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Wetzel's Problem
In mathematics, Wetzel's problem concerns bounds on the cardinality of a set of analytic functions that, for each of their arguments, take on few distinct values. It is named after John Wetzel, a mathematician at the University of Illinois at Urbana–Champaign... Let ''F'' be a family of distinct analytic functions on a given domain with the property that, for each ''x'' in the domain, the functions in ''F'' map ''x'' to a countable set of values. In his doctoral dissertation, Wetzel asked whether this assumption implies that ''F'' is necessarily itself countable. Paul Erdős in turn learned about the problem at the University of Michigan, likely via Lee Albert Rubel. In his paper on the problem, Erdős credited an anonymous mathematician with the observation that, when each ''x'' is mapped to a finite set of values, ''F'' is necessarily finite. However, as Erdős showed, the situation for countable sets is more complicated: the answer to Wetzel's question is yes if and only if t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lee Albert Rubel
Lee Albert Rubel ( – ) was a mathematician known for his contributions to analog computing. Career Originally from New York, he held a Doctorate of Mathematics degree from University of Wisconsin-Madison, and was professor of Mathematics at University of Illinois at Urbana-Champaign since 1954. He wrote for several scientific publications like the ''Complex Variables and Elliptic Equations International Journal'', the '' Constructive Approximation'' mathematical journal, the ''American Mathematical Monthly'', the ''Journal of Differential Equations'', the ''Journal of Approximation Theory'', the ''Journal of Symbolic Logic'', the '' Journal of the Australian Mathematical Society''. He also collaborated to the ''Functional Analysis'' periodical, the '' Tohoku Mathematical'', the '' Mathematical Proceedings of the Cambridge Philosophical Society'', the ''Franklin Institute-engineering and Applied Mathematics'', ''Combinatorica'', ''Israel Journal of Mathematics'', and ''Journa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saharon Shelah
Saharon Shelah (; , ; born July 3, 1945) is an Israeli mathematician. He is a professor of mathematics at the Hebrew University of Jerusalem and Rutgers University in New Jersey. Biography Shelah was born in Jerusalem on July 3, 1945. He is the son of the Hebrew poet and Canaanist political activist Yonatan Ratosh. He received his PhD for his work on stable theories in 1969 from the Hebrew University. Shelah is married to Yael, and has three children. His brother, magistrate judge Hamman Shelah was murdered along with his wife and daughter by an Egyptian soldier in the Ras Burqa massacre in 1985. Shelah planned to be a scientist while at primary school, but initially was attracted to physics and biology, not mathematics. Later he found mathematical beauty in studying geometry: He said, "But when I reached the ninth grade I began studying geometry and my eyes opened to that beauty—a system of demonstration and theorems based on a very small number of axioms which impress ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proofs From THE BOOK
''Proofs from THE BOOK'' is a book of mathematical proofs by Martin Aigner and Günter M. Ziegler. The book is dedicated to the mathematician Paul Erdős, who often referred to "The Book" in which God keeps the most elegant proof of each mathematical theorem. During a lecture in 1985, Erdős said, "You don't have to believe in God, but you should believe in The Book." Content ''Proofs from THE BOOK'' contains 32 sections (45 in the sixth edition), each devoted to one theorem but often containing multiple proofs and related results. It spans a broad range of mathematical fields: number theory, geometry, Mathematical analysis, analysis, combinatorics and graph theory. Erdős himself made many suggestions for the book, but died before its publication. The book is illustrated by . It has gone through six editions in English, and has been translated into Persian, French, German, Hungarian, Italian, Japanese, Chinese, Polish, Portuguese, Korean, Turkish, Russian, Spanish and Greek. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zermelo–Fraenkel Set Theory
In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the historically controversial axiom of choice (AC) included, is the standard form of axiomatic set theory and as such is the most common foundation of mathematics. Zermelo–Fraenkel set theory with the axiom of choice included is abbreviated ZFC, where C stands for "choice", and ZF refers to the axioms of Zermelo–Fraenkel set theory with the axiom of choice excluded. Informally, Zermelo–Fraenkel set theory is intended to formalize a single primitive notion, that of a hereditary well-founded set, so that all entities in the universe of discourse are such sets. Thus the axioms of Zermelo–Fraenkel set theory refer only to pure sets and prevent its models fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Cohen (mathematician)
Paul Joseph Cohen (April 2, 1934 – March 23, 2007) was an American mathematician, best known for his proofs that the continuum hypothesis and the axiom of choice are independence (mathematical logic), independent from Zermelo–Fraenkel set theory, for which he was awarded a Fields Medal. Early life and education Cohen was born in Long Branch, New Jersey in 1934, into a Jews, Jewish family that had immigrated to the United States from what is now Poland; he grew up in Brooklyn.. He graduated in 1950, at age 16, from Stuyvesant High School in New York City. Cohen next studied at the Brooklyn College from 1950 to 1953, but he left without earning his bachelor's degree when he learned that he could start his graduate studies at the University of Chicago with just two years of college. At Chicago, Cohen completed his master's degree in mathematics in 1954 and his Doctor of Philosophy degree in 1958, under supervision of Antoni Zygmund. The title of his doctoral thesis was ''Topics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Hypothesis
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states: Or equivalently: In Zermelo–Fraenkel set theory with the axiom of choice (ZFC), this is equivalent to the following equation in aleph numbers: 2^=\aleph_1, or even shorter with beth numbers: \beth_1 = \aleph_1. The continuum hypothesis was advanced by Georg Cantor in 1878, and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940. The name of the hypothesis comes from the term '' continuum'' for the real numbers. History Cantor believed the continuum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Michigan
The University of Michigan (U-M, U of M, or Michigan) is a public university, public research university in Ann Arbor, Michigan, United States. Founded in 1817, it is the oldest institution of higher education in the state. The University of Michigan is one of the earliest American research universities and is a founding member of the Association of American Universities. In the fall of 2023, the university employed 8,189 faculty members and enrolled 52,065 students in its programs. The university is Carnegie Classification of Institutions of Higher Education, classified among "R1: Doctoral Universities – Very high research activity". It consists of nineteen colleges and offers 250 degree programs at the undergraduate and graduate levels. The university is Higher education accreditation in the United States, accredited by the Higher Learning Commission. In 2021, it ranked third among American universities in List of countries by research and development spending, research expe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cardinality
The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thumb is ''pollex'' (compare ''hallux'' for big toe), and the corresponding adjective for thumb is ''pollical''. Definition Thumb and fingers The English word ''finger'' has two senses, even in the context of appendages of a single typical human hand: 1) Any of the five terminal members of the hand. 2) Any of the four terminal members of the hand, other than the thumb. Linguistically, it appears that the original sense was the first of these two: (also rendered as ) was, in the inferred Proto-Indo-European language, a suffixed form of (or ), which has given rise to many Indo-European-family words (tens of them defined in English dictionaries) that involve, or stem from, concepts of fiveness. The thumb shares the following with each of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( ; 26March 191320September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered on discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He was known both for his social practice of mathematics, working with more than 500 collaborators, and for his eccentric lifestyle; ''Time'' magazine called him "The Oddball's Oddba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable Set
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (the number of elements of the set) is not greater than that of the natural numbers. A countable set that is not finite is said to be countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
