Что такое однородная функция
Однородная функция
СОДЕРЖАНИЕ
Примеры [ править ]
Пример 1 [ править ]
Например, предположим, что x = 2, y = 4 и t = 5. Тогда
Линейные функции [ править ]
Любое линейное отображение ƒ : V → W однородно степени 1, поскольку по определению линейности
Аналогично, любая полилинейная функция ƒ : V 1 × V 2 × ⋯ × V n → W однородна степени n, поскольку по определению полилинейности
Однородные многочлены [ править ]
однородна степени 10, так как
Однородный многочлен является многочленом из суммы одночленов той же степени. Например,
x 5 + 2 x 3 y 2 + 9 x y 4 <\displaystyle x^<5>+2x^<3>y^<2>+9xy^<4>\,> 
является однородным многочленом степени 5. Однородные многочлены также определяют однородные функции.
( x k + y k + z k ) 1 k <\displaystyle \left(x^
Мин. / Макс. [ Редактировать ]
Поляризация [ править ]
Рациональные функции [ править ]
Не примеры [ править ]
Логарифмы [ править ]
Аффинные функции [ править ]
Положительная однородность [ править ]
В частном случае векторных пространств над действительными числами понятие положительной однородности часто играет более важную роль, чем однородность в указанном выше смысле.
Все приведенные выше определения можно обобщить, заменив равенство f ( rx ) = r f ( x ) на f ( rx ) = | г | f ( x ), и в этом случае мы ставим перед этим определением слово « абсолютный » или « абсолютно ». Например,
Обобщения [ править ]
Моноиды и моноидные действия [ править ]
Однородность [ править ]
Если мы говорим, что функция однородна над M (соответственно, абсолютно однородна над M ), мы имеем в виду, что она однородна степени 1 над M (соответственно абсолютно однородна степени 1 над M ).
Понятие бытия абсолютно однородна степени к над М обобщена аналогично.
Теорема Эйлера об однородных функциях [ править ]
Непрерывно дифференцируемые положительно однородные функции характеризуются следующей теоремой:
Теорема может быть специализирована для случая функции одной действительной переменной ( n = 1 ), и в этом случае функция удовлетворяет обыкновенному дифференциальному уравнению
f ′ ( x ) − k x f ( x ) = 0. <\displaystyle f'(x)-<\frac
Однородные распределения [ править ]
t − n ∫ R n f ( y ) φ ( y t ) d y = t k ∫ R n f ( y ) φ ( y ) d y <\displaystyle t^<-n>\int _ <\mathbb
Приложение к дифференциальным уравнениям [ править ]
Подстановка v = y / x преобразует обыкновенное дифференциальное уравнение
Однородная функция
Смотреть что такое «Однородная функция» в других словарях:
Однородная функция — степени числовая функция такая, что для любого и выполняется равенство: причём называют порядком однородности. Различают также положительно однородные функции, для которых равенство … Википедия
ОДНОРОДНАЯ ФУНКЦИЯ — (homogeneous function) Функция, в которой умножение всех аргументов на любую постоянную λ умножает значение функции на λа является однородной порядка а. Так, например, функция y=(βx+γz)/(δx+εz) является однородной нулевого порядка, поскольку… … Экономический словарь
ОДНОРОДНАЯ ФУНКЦИЯ — степени числовая функция такая, что для всех точек из области ее определения и всех действительных t> 0 выполняется равенство где действительное число; при этом предполагается, что вместе с каждой точкой из области определения функции f при… … Математическая энциклопедия
Положительно однородная функция — Однородная функция степени q числовая функция такая, что для любого и выполняется равенство: причём q называют порядком однородности. Различают также положительно однородные функции, для которых равенство ( * ) выполняется только для… … Википедия
функция управляющего объекта — Совокупность действий управляющего объекта, относительно однородная по некоторому признаку, направленная на достижение частной цели, подчиненной общей цели управления. Примечание К числу функций управляющих объектов, например, относят: функцию… … Справочник технического переводчика
ОДНОРОДНАЯ ПРОИЗВОДСТВЕННАЯ ФУНКЦИЯ — HOMOGENEOUS PRODUCTION FUNCTIONФункциональная взаимосвязь между затратами и уровнем выпуска продукции. Общая производственная функция может быть выражена формулойQ = f(K, L),где продукция (Q) является функцией капитала (K) и труда (L), еслиf(kK,… … Энциклопедия банковского дела и финансов
функция управляющего объекта — Совокупность действий управляющего объекта, относительно однородная по некоторому признаку, направленная на достижение частной цели, подчиненной общей цели управления … Политехнический терминологический толковый словарь
ОПОРНАЯ ФУНКЦИЯ — опорный функционал, множества А, лежащего в векторном пространстве X, функция sA, задаваемая в находящемся с ним в двойственности векторном пространстве Y соотношением Напр., О. ф. единичного тара в нормированном пространстве, рассматриваемом в… … Математическая энциклопедия
Leonhard Euler
Salome Abigail Gsell (1776–1783)
Berlin Academy
Stepan Rumovsky
Joseph-Louis Lagrange (epistolary correspondent)
Leonhard Euler ( /ˈɔɪlər/ OY-lər; [2] German: [ˈɔʏlɐ] ( listen ) ; [3] 15 April 1707 – 18 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function. [4] He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. [5]
Euler is held to be one of the greatest mathematicians in history. A 1988 Mathematical Intelligencer poll ranked 3 of his results in the top 5 most beautiful equations ever (his iconic identity was ranked number one). [6] A statement attributed to Pierre-Simon Laplace expresses Euler’s influence on mathematics: «Read Euler, read Euler, he is the master of us all.» [7] [8] He is also widely considered to be the most prolific, as his collected works fill 92 volumes, [9] more than anyone else in the field. He spent most of his adult life in Saint Petersburg, Russia, and in Berlin, then the capital of Prussia.
Amongst his many discoveries and developments, Euler is credited for: Popularizing the Greek letter π (lowercase pi) to denote the Archimedes’ constant (the ratio of a circle’s circumference to its diameter); First employing the term f(x) to describe a function’s y-axis; The letter i to express the imaginary unit equivalent to √-1; The Greek letter Σ (uppercase sigma) to express summations; For developing e, a new mathematical constant (commonly known as Euler’s Number) which is roughly equivalent to 2.71828, representing a logarithm’s natural base, having several applications such as calculating compound interest in financial engineering. [10]
Euler also revolutionized the field of physics by reformulating Newton’s classic laws of physics into new laws that could explain the motion of rigid bodies more easily, and made significant contributions to the study of elastic deformations of solid objects.
Contents
Early life
Leonhard Euler was born on 15 April 1707, in Basel, Switzerland, to Paul III Euler, a pastor of the Reformed Church, and Marguerite née Brucker, another pastor’s daughter. He had two younger sisters, Anna Maria and Maria Magdalena, and a younger brother, Johann Heinrich. [11] Soon after the birth of Leonhard, the Euler family moved from Basel to the town of Riehen, Switzerland, where Leonhard spent most of his childhood. Paul was a friend of the Bernoulli family; Johann Bernoulli, then regarded as Europe’s foremost mathematician, would eventually be the most important influence on young Leonhard.
Euler’s formal education started in Basel, where he was sent to live with his maternal grandmother. In 1720, at age thirteen, he enrolled at the University of Basel. In 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who quickly discovered his new pupil’s incredible talent for mathematics. [12] At that time Euler’s main studies included theology, Greek and Hebrew at his father’s urging to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician.
In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. [13] At that time, he was unsuccessfully attempting to obtain a position at the University of Basel. In 1727, he first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship. Pierre Bouguer, who became known as «the father of naval architecture», won and Euler took second place. Euler later won this annual prize twelve times. [14]
Career
Saint Petersburg
Around this time Johann Bernoulli’s two sons, Daniel and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia. [15] [16] When Daniel assumed his brother’s position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. [17]
Euler arrived in Saint Petersburg on 17 May 1727. He was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he often worked in close collaboration. Euler mastered Russian and settled into life in Saint Petersburg. He also took on an additional job as a medic in the Russian Navy. [18]
The Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made especially attractive to foreign scholars like Euler. The academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Very few students were enrolled in the academy to lessen the faculty’s teaching burden. The academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions. [14]
The Academy’s benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler’s arrival. The Russian nobility then gained power upon the ascension of the twelve-year-old Peter II. The nobility, suspicious of the academy’s foreign scientists, cut funding and caused other difficulties for Euler and his colleagues.
Conditions improved slightly after the death of Peter II, and Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years later, Daniel Bernoulli, who was fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. [19]
On 7 January 1734, he married Katharina Gsell (1707–1773), a daughter of Georg Gsell, a painter from the Academy Gymnasium. [20] The young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. [21]
Berlin
Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia. He lived for 25 years in Berlin, where he wrote over 380 articles. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum, a text on functions published in 1748, and the Institutiones calculi differentialis, [22] published in 1755 on differential calculus. [23] In 1755, he was elected a foreign member of the Royal Swedish Academy of Sciences.
In addition, Euler was asked to tutor Friederike Charlotte of Brandenburg-Schwedt, the Princess of Anhalt-Dessau and Frederick’s niece. Euler wrote over 200 letters to her in the early 1760s, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess. [24] This work contained Euler’s exposition on various subjects pertaining to physics and mathematics, as well as offering valuable insights into Euler’s personality and religious beliefs. This book became more widely read than any of his mathematical works and was published across Europe and in the United States. The popularity of the «Letters» testifies to Euler’s ability to communicate scientific matters effectively to a lay audience, a rare ability for a dedicated research scientist. [23]
Despite Euler’s immense contribution to the Academy’s prestige, he eventually incurred the ire of Frederick and ended up having to leave Berlin. The Prussian king had a large circle of intellectuals in his court, and he found the mathematician unsophisticated and ill-informed on matters beyond numbers and figures. Euler was a simple, devoutly religious man who never questioned the existing social order or conventional beliefs, in many ways the polar opposite of Voltaire, who enjoyed a high place of prestige at Frederick’s court. Euler was not a skilled debater and often made it a point to argue subjects that he knew little about, making him the frequent target of Voltaire’s wit. [23] Frederick also expressed disappointment with Euler’s practical engineering abilities:
I wanted to have a water jet in my garden: Euler calculated the force of the wheels necessary to raise the water to a reservoir, from where it should fall back through channels, finally spurting out in Sanssouci. My mill was carried out geometrically and could not raise a mouthful of water closer than fifty paces to the reservoir. Vanity of vanities! Vanity of geometry! [25]
Personal life
Eyesight deterioration
Euler’s eyesight worsened throughout his mathematical career. In 1738, three years after nearly expiring from fever, he became almost blind in his right eye, but Euler rather blamed the painstaking work on cartography he performed for the St. Petersburg Academy for his condition. Euler’s vision in that eye worsened throughout his stay in Germany, to the extent that Frederick referred to him as «Cyclops». Euler remarked on his loss of vision, «Now I will have fewer distractions.» [26] He later developed a cataract in his left eye, which was discovered in 1766. Just a few weeks after its discovery, a failed surgical restoration rendered him almost totally blind. He was 59 years old then. However, his condition appeared to have little effect on his productivity, as he compensated for it with his mental calculation skills and exceptional memory. For example, Euler could repeat the Aeneid of Virgil from beginning to end without hesitation, and for every page in the edition he could indicate which line was the first and which the last. With the aid of his scribes, Euler’s productivity in many areas of study actually increased. He produced, on average, one mathematical paper every week in the year 1775. [27] The Eulers bore a double name, Euler-Schölpi, the latter of which derives from schelb and schief, signifying squint-eyed, cross-eyed, or crooked. This suggests that the Eulers had a susceptibility to eye problems. [28]
Return to Russia and death
In 1760, with the Seven Years’ War raging, Euler’s farm in Charlottenburg was sacked by advancing Russian troops. Upon learning of this event, General Ivan Petrovich Saltykov paid compensation for the damage caused to Euler’s estate, with Empress Elizabeth of Russia later adding a further payment of 4000 roubles—an exorbitant amount at the time. [29] The political situation in Russia stabilized after Catherine the Great’s accession to the throne, so in 1766 Euler accepted an invitation to return to the St. Petersburg Academy. His conditions were quite exorbitant—a 3000 ruble annual salary, a pension for his wife, and the promise of high-ranking appointments for his sons. All of these requests were granted. He spent the rest of his life in Russia. However, his second stay in the country was marred by tragedy. A fire in St. Petersburg in 1771 cost him his home, and almost his life. In 1773, he lost his wife Katharina after 40 years of marriage.
Three years after his wife’s death, Euler married her half-sister, Salome Abigail Gsell (1723–1794). [30] This marriage lasted until his death. In 1782 he was elected a Foreign Honorary Member of the American Academy of Arts and Sciences. [31]
In St. Petersburg on 18 September 1783, after a lunch with his family, Euler was discussing the newly discovered planet Uranus and its orbit with a fellow academician Anders Johan Lexell, when he collapsed from a brain hemorrhage. He died a few hours later. [32] Jacob von Staehlin-Storcksburg wrote a short obituary for the Russian Academy of Sciences and Russian mathematician Nicolas Fuss, one of Euler’s disciples, wrote a more detailed eulogy, [33] which he delivered at a memorial meeting. In his eulogy for the French Academy, French mathematician and philosopher Marquis de Condorcet, wrote:
Euler was buried next to Katharina at the Smolensk Lutheran Cemetery on Goloday Island. In 1785, the Russian Academy of Sciences put a marble bust of Leonhard Euler on a pedestal next to the Director’s seat and, in 1837, placed a headstone on Euler’s grave. To commemorate the 250th anniversary of Euler’s birth, the headstone was moved in 1956, together with his remains, to the 18th-century necropolis at the Alexander Nevsky Monastery.
Contributions to mathematics and physics
Euler worked in almost all areas of mathematics, such as geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics. He is a seminal figure in the history of mathematics; if printed, his works, many of which are of fundamental interest, would occupy between 60 and 80 quarto volumes. [27] Euler’s name is associated with a large number of topics.
Euler is the only mathematician to have two numbers named after him: the important Euler’s number in calculus, e, approximately equal to 2.71828, and the Euler–Mascheroni constant γ (gamma) sometimes referred to as just «Euler’s constant», approximately equal to 0.57721. It is not known whether γ is rational or irrational. [35]
Mathematical notation
Euler introduced and popularized several notational conventions through his numerous and widely circulated textbooks. Most notably, he introduced the concept of a function [4] and was the first to write f(x) to denote the function f applied to the argument x. He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler’s number), the Greek letter Σ for summations and the letter i to denote the imaginary unit. [36] The use of the Greek letter π to denote the ratio of a circle’s circumference to its diameter was also popularized by Euler, although it originated with Welsh mathematician William Jones. [37]
Analysis
The development of infinitesimal calculus was at the forefront of 18th-century mathematical research, and the Bernoullis—family friends of Euler—were responsible for much of the early progress in the field. Thanks to their influence, studying calculus became the major focus of Euler’s work. While some of Euler’s proofs are not acceptable by modern standards of mathematical rigour [38] (in particular his reliance on the principle of the generality of algebra), his ideas led to many great advances. Euler is well known in analysis for his frequent use and development of power series, the expression of functions as sums of infinitely many terms, such as
Euler directly proved the power series expansions for e and the inverse tangent function. (Indirect proof via the inverse power series technique was given by Newton and Leibniz between 1670 and 1680.) His daring use of power series enabled him to solve the famous Basel problem in 1735 (he provided a more elaborate argument in 1741): [38]
Euler introduced the use of the exponential function and logarithms in analytic proofs. He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers, thus greatly expanding the scope of mathematical applications of logarithms. [36] He also defined the exponential function for complex numbers and discovered its relation to the trigonometric functions. For any real number φ (taken to be radians), Euler’s formula states that the complex exponential function satisfies
A special case of the above formula is known as Euler’s identity,
De Moivre’s formula is a direct consequence of Euler’s formula.
Euler elaborated the theory of higher transcendental functions by introducing the gamma function and introduced a new method for solving quartic equations. He found a way to calculate integrals with complex limits, foreshadowing the development of modern complex analysis. He invented the calculus of variations including its best-known result, the Euler–Lagrange equation.
Euler pioneered the use of analytic methods to solve number theory problems. In doing so, he united two disparate branches of mathematics and introduced a new field of study, analytic number theory. In breaking ground for this new field, Euler created the theory of hypergeometric series, q-series, hyperbolic trigonometric functions and the analytic theory of continued fractions. For example, he proved the infinitude of primes using the divergence of the harmonic series, and he used analytic methods to gain some understanding of the way prime numbers are distributed. Euler’s work in this area led to the development of the prime number theorem. [41]
Number theory
Euler’s interest in number theory can be traced to the influence of Christian Goldbach, his friend in the St. Petersburg Academy. A lot of Euler’s early work on number theory was based on the works of Pierre de Fermat. Euler developed some of Fermat’s ideas and disproved some of his conjectures.
Euler linked the nature of prime distribution with ideas in analysis. He proved that the sum of the reciprocals of the primes diverges. In doing so, he discovered the connection between the Riemann zeta function and the prime numbers; this is known as the Euler product formula for the Riemann zeta function.
Euler proved Newton’s identities, Fermat’s little theorem, Fermat’s theorem on sums of two squares, and he made distinct contributions to Lagrange’s four-square theorem. He also invented the totient function φ(n), the number of positive integers less than or equal to the integer n that are coprime to n. Using properties of this function, he generalized Fermat’s little theorem to what is now known as Euler’s theorem. He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. He proved that the relationship shown between even perfect numbers and Mersenne primes earlier proved by Euclid was one-to-one, a result otherwise known as the Euclid–Euler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem of number theory, and his ideas paved the way for the work of Carl Friedrich Gauss. [42] By 1772 Euler had proved that 2 31 − 1 = 2,147,483,647 is a Mersenne prime. It may have remained the largest known prime until 1867. [43]
Graph theory
In 1735, Euler presented a solution to the problem known as the Seven Bridges of Königsberg. [44] The city of Königsberg, Prussia was set on the Pregel River, and included two large islands that were connected to each other and the mainland by seven bridges. The problem is to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. It is not possible: there is no Eulerian circuit. This solution is considered to be the first theorem of graph theory, specifically of planar graph theory. [44]
Applied mathematics
Some of Euler’s greatest successes were in solving real-world problems analytically, and in describing numerous applications of the Bernoulli numbers, Fourier series, Euler numbers, the constants e and π, continued fractions and integrals. He integrated Leibniz’s differential calculus with Newton’s Method of Fluxions, and developed tools that made it easier to apply calculus to physical problems. He made great strides in improving the numerical approximation of integrals, inventing what are now known as the Euler approximations. The most notable of these approximations are Euler’s method and the Euler–Maclaurin formula. He also facilitated the use of differential equations, in particular introducing the Euler–Mascheroni constant:
One of Euler’s more unusual interests was the application of mathematical ideas in music. In 1739 he wrote the Tentamen novae theoriae musicae, hoping to eventually incorporate musical theory as part of mathematics. This part of his work, however, did not receive wide attention and was once described as too mathematical for musicians and too musical for mathematicians. [49]
In 1911, almost 130 years after Euler’s death, Alfred J. Lotka used Euler’s work to derive the Euler–Lotka equation for calculating rates of population growth for age-structured populations, a fundamental method that is commonly used in population biology and ecology.