Home Random Page


CATEGORIES:

BiologyChemistryConstructionCultureEcologyEconomyElectronicsFinanceGeographyHistoryInformaticsLawMathematicsMechanicsMedicineOtherPedagogyPhilosophyPhysicsPolicyPsychologySociologySportTourism






Riemann

When Gauss died in 1855, his post at Göttingen was taken by Peter Gustav Lejeune Dirichlet. One mathematician who found the presence of Dirichlet a stimulus to research was Riemann, and his few short contributions to mathematics were among the most influential of the century. Riemann's first paper, his doctoral thesis (1851) on the theory of complex functions, provided the foundations for a geometric treatment of functions of a complex variable. His main result guaranteed the existence of a wide class of complex functions satisfying only modest general requirements and so made it clear that complex functions could be expected to occur widely in mathematics. More importantly, it was achieved by yoking together the complex theory with the theory of harmonic functions and potential theory. The theories of complex and harmonic functions were henceforth inseparable.

Riemann then wrote on the theory of Fourier series and their integrability. His paper was directly in the tradition that ran from Cauchy and Fourier to Dirichlet, and it marked a considerable step forward in the precision with which the concept of integral can be defined. In 1854 he took up a subject that much interested Gauss, the hypotheses lying at the basis of geometry.

The study of geometry has always been one of the central concerns of mathematicians. It was the language, and the principal subject matter, of Greek mathematics, the mainstay of elementary education in the subject, and it has an obvious visual appeal. It seems easy to apply, for one can proceed from a base of naively intelligible concepts. In keeping with the general trends of the century, however, it was just the naive concepts that Riemann chose to refine. What he proposed as the basis of geometry was far more radical and fundamental than anything that had gone before.

Riemann took his inspiration from Gauss's discovery that the curvature of a surface is intrinsic, and he argued that one should therefore ignore Euclidean space and treat each surface by itself. A geometric property, he argued, was one that was intrinsic to the surface. To do geometry, it was enough to be given a set of points and a way of measuring lengths along curves in the surface. For this, traditional ways of applying the calculus to the study of curves could be made to suffice. But Riemann did not stop with surfaces. He proposed that geometers study spaces of any dimension in this spirit, even, he said, spaces of infinite dimension.

Several profound consequences followed from this view. It dethroned Euclidean geometry, which now became just one of many geometries. It allowed the geometry of Bolyai and Lobachevsky to be recognized as the geometry of a surface of constant negative curvature, thus resolving doubts about the logical consistency of their work. It highlighted the importance of intrinsic concepts in geometry. It helped open the way to the study of spaces of many dimensions. Last, but not least, Riemann's work ensured that any investigation of the geometric nature of physical space would thereafter have to be partly empirical. One could no longer say that physical space is Euclidean because there is no geometry but Euclid's. This finally destroyed any hope that questions about the world could be answered by a priori reasoning.



In 1857 Riemann published several papers applying his very general methods for the study of complex functions to various parts of mathematics. One of these papers solved the outstanding problem of extending the theory of elliptic functions to the integration of any algebraic function. It opened up the theory of complex functions of several variables and showed how Riemann's novel topological ideas were essential in the study of complex functions. (In subsequent lectures Riemann showed how the special case of the theory of elliptic functions could be regarded as the study of complex functions on a torus.)

Another paper dealt with the question of how many prime numbers there are that are less than any given number x. The answer is a function of x, and Gauss had conjectured on the basis of extensive numerical evidence that this function was approximately x/ln(x). This turned out to be true, but it was not proved until 1896, when both the Belgian mathematician Charles Jean de la Vallée-Poussin and the French mathematician Jacques-Salomon Hadamard independently proved it. It is remarkable that a question about integers led to a discussion of functions of a complex variable, but similar connections had previously been made by Dirichlet. Riemann took the expression Π(1 - p-s)-1 = Σn-s, introduced by Euler the century before, where the infinite product is taken over all prime numbers p and the sum over all whole numbers n and treated it as a function of s. The infinite sum makes sense whenever s is real and greater than 1. Riemann proceeded to study this function when s is complex (now called the Riemann zeta function), and he thereby not only helped clarify the question of the distribution of primes but also was led to several other remarks that later mathematicians were to find of exceptional interest. One remark has continued to elude proof and remains one of the greatest conjectures in mathematics: the claim that the nonreal zeros of the zeta function are complex numbers whose real part is always equal to 1/2.

 

I. Analyse all the uses of the pronoun "it" in the article and suggest their correct translation into Russian.

 

II. Define the functions of "one" and translate the sentences into Russian:

1. One mathematician who found the presence of Dirichlet a stimulus to research was Riemann…

2. 1.…and he argued that one should therefore ignore Euclidean space and treat each surface by itself.

3. A geometric property, he argued, was one that was intrinsic to the surface.

4. The study of geometry has always been one of the central concerns of mathematicians.

5. It seems easy to apply, for one can proceed from a base of naively intelligible concepts.

By the way , what is the function of "for" in this sentence? And what is meant by the pronoun "it" in it?

6. It dethroned Euclidean geometry, which now became just one of many geometries.

7. One could no longer say that physical space is Euclidean because there is no geometry but Euclid's.

8. One of these papers solved the outstanding problem of extending the theory of elliptic functions…

9. One remark has continued to elude proof and remains one of the greatest conjectures in mathematics

 

III. Are the following nouns in the plural or in the singular? Supply the missing forms.

thesis, stimulus, series, basis, calculus, torus, hypotheses.

 

IV. Explain the form of the italicized verb and provide its right translation:

He proposed that geometers study spaces of any dimension in this spirit, even, he said, spaces of infinite dimension.

 

V. Suggest the best translation of the Subjective Infinitive Construction in the sentence:

…made it clear that complex functions could be expected to occur widely in mathematics

 

VI. Provide the best translation of the subject subordinate clause:

What he proposed as the basis of geometry was far more radical and fundamental than anything that had gone before.

 

VII. Read and translate the whole article into Russian taking into account the preceding tasks.

VIII. Read the article again and pick out the information about Riemann's contribution to mathematics.

IX. Suggest the Russian counterparts for the following words and phrases:

a doctoral thesis, the theory of complex functions, complex variable, to satisfy requirements, harmonic functions, the potential theory, the theory of Fourier series, to take up a subject, the principal subject matter, a visual appeal, to be in keeping with, the curvature of a surface, a geometric property, to measure lengths, constant negative curvature, to resolve doubts, logical consistency, to highlight the importance, a priori, the theory of elliptic functions, on the basis of, the infinite product, to make sense, to clarify the question, the distribution of primes, the zeta function.


Date: 2015-01-29; view: 526


<== previous page | next page ==>
The theory of numbers | Differential equations
doclecture.net - lectures - 2014-2019 year. Copyright infringement or personal data (0.003 sec.)