, there is some noise in this amplifier. Lumped amplification method is realized by erbium-doped fiber amplifier, and its stability has been proved by theory and experiment, which has become the main amplification method of soliton communication at present. Optical amplification is considered as the core problem of all-optical soliton communication.
Optical soliton switching technology. When designing all-optical switch, the whole design can be optimized by using optical soliton pulse as input signal. The biggest characteristics of optical soliton switch are high switching speed (up to 10-2s), high switching conversion rate (up to 100%) and good selection performance.
development prospect
All-optical soliton communication is a new generation of ultra-long distance and ultra-high bit rate optical fiber communication system, which is recognized as the most promising and pioneering frontier topic in optical fiber communication. Compared with linear optical fiber communication, optical soliton communication has a series of remarkable advantages: first, the transmission capacity is 1 ~2 orders of magnitude larger than that of the best linear communication system; Second, all-optical relay can be carried out. Because of the special properties of soliton pulse, the relay process is simplified as adiabatic amplification process, which greatly simplifies the relay equipment and is efficient, simple and economical. Compared with linear optical fiber communication, optical soliton communication has obvious advantages in technology and economy. Optical soliton communication is superior to light intensity modulation/direct detection and coherent light communication in high fidelity and long distance transmission.
It is precisely because of these advantages and potential development prospects of optical soliton communication technology that international and domestic efforts have been made to study and develop this technology in recent years. These studies have laid a theoretical, technical and material foundation for realizing ultra-high speed and ultra-long distance unrepeatered optical soliton communication system:
The invariance of 1. soliton pulse determines that relay is not needed;
2. Fiber amplifiers, especially erbium-doped fiber amplifiers pumped by laser diodes, compensate losses;
Thirdly, the stability of optical solitons after collision separation provides convenience for designing wavelength division multiplexing;
4. Using pre-emphasis technology and dispersion-shifted fiber transmission to amplify the lumped signal of erbium-doped fiber, weaken the influence of ASE and extend the relay distance under the condition of low gain;
Fifth, the pilot filter effectively reduces the soliton time jitter caused by noise in the ultra-long distance;
6. The new concept of eigenvalue communication broadens soliton communication from using only basic solitons to using higher-order solitons, thus increasing the amount of information carried by each pulse.
This series of progress in optical soliton communication makes the current experiment of soliton communication system reach the level of transmission rate 10~20Gbit/s and transmission distance 13000~20000 km.
The future prospect of optical soliton technology is: using ultra-long distance high-speed communication, ultra-short pulse control technology in time domain and frequency domain, ultra-short pulse generation and application technology, the current rate will be increased from 10~20Gbit/s to above 100Gbit/s; In the aspect of increasing transmission distance, retiming, shaping and regeneration technologies are adopted to reduce ASE, and optical filtering is adopted to increase the transmission distance to more than 100000 km. In the aspect of high performance EDFA, it is to obtain low noise and high output EDFA. Of course, there are still many technical problems in actual optical soliton communication, but the breakthrough progress made at present makes us believe that optical soliton communication has a bright development prospect in long-distance, high-speed and large-capacity all-optical communication, especially in submarine optical communication systems.