Stefan Banach

Stefan Banach was born on March 30, 1892, at St. Lazarus General Hospital in Kraków, then part of Austro-Hungarian Galicia. Banach's parents were Stefan Greczek and one Katarzyna Banach, both natives of the Podhale region.^[2] Stefan Greczek was born in Ostrowsko near the town of Nowy Targ^[3] and at one time was a soldier in the Austro-Hungarian Army stationed in Kraków. Stefan Greczek's father, Józef, was a farmer and a village mayor and Józef's wife, Antonina (née Borkowska) bore the Pomian coat of arms.^[3]

Banach's mother left him after baptizing him when he was four days old. Her name on the birth certificate is Katarzyna Banach. Later in life Banach would ask his father to tell him his mother's actual identity but would only be told that he had taken an oath of secrecy about it.^[4] Stefan Greczek would go on to marry twice and have a son by his first wife and four children by the second.

Unusually, Stefan's surname was that of his mother instead of his father, though he received his father's given name. Since Banach's father was a private and was prevented by military regulations from marrying, and the mother was too poor to support the child, the couple decided that he should be reared by family and friends. Family legend says that Banach spent his early childhood in Ostrowsko with his grandmother, to whom he was very close. When she became ill, his father sent him to Kraków to live with Franciszka Płowa and her daughter, Maria, although Banach would continue to visit his grandmother up to her funeral.^[5] Franciszka worked in a branch of the Tęcza laundries while her husband was the manager of the Krakowski Hotel. Together, they were able to give Banach what was a relatively comfortable life for the time.^[2][6] Contacts between Banach and his father were polite and cordial; though Banach loved his father, he did not show him much warmth or filial affection.^[7]

As a child, Banach was introduced to Juliusz Mien, a French intellectual who had moved to Kraków in 1870 and who was a guardian of Maria Płowa. Mien guided Banach by teaching him French and supervising his education without charge. Mien likely nurtured Banach's early mathematical skills, and he taught him to speak French so fluently that later in life Banach was able to impress foreign colleagues with his knowledge of the language.^[6]

In 1902 Banach, aged 10, enrolled in Kraków's Henryk Sienkiewicz Gymnasium no. IV where he became known as a prodigy. The school specialized in the humanities, including languages such as Latin, Greek, and German as well as subjects such as History and Geography along side some Mathematics. Despite this shortcoming, Banach and his best friend Witold Wiłkosz, a future mathematician, would regularly work on mathematics problems during school breaks and after school. In 1906 Banach, aged 14, was studying higher mathematics and two years later he had started in on several languages, both western and eastern,^[8] however he was especially fond of Latin. After obtaining his matura at age 18 in 1910, Banach went with Witold Wiłkosz to Lviv, then the capital of Galicia, intending to enroll in engineering at the Lwów Polytechnic. However, as Banach had to earn money to support his studies, it was not until 1914 that he finally, at age 22, passed his half-diploma exams.^[9]

When World War I broke out, Banach was excused from military service due to his left-handedness and poor vision. When the Russian Army opened its offensive toward Lwów, Banach left for Kraków, to spend the rest of the war there and in other Galician towns. He made his living tutoring at local gymnasiums and working in a bookshop. He may have attended lectures at the Jagiellonian University, but little is known of that period in his life.^[10]

In 1916, in Kraków's Planty gardens, Banach encountered Professor Hugo Steinhaus, one of the renowned mathematicians of the age. Steinhaus became fascinated with the self-taught young mathematician. The encounter resulted in a long-lasting collaboration and friendship. It was also through Steinhaus that Banach met his future wife, Łucja Braus.

Steinhaus introduced Banach to academic circles and substantially accelerated his career. After Poland regained independence, in 1920 Banach was given an assistantship at Kraków's Jagiellonian University. Steinhaus' backing also allowed him to receive a doctorate without actually graduating from a university. The doctoral thesis, accepted by King John II Casimir University of Lwów and published in 1922,^[11] included the basic ideas of functional analysis, which was soon to become an entirely new branch of mathematics. The thesis was widely discussed in academic circles and allowed him in 1922 to become a professor at the Lwów Polytechnic. Initially an assistant to Professor Antoni Łomnicki, in 1927 Banach received his own chair. In 1924 he was also accepted as a member of the Polish Academy of Learning. At the same time, from 1922, Banach also headed the second Chair of Mathematics at University of Lwów. Young and talented, Banach gathered around him a large group of mathematicians. The group, meeting in the Scottish Café, soon gave birth to the "Lwów School of Mathematics." In 1929 the group began publishing its own journal, Studia Mathematica, devoted primarily to Banach's field of study — functional analysis. Around that time, Banach also began working on his best-known work, the first monograph on the general theory of linear-metric space. First published in Polish in 1931,^[12] the following year it was also translated into French and gained wider recognition in European academic circles.^[13] The book was also the first in a long series of mathematics monographs edited by Banach and his circle.

Following the invasion of Poland by Nazi Germany and the Soviet Union, Lwów came under the control of the Soviet Union for almost two years. Banach, from 1939 a corresponding member of the Academy of Sciences of Ukraine, and on good terms with Soviet mathematicians,^{[citation needed]} had to promise to learn Ukrainian to be allowed to keep his chair and continue his academic activities.^[14] Following the German takeover of Lwów in 1941 in Operation Barbarossa, all universities were closed and Banach, along with many colleagues and his son, was employed as lice feeder at Professor Rudolf Weigl's Typhus Research Institute. Employment in Weigl's Institute provided many unemployed university professors and their associates protection from random arrest and deportation to Nazi concentration camps.

After the Red Army recaptured Lviv in the Lvov–Sandomierz Offensive of 1944, Banach returned to the University and helped re-establish it after the war years. However, because the Soviets were removing Poles from annexed formerly Polish territories, Banach began preparing to leave the city and settle in Kraków, Poland, where he had been promised a chair at the Jagiellonian University.^[4] He was also considered a candidate for Minister of Education of Poland.^[15] In January 1945, however, he was diagnosed with lung cancer and was allowed to stay in Lwów. He died on August 31, 1945, aged 53. His funeral at the Lychakiv Cemetery turned into a patriotic demonstration by the Poles who still remained in the city.
Vahid Damanafshan
Reference: Wikipedia

Measure theory

In mathematics, more specifically in measure theory, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area and volume of Euclidean geometry to suitable subsets of Rⁿ, n=1,2,3,.... For instance, the Lebesgue measure of [0,1] in the real numbers is its length in the non-formal sense of the word, specifically 1.

To qualify as a measure (see Definition below), a function that assigns a non-negative real number or infinity to a set's subsets must satisfy a few conditions. One important condition is countable additivity. This condition states that the size of the union of a sequence of disjoint subsets is equal to the sum of the sizes of the subsets. However, it is in general impossible to consistently associate a size to each subset of a given set and also satisfy the other axioms of a measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the subsets on which the measure is to be defined are called measurable and they are required to form a sigma-algebra, meaning that unions, intersections and complements of sequences of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be consistently defined, are necessarily complex to the point of incomprehensibility, in a sense badly mixed up with their complement; indeed, their existence is a non-trivial consequence of the axiom of choice.

Measure theory was developed in successive stages during the late 19th and early 20th century by Emile Borel, Henri Lebesgue, Johann Radon and Maurice Fréchet, among others. The main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space; moreover, integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor, the Riemann integral. Probability theory considers measures that assign to the whole set the size 1, and considers measurable subsets to be events whose probability is given by the measure.

Vahid Damanafshan

Reference: Gigapedia

Power domains

In denotational semantics and domain theory, power domains are domains of nondeterministic and concurrent computations.

The idea of power domains for functions is that a nondeterministic function may be described as a deterministic set-valued function, where the set contains all values the nondeterministic function can take for a given argument. For concurrent systems, the idea is to express the set of all possible computations.

Roughly speaking, a power domain is a domain whose elements are certain subsets of a domain. Taking this approach naively, though, often gives rise to domains that don't quite have the desired properties, and so one is led to increasingly complicated notions of power domain. There are three common variants: the Plotkin, upper, and lower power domains. One way to understand these concepts is as free models of theories of nondeterminism.

For most of this article we use the terms "domain" and of "continuous function" quite loosely, meaning respectively some kind of ordered structure and some kind of limit-preserving function. This flexibility is genuine; for example, in some concurrent systems it is natural to impose the condition that every message sent must eventually be delivered. However, the limit of a chain of approximations in which a message was not delivered, would be a completed computation in which the message was never delivered!

A modern reference to this subject is the chapter of Abramsky and Jung [1994]. Older references include those of Plotkin [1983, Chapter 8] and Smyth [1978].

Reference: Wikipedia

Domain theory

Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and has close relations to topology. An alternative important approach to denotational semantics in computer science is that of metric spaces.
Reference: Wikipedia

Real Analysis

Mathematical analysis can be applied in the study of classical concepts of real numbers, such as the complex variables, trigonometric functions, and algorithms, or of non-classical concepts like constructivism, harmonics, infinity, and vectors.

History:

Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. In India, the 12th century mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem.

In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent. Alongside his development of the Taylor series of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century.

In Europe, during the later half of the 17th century, Newton and Leibniz independently developed calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.

In the 18th century, Euler introduced the notion of mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816. In the 19th century, Cauchy helped to put calculus on a firm logical foundation by introducing the concept of the Cauchy sequence. He also started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the modern notion of mathematical rigor, thus founding the field of mathematical analysis (at least in the modern sense).

In the middle of the century Riemann introduced his theory of integration. The last third of the 19th century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the "epsilon-delta" definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which a mathematician creates irrational numbers that serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.

Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be created. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

Reference: Wikipedia

Hi!
This is my blog.