Lee Liu
China was a great mathematician in Wei and Jin Dynasties and one of the founders of China's classical mathematical theory. In 263 AD, Liu Hui annotated Nine Chapters of Arithmetic. He comprehensively demonstrated the methods and formulas of Nine Chapters Arithmetic, pointed out and corrected the mistakes, and made outstanding contributions to mathematical methods and theories. Liu Hui creatively used limit thought to prove the formula of circular area and put forward the calculation method of pi.
He cut a circle from a circle with a diameter of 2 feet by connecting a regular hexagon, and then he got a regular hexagon 12 and a regular hexagon 24 ... The thinner he cut, the smaller the difference between the area of a regular polygon and the area of a circle. In his original words, it is "cut carefully, and the loss is not big. If you cut again, there will be no loss." He calculated the area of 3072 polygons and verified this value. The scientific method of calculating pi put forward by Liu Hui has established China's leading position in the world for more than 1000 years.
Liu Hui has made great contributions to mathematics, and put forward the idea of "seeking the number of emblems" among endless problems. This method is consistent with the later method of finding the approximate value of irrational roots. It is not only a necessary condition for accurate calculation of pi, but also promotes the generation of decimals. In the solution of linear equations, he created a simpler method of mutual multiplication and elimination than direct division, which is basically consistent with the current solution; And put forward the problem of indefinite equation for the first time in the history of Chinese mathematics; He also established the summation formula of arithmetic progression's first n terms; Many mathematical concepts are put forward and defined, such as power (area); Equation (linear equation); Positive and negative numbers and so on. Liu Hui also put forward many generally accepted and correct judgments as the premise of proof. Most of his reasoning and proof are logically rigorous, thus establishing nine chapters of arithmetic and his own solutions and formulas on the basis of inevitability. Although Liu Hui did not write a self-contained book, he noticed that the mathematical knowledge used in Nine Chapters Arithmetic has actually formed a unique set, including concepts and judgments, and is based on mathematical proof.
Chungchi Tsu
Zu Chongzhi (429-500), a scientist in the Song and Qi Dynasties in the Southern Dynasties, was also named Fan (now Laishui, Hebei Province). He is knowledgeable and versatile, especially in astronomy and mathematics. He widely collects and reads books and documents about astronomy and mathematics. Often, "personally measure the ruler, observe the leakage of the instrument, do your best and plan with your heart" to achieve accurate measurement and careful calculation. Through hard work, he calculated the value of pi (л) to seven decimal places for the first time in the history of world mathematics, that is, between 3. 14 15926 and 3. 14 15927. He proposed an agreed rate of 22/7 and a secret rate of 355/ 1 13. This secret rate value was first put forward in the world, more than 1000 years earlier than that in Europe, so some people advocate calling it "ancestral rate". He compiled his own mathematical research and achievements into a book called seal script, which was once regarded as a mathematics textbook by Chinese studies in the Tang Dynasty. His Da Ming Li introduced precession into the calendar for the first time. It is suggested that 39 1 year set 144 leap month. The length of a tropical year is 365.338+04438+0 days, and the error is only about 50 seconds. He is not only an outstanding mathematician and astronomer, but also an outstanding mechanic expert. He recreated all kinds of exquisite machinery, such as the long-lost south guide car and the thousand-mile ship. Besides, he also studies music. His works, such as Explaining the Analects of Confucius, Explaining the Classic of Filial Piety, Yi, Lao, Zhuang and the novel Yi Shu Ji, have long been lost.
Hua
Hua (19 10 ~ 1985), mathematician, academician of China Academy of Sciences. 191010 65438 was born in Jintan, Jiangsu province, and 1985 12 died in Tokyo, Japan.
1924 graduated from Jintan middle school and studied hard. 1930, taught in Tsinghua University. 1936 Visiting study at Cambridge University, UK. 1938 became a professor in The National SouthWest Associated University after returning to China. From 65438 to 0946, he went to the United States and served as a researcher at Princeton Institute of Mathematics, a professor at Princeton University and the University of Illinois, and returned to China from 65438 to 0950. He has served as Professor Tsinghua University, director and honorary director of Institute of Mathematics and Institute of Applied Mathematics of China Academy of Sciences, chairman and honorary chairman of Chinese Mathematical Society, director of National Mathematical Competition Committee, foreign academician of American National Academy of Sciences, academician of Third World Academy of Sciences, academician of Bavarian Academy of Sciences of the Federal Republic of Germany, deputy director, vice president and presidium member of physics department, mathematics and chemistry of China Academy of Sciences, director and vice president of mathematics department of China University of Science and Technology, vice chairman of China Association for Science and Technology, and member of the State Council Academic Degree Committee. He was a member of the first to sixth the NPC Standing Committee and vice-chairman of the sixth China People's Political Consultative Conference. He was awarded honorary doctorates by Nancy University in France, The Chinese University of Hong Kong and the University of Illinois in the United States. Mainly engaged in the research and teaching of analytic number theory, matrix geometry, typical groups, automorphic function theory, multiple complex variable function theory, partial differential equations, high-dimensional numerical integration and other fields, and has made outstanding achievements. In the 1940s, the historical problem of Gaussian complete trigonometric sum estimation was solved, and the best error order estimation was obtained (this result is widely used in number theory). The results of G.H. Hardy and J.E. Littlewood on the Welling problem and E. Wright on the Tully problem have been greatly improved and are still the best records.
In algebra, the basic theorem of one-dimensional projective geometry left over from history for a long time is proved. This paper gives a simple and direct proof that the normal child of an object must be contained in its center, which is Hua theorem. His monograph "On Prime Numbers of Pile Foundations" systematically summarizes, develops and perfects Hardy and Littlewood's circle method, vinogradov's triangle sum estimation method and his own method. Its main achievements still occupy the leading position in the world after more than 40 years of publication, and have been translated into Russian, Hungarian, Japanese, German and English, becoming one of the classic works of number theory in the 20th century. His monograph "Harmonic Analysis on Typical Fields of Multiple Complex Variables" gives the complete orthogonal system of typical fields with accurate analysis and matrix skills, combined with group representation theory, and thus gives the expressions of Cauchy and Poisson kernel. This work has a wide and deep influence on harmonic analysis, complex analysis and differential equations, and won the first prize of China Natural Science Award. Advocating the development of applied mathematics and computer, he has published many works such as Master Planning Method and Optimization Research, which have been popularized in China. In cooperation with Professor Wang Yuan, he has made important achievements in the application research of modern number theory methods, which is called "Hua Wang Fa". He made great contributions to the development of mathematics education and the popularization of science. He has published more than 200 research papers and dozens of monographs and popular science works.
Three (or four) greatest mathematicians
Gauss
Gauss (C.F. Gauss,1777.4.30-1855.2.23) is a German mathematician, physicist and astronomer, who was born in a poor family in Zwick, Germany. His father, Gerhard Di Drich, worked as a berm, bricklayer and gardener. His first wife lived with him for more than 65,438+00 years and died of illness, leaving him no children. Diderich later married Luo Jieya, and the next year their child Gauss was born, which was their only child. My father is extremely strict with Gauss, even a little too strict. He often likes to plan his life for the young Gauss according to his own experience. Gauss respected his father and inherited his honest and cautious character. De Derrick died in 1806, when Gauss had made many epoch-making achievements.
In the process of growing up, young Gauss mainly paid attention to his mother and uncle. Gauss's grandfather was a stonemason who died of tuberculosis at the age of 30, leaving two children: Gauss's mother Luo Jieya and his uncle Flier. Flier Ritchie is smart, enthusiastic, intelligent and capable, and devoted himself to the textile trade with remarkable achievements. He found his sister's son clever, so he spent part of his energy on this little genius and developed Gauss's intelligence in a lively way. A few years later, Gauss, who was an adult and achieved great success, recalled what his uncle had done for him and felt that it was crucial to his success. He remembered his prolific thoughts and said sadly, "We lost a genius because of the death of our uncle". It is precisely because Flier Ritchie has an eye for talents and often persuades her brother-in-law to let her children develop into scholars that Gauss didn't become a gardener or a mason.
In the history of mathematics, few people are as lucky as Gauss to have a mother who strongly supports his success. Luo Jieya got married at the age of 34 and was 35 when she gave birth to Gauss. He has a strong personality, wisdom and sense of humor. Since his birth, Gauss has been very curious about all phenomena and things, and he is determined to get to the bottom of it, which is beyond the scope allowed by a child. When the husband reprimands the child for this, he always supports Gauss and resolutely opposes the stubborn husband who wants his son to be as ignorant as he is.
Luo Jieya sincerely hopes that his son can do something great and cherish Gauss's talent. However, he was afraid to put his son into mathematics research that could not support his family at that time. /kloc-when she was 0/9 years old, although Gauss had made many great achievements in mathematics, she still asked her friend W. Bolyai (the father of J. Bolyai, one of the founders of non-Euclidean geometry): Will Gauss have a future? W Bolyai said that her son would become "the greatest mathematician in Europe", and her eyes were filled with tears.
At the age of seven, Gauss went to school for the first time. Nothing special happened in the first two years. 1787 years old, Gauss 10. He entered the first math class. Children have never heard of such a course as arithmetic before. The math teacher is Buttner, who also played a certain role in the growth of Gauss.
A story that is widely circulated all over the world says that when Gauss was at 10, by adding all the integers from 1 to 100, he worked out the arithmetic problem that Butner gave to the students. As soon as Butner described the question, Gauss got the correct answer. However, this is probably an untrue legend. According to the research of E·T· Bell, a famous mathematical historian who has studied Gauss, Butner gave the children a more difficult addition problem: 81297+81495+81693+…+100899.
Of course, this is also a summation problem of arithmetic progression (the tolerance is 198 and the number of items is 100). As soon as Butner finished writing, Gauss finished the calculation and handed in the small tablet with the answers written on it. E. T. Bell wrote that in his later years, Gauss often liked to talk about this matter with people, saying that only his answer was correct at that time, and all the other children were wrong. Gauss didn't specify how he solved the problem so quickly. Mathematical historians tend to think that Gauss had mastered arithmetic progression's summation method at that time. For a child as young as 10, it is unusual to discover this mathematical method independently. The historical facts described by Bell according to Gauss's own account in his later years should be more credible. Moreover, it can better reflect the characteristics that Gauss paid attention to mastering more essential mathematical methods since he was a child.
Gauss's computing ability, mainly his unique mathematical methods and extraordinary creativity, made Butner sit up and take notice of him. He specially bought Gauss the best arithmetic book from Hamburg and said, "You have surpassed me, and I have nothing to teach you." Then Gauss and Bater's assistant Bater established a sincere friendship until Bater died. They studied together and helped each other, and Gauss began real mathematics research.
1788, 1 1 year-old gauss entered a liberal arts school. In his new school, all his classes are excellent, especially classical literature and mathematics. On the recommendation of Bater and others, the Duke of zwick summoned Gauss, who was 14 years old. This simple, clever but poor child won the sympathy of the Duke, who generously offered to be Gauss' patron and let him continue his studies.
Duke Brunswick played an important role in Gauss's success. Moreover, this function actually reflects a model of scientific development in modern Europe, indicating that private funding was one of the important driving factors for scientific development before the socialization of scientific research. Gauss is in the transition period of privately funded scientific research and socialization of scientific research.
1792, Gauss entered Caroline College in Brunswick for further study. 1795, the duke paid various expenses for him and sent him to the famous German family in G? ttingen, which made Gauss study hard and started creative research according to his own ideals. 1799, Gauss finished his doctoral thesis and returned to his hometown of Bren-Zwick. Just when he fell ill because he was worried about his future and livelihood-although his doctoral thesis was successfully passed, he was awarded a doctorate and obtained a lecturer position, but he failed to attract students and had to return to his hometown-the duke extended a helping hand. The Duke paid for the printing of Gauss's long doctoral thesis, gave him an apartment, and printed Arithmetic Research for him, so that the book could be published in 180 1. Also bear all the living expenses of Gauss. All this moved Gauss very much. In his doctoral thesis and arithmetic research, he wrote a sincere dedication: "To Dagong" and "Your kindness relieved me of all troubles and enabled me to engage in this unique research".
1806, the duke was killed while resisting the French army commanded by Napoleon, which dealt a heavy blow to Gauss. He is heartbroken and has long been deeply hostile to the French. The death of Dagong brought economic difficulties to Gauss, the misfortune that Germany was enslaved by the French army, and the death of his first wife, all of which made Gauss somewhat disheartened, but he was a strong man and never revealed his predicament to others, nor did he let his friends comfort his misfortune. It was not until19th century that people knew his state of mind at that time when sorting out his unpublished mathematical manuscripts. In a discussion of elliptic functions, a subtle pencil word was suddenly inserted: "For me, it is better to die than to live like this."
The generous and kind benefactor died, and Gauss had to find a suitable job to support his family. Because of Gauss's outstanding work in astronomy and mathematics, his fame spread all over Europe from 1802. The Academy of Sciences in Petersburg has continuously hinted that since Euler's death in 1783, Euler's position in the Academy of Sciences in Petersburg has been waiting for a genius like Gauss. When the Duke was alive, he strongly discouraged Gauss from going to Russia. He is even willing to raise his salary and set up an observatory for him. Now, Gauss is facing a new choice in life.
In order not to lose Germany's greatest genius, B.A. von von humboldt, a famous German scholar, joined other scholars and politicians to win Gauss the privileged positions of professor of mathematics and astronomy at the University of G? ttingen and director of the G? ttingen Observatory. 1807, Gauss went to Kottingen to take office, and his family moved here. Since then, he has lived in G? ttingen except for attending a scientific conference in Berlin. The efforts of Humboldt and others not only made the Gauss family have a comfortable living environment, but also enabled Gauss himself to give full play to his genius, and created conditions for the establishment of Gottingen Mathematics School and Germany to become a world science center and mathematics center. At the same time, it also marks a good beginning of scientific research socialization.
Gauss's academic position has always been highly respected by people. He has the reputation of "prince of mathematics" and "king of mathematicians" and is considered as "one of the three (or four) greatest mathematicians in human history" (Archimedes, Newton, Gauss or Euler). People also praised Gauss as "the pride of mankind". Genius, precocity, high yield, persistent creativity, ..., almost all the praises in the field of human intelligence are not too much for Gauss.
Gauss's research field covers all fields of pure mathematics and applied mathematics, and has opened up many new fields of mathematics, from the most abstract algebraic number theory to intrinsic geometry, leaving his footprints. Judging from the research style, methods and even concrete achievements, he is the backbone of 18- 19 century. If we imagine mathematicians in the18th century as a series of high mountains, the last awe-inspiring peak is Gauss; If mathematicians in the19th century are imagined as rivers, then their source is Gauss.
Although mathematical research and scientific work did not become an enviable career at the end of 18, Gauss was born at the right time, because the development of European capitalism made governments around the world pay attention to scientific research when he was almost 30 years old. With Napoleon's emphasis on French scientists and scientific research, Russian czars and many European monarchs began to look at scientists and scientific research with new eyes. The socialization process of scientific research is accelerating and the status of science is improving. As the greatest scientist at that time, Gauss won many honors, and many world-famous scientists regarded Gauss as their teacher.
1802, Gauss was elected as an academician of communication and a professor of Kazan University by the Academy of Sciences in Petersburg, Russia. 1877, the Danish government appointed him as a scientific adviser, and the Hanover government of Germany also hired him as a government scientific adviser.
Gauss's life is a typical scholar's life. He has always maintained the frugality of a farmer, making it hard to imagine that he is a great professor and the greatest mathematician in the world. He was married twice, and several children annoyed him. However, these have little influence on his scientific creation. After gaining a high reputation and German mathematics began to dominate the world, a generation of Tianjiao completed the journey of life.
newton
Isaac newton was born in Holthorpe, Lincolnshire, England on Christmas 1642. His father died three months before he was born. After his mother remarried, he had to be raised by his grandmother and uncle. Newton, a teenager, was mediocre at learning, but he liked handmade very much. At the same time, he has an extraordinary talent for painting.
Newton/kloc-started middle school at the age of 0/2. At this time, his hobby developed from manual production to mechanical small production. He realized that learning his lessons well, especially mathematics well, was of great benefit to production. So Newton redoubled his study and made great progress.
Newton 15 years old was forced to drop out of school for family reasons. Newton, who was very eager for knowledge, still seized all the time to study and study hard. Newton's studious spirit touched Newton's uncle. Finally, with the support of my uncle, I went back to school to repeat my studies.
Newton, 166 1 year old, was admitted to the famous Cambridge university. Isaac, Newton's first professor, was at school? Balu had a unique eye and discovered Newton's profound observation and keen understanding, so he taught Newton his mathematical knowledge and guided him to the study of modern natural science. 1664 Newton passed the exam and was elected as Baru's assistant. 1665, graduated from Newton University with a bachelor's degree. When I was preparing to stay in school for further study, a serious plague swept through Britain and Cambridge University was forced to close. Newton's return to his hometown twice to avoid disaster was the most important turning point in his life. Newton was absorbed in mathematics, physics and astronomy in the quiet environment of his hometown. The vitality of the volcano of ideas accumulated for many years finally broke out, and the torrent of wisdom rolled forward. In only 18 months, he gave birth to the basic ideas of flow calculus, the law of universal gravitation and optical analysis. Newton solved the gravity discovered by 1666 by calculation. 1687, at the age of 45, he completed the scientific masterpiece Mathematical Principles of Natural Philosophy, which is rare in the history of human science, inherited Kepler and Galileo, and established a complete classical mechanical system by mathematical methods, which caused a sensation all over the world.
Newton's mathematical contribution is most prominent in three aspects, namely "flow number", binomial theorem and "generalized arithmetic" (algebra) as a special form of calculus.
In order to solve the problem of motion, Newton created a mathematical theory directly related to physical concepts, that is, Newton called it "flow counting" theory, which is actually calculus theory. Newton mentioned "flow counting" in a manuscript on May 20th, 1665, so Newton started calculus about 10 years earlier than the German mathematician Leibniz, the founder of modern calculus, but from the time of official publication, Newton was later than Leibniz. In fact, the two of them independently established calculus. However, Newton's "flow counting" still has some defects.
Newton began to study binomial on the eve of returning home from Cambridge University to avoid the plague. He further clarified the meaning of negative index on the basis of his predecessor Varis. Newton binomial series expansion is a powerful tool to study series theory, function theory, mathematical analysis and equation theory.
Euler
Leonhard euler was born in Basel in 1707- 1783. He went to university of basel to study at the age of 13, and got the most famous mathematician johann bernoulli (1667- 1744).
Euler's profound knowledge, endless creative energy and unprecedented rich works are amazing! He published papers from the age of 19 to the age of 76, and has written countless books and papers for more than half a century. Up to now, Euler's name can be seen in almost every mathematical field, from Euler line of elementary geometry, euler theorem of polyhedron, Euler transformation formula of solid analytic geometry, Euler solution of quartic equation to Euler function in number theory, Euler equation of differential equation, Euler constant of series theory and Euler equation of variational method. Euler formula of complex variable function, etc. , is countless. His contribution to mathematical analysis is even more original. Introduction to infinitesimal analysis is his epoch-making masterpiece, and mathematicians call him "the embodiment of analysis" at that time.
Euler is the most prolific outstanding mathematician in the history of science. According to statistics, * * * has written 886 books and papers in his tireless life, of which 40% is analysis, algebra and number theory, 18% is geometry, 28% is physics and mechanics, 1 1% is astronomy, as well as ballistics and navigation.
It is no accident that Euler's works are surprisingly prolific. He can work in any unfavorable environment. He often puts his children on his lap to finish his homework, no matter how noisy they are. His indomitable perseverance and tireless academic spirit made him never stop studying mathematics after he became blind. In the 65,438+07 years after his blindness, he also dictated several books and about 400 papers. 36860.68868888666
Euler's father Paul Euler is also a mathematician. He wants little Euler to study theology and teach him a little at the same time. Because of his talent and extremely diligent spirit, he got johann bernoulli's appreciation and special guidance. When he was 19 years old, he wrote a paper on the mast and won a prize from the Paris Academy of Sciences. His father no longer opposed him to study mathematics.
Johann bernoulli's son daniel bernoulli went to Russia on 1725 and recommended Euler to czar Cadling I. In this way, Euler came to Petersburg on 17 and 1733. At the age of 26, Euler became a professor of mathematics at the Academy of Sciences in Petersburg. 1735, Euler solved a problem. It took several famous mathematicians several months to solve this problem, but Euler finished it in three days with his own invented method. However, due to overwork, he got an eye disease and unfortunately lost his right eye. At this time, he was only 28 years old. At the invitation of Prussian frederick the great, Euler went to Berlin as the director of the Institute of Physics and Mathematics of the Chinese Academy of Sciences until 1768. Later, at the sincere urging of Tsar Cadling II, he returned to Petersburg. Unexpectedly, not long after, his left eye vision decreased and he was completely blind. Unfortunately, the fire in Petersburg in 177 1 affected Euler's residence. 64-year-old Euler was blinded by illness and was trapped in the fire. Although he was saved from the fire by others, his research and a lot of research results were reduced to ashes.
The heavy blow still didn't knock Euler down. He vowed to recover the loss. Before he was completely blind, he could still see vaguely. He seized this last moment, scribbled down the formula he found on a big blackboard, and then dictated its contents, which were recorded by his students, especially his eldest son A. Euler (mathematician and physicist). After Euler was completely blind, he still struggled with the darkness with amazing perseverance and studied it with memory and mental arithmetic.
Euler's memory and mental arithmetic are rare. He can retell the contents of his notes when he was young. Mental arithmetic is not limited to simple operations, and advanced mathematics can also be done by heart. An example is enough to illustrate his skill. Two students of Euler added the term 17 of a complex convergence series to the 50th place, and the difference between them was one unit. In order to determine who is right, Euler calculated all the errors in his mind and finally put them into the errors. It also solved Newton's headache-the problem of starting from the moon and many complicated analysis problems.
Euler has a high style. Lagrange is a great mathematician after Euler. Since the age of 19, he has been communicating with Euler to discuss the general solution of isoperimetric problems, from which the variational method was born. The isoperimetric problem is a problem that Euler has painstakingly considered for many years. Lagrange's solution won warm praise from Euler. 1February 2, 759, Euler praised Lagrange's achievements in his reply. He modestly suppressed his immature works in this respect from being published for the time being, so that the works of young Lagrange could be published and circulated, and won great reputation. In his later years, all mathematicians in Europe regarded him as a teacher. The famous mathematician Laplace once said, "Euler is our mentor." Euler's energy was maintained until the last moment, in the afternoon of September 1783. In order to celebrate his successful calculation of the law of balloon rising, Euler invited friends to dinner. Soon after Uranus was discovered, Euler wrote the essentials of calculating Uranus' orbit and made fun of his grandson. After drinking tea, he suddenly fell ill, and his pipe fell out of his hand, muttering "I'm dead". Finally, Euler "stopped living and calculating".
Euler's life is a life of struggle for the development of mathematics. His outstanding wisdom, tenacious perseverance, tireless spirit of struggle and noble scientific ethics are always worth learning. 【 Euler also created many mathematical symbols, such as π( 1736), I( 1777) and E (1744). Tg( 1753), △x( 1755), ∑( 1755), f(x)( 1734), etc.
Archimedes
Archimedes was born in Syracuse, Sicily, at the southern tip of the Italian peninsula in 287 BC. Father is a mathematician and astronomer. Archimedes had a good family upbringing since childhood. 1 1 years old, was sent to study in Alexandria, the cultural center of Greece. In this famous city known as the "Capital of Wisdom", Archimedes Job collected books and learned a lot of knowledge, and became a protege of Euclid students erato Sese and Cannon, studying geometric elements.
Later, Archimedes became a great scholar who was both a mathematician and a mechanic, enjoying the reputation of "the father of mechanics". The reason is that he discovered the lever principle through a lot of experiments, and then deduced many lever propositions through geometric derivation and gave strict proofs. Among them is the famous Archimedes principle, and he has also made brilliant achievements in mathematics. Although there are only a dozen works by Archimedes, most of them are geometric works, which have played a decisive role in promoting the development of mathematics. He wrote Calculation of Sand, Measurement of Circle, Ball and Cylinder, Parabolic Quadrature Method, On Spiral, Plane Balance, Floating Body, On Cone and Sphere, etc.
1906, Danish mathematician Heiberg discovered a copy of Archimedes' letter to erato Sese and some other works of Archimedes. Through research, it is found that these letters and transcripts contain the idea of calculus. What he lacks is the concept of no limit, but the essence of his thought extends to the field of infinitesimal analysis, which is maturing in the17th century, and predicts the birth of calculus.
Because of his outstanding contribution, American E.T. Bell commented on Archimedes in Mathematical Figures: Any open list of the three greatest mathematicians of all time will definitely include Archimedes, while the other two are usually Newton and Gauss. However, compared with his brilliant achievements and background of the times, or his far-reaching influence on contemporary and future generations, Archimedes should be the first to be respected.