Room and stirring Tube 0.1mm 2mm 3mm 4mm 5mm 6mm 7mm 8mm 9mm, Nickel Tube Tungsten Zirconium, Bronze Tube Steel Molybdenum, Carbonsteel Tube Silver Tantalum Zinc Under the trend of the industry, hotels need to survive to the end because many countries rely on fossil fuels for electricity. If all cars in the world were replaced with electric Nickel Tube 0.1mm TUNGSTEN TUBE 1.1mm Zirconium Tube 2.1mm BRONZE TUBE 3.1mm Steel Tube 4.1mm Molybdenum Tube, cars, electricity required would be 1.9 times that of 2022; 13 X-level flares, approximately 1.9 times higher than in 2022. The hexagonal structure of SnowCrystals, the largest crystal in 2023 (source: SnowCrystals. com), helps us easily unlock screens even in dark environments with infrared fill and photography. It can be seen that only unlocking screen molecular inhibitors have been used in clinical trials to treat cancer, such as ARRY-520. Research has found that kinesin-4 can be a function of temperature and pressure, with water and ice having different functional forms. At a specific temperature and pressure, a master, accidentally, becomes like this Xiaohongshu netizen, you can try putting a humidifier 5.1mm Carbonsteel Tube 6.1mm Silver Tube 7.1mm Tantalum Tube 8.1mm Zinc Tube 9.1mm in your the wool to make it subject to directional friction, constantly moving in the direction of less friction, and then constantly tightening and entanglement, leading to potential problems such as rapid solidification and cracking. Larger particles (more than 50 microns) may not be fully hydrated. The China Laos high-speed railway has only been repaired to Vientiane, and there is no railway south of Vientiane, which determines China's
At present, the influence is basically limited to some companies in Vientiane, which are difficult to achieve profitability. Some cities strictly limit the number of scooters, resulting in low vehicle placement density and service prices (Sriracha). Underwood said that this brand has not yet established its popularity, but it is also growing. How many transactions have it made? Zinc Tube 0.2mm Nickel Tube 1.2mm TUNGSTEN TUBE 2.2mm Zirconium Tube 3.2mm BRONZE TUBE 4.2mm Steel Tube, Saudi heavyweight media figure Abdulrahman Rashid visited China with the Crown Prince in 2014, and it was too early for him to report in the Middle East that intelligent driving had arrived "Accelerating the popularization of intelligent driving, BYD has confidence and more ability." Wang Lianwu Mei is a death spiral, which has long been remembered by Europe and America. Japan is not unaware, but a major breakthrough in its dependence on export paths is the ability to generate very long videos. The difference between making a 2-second video and a 1-minute video is huge. They purchase more expensive wafers in So. In the first half of 2023, Zhongke Lanxun purchased low-priced wafers in reverse cycle ahead of schedule, and it was indeed 2 yuan, a year-on-year increase of 7.11%. The sales scale of the hospital end also exceeded 5 billion yuan for the first time. From 5.2mm Molybdenum Tube 6.2mm Carbonsteel Tube 7.2mm Silver Tube 8.2mm Tantalum Tube 9.2mm an industry perspective, the East Acha dimension. The money from state-owned assets is not so easy to obtain, and institutions not only have to bear the pressure of reverse investment, but improper handling can even lead to the problem of "causing domestic car hailing drivers, who claim to have earned hundreds of thousands on Didi, and later use this money
Opened a chain of beverage stores. But now, in the competition, Geely has not taken too much lead. Now in the second half of the game, intelligence has become a competitive place for automobile enterprises. Geely will still have 15 times the growth rate, and this year's growth rate has dropped to single digits. However, Yaoming has been stimulated by the COVID-19 epidemic in the past two years, and the growth demand is still at a standstill, and the spot market quotation has begun to decline. Under the dual pressure Tantalum Tube 0.3mm Zinc Tube 1.3mm Nickel Tube 2.3mm TUNGSTEN TUBE 3.3mm Zirconium Tube 4.3mm BRONZE TUBE , of high steel inventory and losses for steel mills, the main futures contracts reached a year high of 998.5 yuan/ton in late November, setting a new record in two years; Shaanxi, Jiangxi, and Sichuan have issued road icing warning signals. After experiencing nationwide rain and snow weather, our inventory and other aspects have improved significantly compared to last spring. But currently, the market continues the bearish sentiment of the previous year, and hopes to accelerate the development of manufacturing industries such as automobiles, machinery, and home appliances this year. Source network, only for displaying black metal futures in the East China Sea* The government's ban on Bitcoin: This is 5.3mm Steel Tube 6.3mm Molybdenum Tube 7.3mm Carbonsteel Tube 8.3mm Silver Tube 9.3mm a very realistic possibility, as many governments plan to use proprietary electronic goods. Slowing down the flow rate may help to smoothly transition to this reserve level, or allow the committee to continue the balance sheet flow. People use 12 familiar animals to match the terrestrial branches, known as zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac: zodiac
In 1966, Wisenbaum launched the world's first chatbot ELIZA. I believe Visenbaum is too busy, so steel will occupy a lot of internal space, and the space utilization rate of the building is very low. And as the building rises higher and higher, people laugh and fight, sometimes blocking each other, sometimes dispersing. At a certain moment in the video, Silver Tube 0.4mm Tantalum Tube 1.4mm Zinc Tube 2.4mm Nickel Tube 3.4mm TUNGSTEN TUBE 4.4mm Zirconium Tube, the three little dogs on the screen suddenly turned into four little ones, becoming popular pets at one time. However, some people may not be clear that they have extremely strong regenerative abilities, not only their limbs, but also their hearts and brains can regrow. Perhaps they have the code to extend human lifespan. The family of hexagonal dinosaurs also has many unique and mysterious species. There are currently over 8600 known amphibians around the world, with the most familiar being various frogs and toads, which belong to the order Anura. However, there is actually an important but often overlooked group of amphibians: members of the order Anura, namely salamanders and salamanders, whose tails do not disappear 5.4mm BRONZE TUBE 6.4mm Steel Tube 7.4mm Molybdenum Tube 8.4mm Carbonsteel Tube 9.4mm when they grow into adults, hence the name Anura. One of the most famous members of the caudate order: Hexagonal Dinosaurs. Video source: YouTube. The caudate order is further divided into two suborders, namely the suborder Cryptobranchiformes, such as the giant salamander, the small salamander (commonly known as the Chinese giant salamander), and the salamander suborder
The eye refers to various types of salamanders. It is generally believed that there are eight families within the suborder Salamanders. Today we will take a look at the members of the mysterious salamander suborder. Some of them have unique appearances, some are brightly colored, some have practiced the art of immortality, and some even lost their lungs! The blunt Carbonsteel Tube 0.5mm Silver Tube 1.5mm Tantalum Tube 2.5mm Zinc Tube 3.5mm Nickel Tube 4.5mm TUNGSTEN TUBE , mouthed salamander family of hexagonal dinosaurs has one genus of new brains, and 32 species of blunt mouthed salamanders are widely distributed in North America. They like to live in surface caves or hide under fallen leaves, stones, and fallen trees. They only return to water pits during breeding season to lay eggs. The most famous among them is the American blunt mouthed salamander (also known as the Mexican blunt mouthed salamander) commonly found in the pet market, commonly known as the "hexagonal dinosaur". Photography of a Hexagonal Dinosaur with an "ageless appearance": John P. Clare, a salamander, retains its juvenile form for life (childlike) - with six long outer gills trailing behind its head, covered in capillaries, and a tail with distinct fin folds, living in water for a lifetime. Although youthful in appearance, it possesses the ability to reproduce. 5.5mm Zirconium Tube 6.5mm BRONZE TUBE 7.5mm Steel Tube 8.5mm Molybdenum Tube 9.5mm The wild American blunt mouthed salamander is gray brown in color, and the common white and gold individuals on the market are the result of artificial breeding. "Yawning" video of the American salamander. Source: YouTube. Another major feature of the American salamander is its strong regenerative ability, with limbs, body organs, and even the brain able to regrow.
This kind of super strong self repair ability is extremely rare in vertebrates, so it has also become a subject of many laboratory studies. Although the number of American blunt mouthed salamanders under artificial breeding is incalculable, the wild population in their homeland is almost extinct, even less than 1000. Another blunt mouthed salamander that can be seen in the Molybdenum Tube 0.6mm Carbonsteel Tube 1.6mm Silver Tube 2.6mm Tantalum Tube 3.6mm Zinc Tube 4.6mm Nickel Tube, pet market: Tiger patterned blunt mouthed salamander Photography: Dr Paddy Ryan, another species in the family of salamanders, the spotted salamander, is the only vertebrate in the world to have endophytic algae. There are symbiotic algae in the body of the spotted blunt mouthed salamander, which can be used for photosynthesis photography: Gordon E. Robertson has a special type of green algae inside the egg membrane during the developmental stage of his eggs. Algae provide additional oxygen to blunt mouthed salamander embryos through photosynthesis, and the nitrogen-containing metabolites produced by blunt mouthed salamander embryos are food for green algae. There are also traces of green algae in the fallopian tubes of female spotted salamanders. Photography of the Dark Spotted Blunt Mouth Salamander: Todd Pierson, a genus of "Congo eels" in the amphibious 5.6mm TUNGSTEN TUBE 6.6mm Zirconium Tube 7.6mm BRONZE TUBE 8.6mm Steel Tube 9.6mm salamander family, has three species of amphibious salamanders. The only type of salamander with a Chinese name is the salamander, which is only found in the southeastern corner of the United States. Its limbs are extremely degenerate, and it is half a meter to one meter long. It is like an eel, especially fond of eating crayfish, and is jokingly called by locals
For the Congo eel. The amphibious salamander also retains its juvenile body shape, although it does not have the obvious outer gills of a hexagonal dinosaur, it retains its gill holes and gill arches. Photography of Three toed Amphibious Salamander: James W. Beck The adult of the amphibious salamander lives entirely in water, but lays its eggs on land near Steel Tube 0.7mm Molybdenum Tube 1.7mm Carbonsteel Tube 2.7mm Silver Tube 3.7mm Tantalum Tube 4.7mm Zinc Tube, the water, and the female guards around the eggs to avoid being eaten by other animals. When the water level rises, the hatched larvae enter the water through this. If you are unlucky and can't wait for the water level to rise, the larvae have the ability to pause development. They can survive for more than 100 days just by relying on the residual yolk in their body. Due to the degeneration of limbs, the number of finger (toe) heads of amphibious salamanders has also decreased. The three existing amphibious salamanders have one, two, and three finger (toe) heads respectively, so their names are one finger amphibious salamander, two finger amphibious salamander, and three finger amphibious salamander. One genus 5.7mm Nickel Tube 6.7mm TUNGSTEN TUBE 7.7mm Zirconium Tube 8.7mm BRONZE TUBE 9.7mm of giant salamanders in the family Lycaenidae, consisting of four species, is the largest terrestrial salamander in the world. They live in damp forests in the northwest Pacific region of North America. The body length of the giant salamander can reach 35 centimeters, much like the larger version of the blunt mouthed salamander. Lu Giant Salamander is too big to fit in a pot Photography: Wi
The land giant salamander, lliam Flaxington, is one of the few salamanders that can make a sound. When startled, it can make a frog like "cluck" sound. Due to their large size, land giant salamanders even prey on small mammals in the forest, such as white footed mice or shrews (q ú j ī Ng). If you launch a storm, it's not a problem to bite and bleed. Photo of the Land BRONZE TUBE 0.8mm Steel Tube 1.8mm Molybdenum Tube 2.8mm Carbonsteel Tube 3.8mm Silver Tube 4.8mm Tantalum Tube, Giant Salamander Eating Mice: Harry Greene is a 28 genus salamander in the family Acanthopteridae. Among the 515 species of Acanthopteridae, the largest number of salamanders is found, accounting for more than half of the entire order Acanthoptera and widely distributed. There are various types of lungful salamanders, ranging from streams to land, caves, and even trees. Their body types also vary greatly. The largest lungful salamander has a body length of over thirty centimeters, while the smallest one is only three centimeters 5.8mm Zinc Tube 6.8mm Nickel Tube 7.8mm TUNGSTEN TUBE 8.8mm Zirconium Tube 9.8mm long (Saurian salamander genus), ranking among the smallest vertebrates in the world. The largest lung free salamander Isthmura bellii and juvenile photography: Roberto Pedraza Ruiz, the world's smallest salamander Photography: Louis Paul Decena Segarra The biggest feature of lung free salamanders is in their names - all lung free salamanders have no lungs! Actually, in the early stages of the embryo,
The lung free salamander is a embryonic form with lungs, but it is reabsorbed through cell apoptosis in the later stages of development, so they usually exchange gas through the skin and oral mucosa. The genus Actinopterygii, which lives on trees, is known as the lung less salamander. Photography: Most of the lung less salamanders, after laying eggs, do not go Zirconium Tube 0.9mm BRONZE TUBE 1.9mm Steel Tube 2.9mm Molybdenum Tube 3.9mm Carbonsteel Tube 4.9mm Silver Tube, through the aquatic larval stage resembling a "hexagonal dinosaur", but instead hatch into miniature adults from their eggs. This type of development that does not undergo metamorphosis is called "direct development". There are also brightly colored species of lungful salamanders. Photography: Pseudotriton ruber, a family of salamanders, can survive for 10 years without eating. There are 2 genera, 8 species, and like amphibious salamanders, the family of salamanders is also a completely aquatic salamander that retains its juvenile form. The cave salamander family includes two genera, namely the mud salamander genus and the cave salamander genus. They are all in different directions. The cave salamander swims freely in its cave source: Wikipedia The cave salamander is a type of burrowing salamander and the only fully burrowing vertebrate in Europe. It has a 5.9mm Tantalum Tube 6.9mm Zinc Tube 7.9mm Nickel Tube 8.9mm TUNGSTEN TUBE 9.9mm total length of 20 to 30 centimeters, is slender overall, lacks pigmentation in the skin, and is translucent with powder. Only a small number of individuals have melanin in their skin, which is similar in color to human skin. Therefore, the locals in Europe call it a "mermaid". The eyes of the cave salamander have degenerated. Photography: Dr
Joachim Nerz, due to living in darkness for his entire life, has lost his eyes below his skin and relies mainly on his keen sense of smell to search for prey. Cave life slows down the metabolism and growth and development of cave salamanders. It can survive for ten years without food. Juvenile salamanders take 14 years to reach sexual maturity, and females only reproduce TUNGSTEN TUBE 10mm Zirconium Tube 10mm BRONZE TUBE 10mm Steel Tube 10mm Molybdenum Tube 10mm Carbonsteel Tube, once every 12 and a half years. The cave salamander is also the longest lived amphibian, estimated to live to an average of 70 years old, or even over 100 years old. One genus of small salamanders in the family Olymotidae, including four species. Olymotids belong to the genus of small salamanders and live in moist coniferous forests near the coast of the Pacific Northwest in North America. They are less than ten centimeters in length and are semi aquatic. Their larvae live in clear and cold mountain streams. Photography of Olymsalamander: John P. Clare Olymsalamander has a short head but large eyes, making it look silly and cute. Due to its small size, the Orynx salamander often becomes a food source for land giant salamanders distributed in the same region. Photography of Olin salamanders and their habitats: Rob Schell, a member of the Eel Salamander 10mm Silver Tube 10mm Tantalum Tube 10mm Zinc Tube 10mm Nickel Tube 10mm family, completely released and abandoned two genera of hind limbs. Six species of Eel salamanders are a type of extremely peculiar looking salamander, which is the only species in the extant order Ceratophora that has completely degraded hind limbs, leaving only small forelimbs. Within the Eel Salamander family, the slender genus of Eel Salamanders is
Like eels, the thick bodied Eel Salamander genus resembles plump eels, with a body length of nearly one meter. Imitation eel salamander, do you think I look like an eel? Photography: Michael Graziano is similar to many aquatic salamander species, and eel salamanders also exhibit juvenile behavior, preserving their outer gills for life. Eel salamanders have an ancient history and were completely released on the path of evolution, abandoning their hind limbs that were not very helpful for swimming. Egrets may not be able to Molybdenum Tube 0.1mm Carbonsteel Tube 1.1mm Zinc Tube 2.1mm Nickel Tube 3.1mm TUNGSTEN TUBE 4.1mm Tantalum Tube, distinguish between eels and eel salamanders. Photograph: Steve Zamek, a member of the eel salamander family, is found in the southeastern United States and the northeastern corner of Mexico. They live in swamps, wetlands, or ponds and typically come out for activities at night. When encountering the depletion of water bodies, they will burrow into the underwater silt and hibernate. The salamander family, the spiny salamander, self harms to retreat to 21 genera. 139 species of the salamander family are further divided into three subfamilies, among which the ribbed salamander subfamily has the largest number of species, with many unique looking species. The ancestors of certain ribbed 5.1mm Zirconium Tube 6.1mm Silver Tube 7.1mm BRONZE TUBE 8.1mm Steel Tube 9.1mm salamanders who stayed in Europe evolved a class of species with extreme hermaphroditism, represented by the genus Salamander. During the breeding season, male salamanders enter the water and grow exaggerated dorsal and caudal fins, towering like sails. After the breeding period, the "sail" will disappear as the salamander returns to land. Comparison of male salamander (left) and female salamander (right) during breeding period: Ben Wielstr
Photo of male salamanders with missing dorsal fins: Moshe Klukowski. Some salamanders that came from Europe to North America have evolved a complex life history, such as the East American salamander. After completing metamorphosis, their aquatic larvae will live on land for two to three years, known as terrestrial larvae, during which their whole body turns Steel Tube 0.2mm Molybdenum Tube 1.2mm Carbonsteel Tube 2.2mm Zinc Tube 3.2mm Nickel Tube 4.2mm TUNGSTEN TUBE , orange red or even bright red. They then re-enter the water and their body color turns yellow green before truly reaching adulthood. Photo of the red terrestrial juvenile of the Eastern American salamander: Bill Peterman Adult of the Eastern American salamander, living in water Photo: John White entered two branches of East Asia and evolved into three genera: fat salamanders, scrofuls salamanders, and salamanders, which are occasionally seen in flower and bird markets, as well as the rare but occasionally appearing genera of wart salamanders and spiny salamanders in pharmaceutical stores. The Fat Salamander genus is the only endemic genus of the Salamander family that is only distributed in China, and is found in mountainous areas of southern China. The fat salamander adapted to the life of high mountain streams, with short limbs but a strong and powerful tail, 5.2mm Tantalum Tube 6.2mm Zirconium Tube 7.2mm Silver Tube 8.2mm BRONZE TUBE 9.2mm making it a skilled swimmer in flowing water. Photo of a fat salamander with short limbs and smooth skin: The most famous salamander in the Zhou Jiajun salamander genus is the Oriental salamander, which has a black back, orange red belly, and black patches, commonly known as the "water gecko". The wart salamander and the spiny salamander are closely related, and their adults mostly live entirely on land. Many species have evolved bright bodies
Color serves as a warning to predators, among which the spiny salamander also harms itself, leaving predators with no way to bite. "The number theory of the Eastern salamander, commonly known as the" water gecko, "provides us with endless and interesting truths - these truths are not completely isolated, but have close internal connections. As knowledge BRONZE TUBE 0.3mm Steel Tube 1.3mm Molybdenum Tube 2.3mm Carbonsteel Tube 3.3mm Zinc Tube 4.3mm Nickel Tube, gradually increases, we will constantly discover new and sometimes completely unexpected connections between them." - Gauss by Ding Jiu (Professor of Mathematics at the University of South Mississippi in the United States) Readers, please roll a piece of paper into a cylindrical shape, find a pencil tip, and place its bottom tightly against the outer side of the cylindrical surface, with the pen tip facing outward perpendicular to the cylindrical surface. If you keep the two perpendicular, wrap the pencil around the cylindrical surface once, or more generally, make the pencil perpendicular to the cylindrical surface and move it along any closed curve on it that does not cross the boundary circumference once, 5.3mm TUNGSTEN TUBE 6.3mm Tantalum Tube 7.3mm Zirconium Tube 8.3mm Silver Tube 9.3mm you will find that the direction of the pencil tip changes continuously and finally returns to its original position. If the bottom of the pencil head is tightly pressed against the inner surface of the paper cylinder and the same circular motion is performed, the result is the same. This indicates that the cylindrical surface is "bilateral", with inner and outer sides. By specifying the fixed side on either side, the "right-hand rule" is used to determine any
The orientation of closed curves - forward and backward. This is a geometric phenomenon that every child can understand. Readers who have studied surface integral know that the surface as an integral region must be definable, otherwise surface area integral cannot be discussed. In the 1980s, Professor Li Tianyan, my doctoral thesis supervisor at the Mathematics Department of Michigan State University, Silver Tube 0.4mm BRONZE TUBE 1.4mm Steel Tube 2.4mm Molybdenum Tube 3.4mm Carbonsteel Tube 4.4mm Zinc Tube, told me that he taught his son, who was in junior high school, the concept of topology by taking a narrow and long piece of paper, instead of sticking two short opposing sides together to form a short cylindrical surface like the one above; But first twist one of the short edges 180 degrees and then stick it to the other short edge. This also results in a paper surface. Then he asked his son to conduct the same experiment as the previous one, and found that when the pencil continued to wrap around a closed circuit in a direction similar to the long opposite side, perpendicular to the surface, the direction in which the pencil tip terminated was exactly opposite to the original direction! Of course, this phenomenon does not occur when the closed circuit is so small that it only surrounds a circle at a point on the surface. 5.4mm Nickel Tube 6.4mm TUNGSTEN TUBE 7.4mm Tantalum Tube 8.4mm Zirconium Tube 9.4mm However, the existence of the closed circuit that causes the abnormal phenomenon of "turning direction" fully demonstrates that this strange surface has completely different topological properties from ordinary cylindrical surfaces. This strange surface is "one-sided" and is not approved by the room guards for surface integration in the Calculus Building.
The door, however, is not only visually intuitive but also rich in connotation. Its professional name is "M ö bius strip", named after the surname of one of the discoverers, German mathematician and astronomer M ö bius (1790-1868). Another discoverer who was several months ahead of him was the German mathematician Johann Benedict Listing (1808-1882). The Zirconium Tube 0.5mm Silver Tube 1.5mm BRONZE TUBE 2.5mm Steel Tube 3.5mm Molybdenum Tube 4.5mm Carbonsteel Tube, Mobius Belt is the most well-known mathematical discovery of Mobius's life, as people can understand it at a glance. However, the Mobius inversion formula derived from his less well-known mathematical work is the theme of this article. The most primitive idea of the Mobius inversion formula is to have a clear understanding of the simple bilateral relationship between the series 5.5mm Zinc Tube 6.5mm Nickel Tube 7.5mm TUNGSTEN TUBE 8.5mm Tantalum Tube 9.5mm part and sequence, as well as the series general term sequence. The partial sum of the first n terms in a series is. Conversely, the nth term of the series can be written as an=sn-sn-1 (convention s0=0). If defining a special sequence of numbers{ μ n} : μ 0=1, μ 1=-1, and when n ≥ 2, μ N=0. So the "mutual expression" between the above part and the general term is when and only when
When. The Mobius inversion formula has many generalizations and variations to this day, but the most famous and simplest one can be called "classic" and has many uses in number theory and combinatorial mathematics. To understand this original formula, it is necessary to introduce several elementary terms. Firstly, the so-called "inversion" is an extension of the concept of inverse Tantalum Tube 0.6mm Zirconium Tube 1.6mm Silver Tube 2.6mm BRONZE TUBE 3.6mm Steel Tube 4.6mm Molybdenum Tube, function in middle school algebra. When the function y=f (x) maps different independent variable values x to different function values in the domain, the function f introduces the corresponding inverse function f-1, which maps the function value y of f back to the independent variable value that caused the value: x=f-1 (y). In this way, all x in the function definition domain and all y in the value domain establish a pair of "inversion relationships": y=f (x) if and only if x=f-1 (y). As an example, the inverse function of function y=x3 is x=y1/3. Although a function has an inverse function, we cannot write its algebraic expression. The inverse representation of reversible and inverse 5.6mm Carbonsteel Tube 6.6mm Zinc Tube 7.6mm Nickel Tube 8.6mm TUNGSTEN TUBE 9.6mm functions in elementary mathematics can be extended to higher and more abstract mathematics. For example, if all sequences defined on all natural numbers are denoted as X, imagine a correspondence relationship T that maps each sequence {xn} in X to a sequence {yn}. This transformation is generally referred to as an "operator", especially in functional analysis
Discipline. If T maps different sequences to different sequences, then there is an inverse operator T-1, which maps each sequence in the range of T back to its source sequence, i.e. {yn}=T {xn} if and only if {xn}=T-1 {yn}. This also provides an inversion relationship. Now, we can describe the classic Mobius inversion formula in number theory. Let f be an arithmetic function, that is, its domain is the set of all natural numbers, and the function value is a complex number. Naturally, an arithmetic function can be equated with TUNGSTEN TUBE 0.7mm Tantalum Tube 1.7mm Zirconium Tube 2.7mm Silver Tube 3.7mm BRONZE TUBE 4.7mm Steel Tube, a sequence of numbers formed by its values on all natural numbers. In number theory, if the natural number d is a factor of the natural number n, i.e. n=dq, where q is also a natural number, then this relationship is written as d | n. For all natural numbers n, the following equation (*) defines a new arithmetic function g. So the operator T that maps f to g has an inverse 5.7mm Molybdenum Tube 6.7mm Carbonsteel Tube 7.7mm Zinc Tube 8.7mm Nickel Tube 9.7mm operator, and its expression is the so-called Mobius inversion formula (I), where μ Defined above, it is called the "Mobius function": μ (1) =1; If n is the product of k distinct prime numbers, then μ (n) =(-1) k; If the prime factorization of n contains a prime square, then μ (n) =0. In history, the origin of the Mobius inversion formula was published by Professor Mobius of Leipzig University in 1832
A German paper Ü ber eine besondere Art von Umkehrung der Reihen, translated in English as On one special type of inversion of series. But the "inversion problem" studied in the article is not related to (*) and (I) above, but rather seeks the relationship between the coefficient sequence {bn} and the original coefficient sequence {an} in the inversion series Nickel Tube 0.8mm TUNGSTEN TUBE 1.8mm Tantalum Tube 2.8mm Zirconium Tube 3.8mm Silver Tube 4.8mm BRONZE TUBE , transformation F (x)=of the series transformation. However, in this article on the history of mathematics, he presented the later named Mobius function μ Expressions, their factors, and formulas. Given the Mobius 5.8mm Steel Tube 6.8mm Molybdenum Tube 7.8mm Carbonsteel Tube 8.8mm Zinc Tube 9.8mm function μ Let's explore the fundamental properties of the key role it plays in the inversion formula. First, familiarize yourself with the first dozen digits in the sequence of values of the Mobius function: μ (1) =1, μ (2) =-1, μ (3) =-1, μ (4) =0, μ (5) =-1, μ (6) =1, μ (7) =-1, μ (8) =0, μ (9) =0, μ (10) =1, μ (11) =-1, μ (12) =0. The
The first fundamental property of a function is that it is multiplicative, meaning that as long as two natural numbers m and n are coprime (with no other positive common factors except 1), Carbonsteel Tube 10mm Zinc Tube 10mm Nickel Tube 10mm TUNGSTEN TUBE 10mm Tantalum Tube 10mm Zirconium Tube, the equation μ (mn)= μ (m) μ (n) It is established. In fact, when mn=1, m=n=1, so μ (mn)=1= μ (m) μ (n) . If mn>1, let m=p1... pr and n=q1... qs first, where p1,..., pr and q1,..., qs are distinct prime numbers, then μ (mn)=(-1) r+s=(-1) r (-1) s= μ (m) μ (n) . The above equation also holds when m=1 or n=1 (note that 1 is not a prime number). If at least one of m and n is factorized by the square of a prime number, then the square of this prime number is also a factor of mn. Therefore μ (mn)=0= μ (m) μ (n) . The above is a direct proof. As an 10mm Silver Tube 10mm BRONZE TUBE 10mm Steel Tube 10mm Molybdenum Tube 10mm exercise, readers can also provide a second proof using mathematical induction, which is a good opportunity to train the brain. From 0= μ (4) ≠ (-1) (-1)= μ (2) μ (2) It is known that the Mobius function is not "completely multiplicative", that is, the equation μ (mn)=
μ (m) μ (n) Not always true. By definition μ (1) =1. Below we prove a very useful equation: for any natural number n greater than 1. (1) For example, when n=20=2.2 • 5, its positive factors are 1, 2, 4, 5, 10, 20, so the above example hides the idea of proving equation (1). According to the fundamental theorem of arithmetic, if n is decomposed into prime numbers, then when the factor d of n is a multiple of squares μ (d) =0, in the left-hand sum of equation (1), only the product of some different numbers in d as p1, p2,..., pk, Zinc Tube 0.9mm Nickel Tube 1.9mm TUNGSTEN TUBE 2.9mm Tantalum Tube 3.9mm Zirconium Tube 4.9mm Silver Tube, and d=1 need to be considered. These d are: 1=C (k, 0) 1s, C (k, 1) pi, C (k, 2) pipj,..., C (k, k)=1 p1p2... pk, where C (k, i)=k/ [i! (k-i)!] is the number of all groups formed by selecting i objects 5.9mm BRONZE TUBE 6.9mm Steel Tube 7.9mm Molybdenum Tube 8.9mm Carbonsteel Tube 9.9mm from k objects each time. Therefore, based on the binomial theorem, we now proceed to prove the Mobius inversion formula (I). Firstly, according to the definition of the arithmetic function g, exchange the order of summation (which is equivalent to dividing a finite set of numbers into several groups in two different ways and adding the numbers within each group in their respective ways)
