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Posted on September 6, 2017 by mnicholls

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LAGRANGIAN

In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped.

It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets. Historically speaking, the Jordan measure came first, towards the end of the nineteenth century. For historical reasons, the term Jordan measure is now well-established, despite the fact that it is not a true measure in its modern definition, since Jordan-measurable sets do not form a σ-algebra. For example, singleton sets in each have a Jordan measure of 0, while , a countable union of them, is not Jordan-measurable.[1] For this reason, some authors[2] prefer to use the term Jordan content (see the article on content).

The Peano-Jordan measure is named after its originators, the French mathematician Camille Jordan, and the Italian mathematician Giuseppe Peano.[3

Molnár Antal Zeneiskola-Charpentier-TeDeum-20091218

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II. DEFINITIONS AND DESIGNATIONS

Consider the discrete optimization problem (which we refer to as Problem A)

,

where – is a non-decreasing -order-convex function on a partially set .

Let be an optimal solution of Problem A, and let be the point obtained by the following iterative procedure [4]:

which halts on the step if either or is the maximal element of the set (the set contains the zero , as we have stipulated). This point is called

[2] Devyaterikova M.V., Kolokolov A.A. Stability Analysis o Some Discrete Optimization Algorithms // Automation and Remote Control, 2004. № 3. pp. 48-54.

[3] Ramazanov A.B. On stability o the gradient algorithm in convex discrete optimization problems and related questions // J. Discrete Mathematics and Applications, 2011, vol. 21, Issue 4, pp. 465-476.

erential equations, Jordan was led to study the finite subgroups of the general linear group of n × n matrices over the complex numbers. Although there are infinite families of such finite subgroups, Jordan found that they were of a very specific group theoretic structure which he was able to describe.

Another generalisation, this time of work by Hermite on quadratic forms with integral coefficients, led Jordan to consider the special linear group of n × n matrices of determinant 1 over the complex numbers acting on the vector space of complex polynomials in n indeterminates of degree m.

Jordan is best remembered today among analysts and topologists for his proof that a simply closed curve divides a plane into exactly two regions, now called the Jordan curve theorem. It was only his increased understanding of mathematical rigour which made him realise that a proof of such a result was necessary. He also originated the concept of functions of bounded variation and is known especially for his definition of the length of a curve. These concepts appears in his Cours d’analyse de l’École Polytechnique first published in three volumes between 1882 and 1887. The second edition appeared in 1893 while the Jordan curve theorem appeared in the third edition of the text which appeared between 1909 and 1915.

Of course by 1882, when the first volume was published, Jordan was lecturing at the École Polytechnique and the book was written as a text for the students there. In some respects this is a little strange since it is a rigorous analysis text built on top of the attempts to put the topic on a firm foundation begun by Cauchy and given considerable impetus by Weierstrass. However, the courses at the École Polytechnique were supposed to train students to become civil and military engineers and this does not seem to be the approach which one would take trying to teach applications of the calculus to engineers. There had been a tradition of rigorous analysis at the École Polytechnique begun, of course, by Cauchy himself. Jordan was aware that his work was at a level that would be somewhat inappropriate for engineering students for he once said to Lebesgue that he called it “École Polytechnique analysis course” since:-

… one puts that on the cover to please the publisher…

Gispert-Chambaz in [7] contrasts the way that topological concepts are treated by Jordan in the first and second editions of the book. In the first edition most of the topological concepts are dealt with in a supplement to Volume 3. However between the editions Jordan had taught more advanced courses on analysis at the Collège de France and this may have influenced him to put set topology right up front in the second edition. In this respect one can see the second edition as setting a tone for analysis textbooks which continues today.

Among Jordan’s many contributions to analysis we should also mention his generalisation of the criteria for the convergence of a Fourier series.

The Journal de Mathématiques Pure et Appliquées was a leading mathematical journal and played a very significant part in the development of mathematics throughout the 19th century. It was usually known as the Journal de Liouville since Liouville had founded the journal in 1836. Liouville died in 1882 and in 1885 Jordan became editor of the Journal, a role he kept for over 35 years until his death.

In 1912

Euclidean geometry (say Hilbert’s axioms, which are studied in a course on the foundations of geometry; Euclid himself simply proceeded with blind faith that the constructions he performed did not stumble into any holes). And how do we know there is a model of Euclidean geometry? The canonical model for Euclidean geometry is the Cartesian plane consisting of ordered pairs of real numbers, and the verification of the axioms of Euclidean geometry depends on the properties of the real number line. If we follow this route to construct the real numbers from a Euclidean straight line, we find we have traveled in a logical circle. as work prof dr constantin Udriste

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