|
Antoni Zygmund
Antoni Zygmund (December 26, 1900 – May 30, 1992) was a Polish-American mathematician. He worked mostly in the area of mathematical analysis, including harmonic analysis, and he is considered one of the greatest analysts of the 20th century. Zygmund was responsible for creating the Chicago school of mathematical analysis together with his doctoral student Alberto Calderón, for which he was awarded the National Medal of Science in 1986. Biography Born in Warsaw, Zygmund obtained his Ph.D. from the University of Warsaw (1923) and was a professor at Stefan Batory University at Wilno from 1930 to 1939, when World War II broke out and Poland was occupied. In 1940 he managed to emigrate to the United States, where he became a professor at Mount Holyoke College in South Hadley, Massachusetts. In 1945–1947 he was a professor at the University of Pennsylvania, and from 1947, until his retirement, at the University of Chicago. He was a member of several scientific societies. Fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warsaw
Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at 1.86 million residents within a Warsaw metropolitan area, greater metropolitan area of 3.27 million residents, which makes Warsaw the List of cities in the European Union by population within city limits, 6th most-populous city in the European Union. The city area measures and comprises List of districts and neighbourhoods of Warsaw, 18 districts, while the metropolitan area covers . Warsaw is classified as an Globalization and World Cities Research Network#Alpha 2, alpha global city, a major political, economic and cultural hub, and the country's seat of government. It is also the capital of the Masovian Voivodeship. Warsaw traces its origins to a small fishing town in Masovia. The city rose to prominence in the late 16th cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Józef Marcinkiewicz
Józef Marcinkiewicz (; 30 March 1910 in Cimoszka, near Białystok, Poland – 1940 in Kharkiv, USSR) was a Polish mathematician. Life and career He was a student of Antoni Zygmund; and later worked with Juliusz Schauder, Stefan Kaczmarz and Raphaël Salem. He was a professor of the Stefan Batory University in Wilno. He enlisted in the Polish Army during the German invasion of Poland. In the aftermath of the simultaneous Soviet invasion of Poland, Marcinkiewicz was taken as a Polish POW to a Soviet camp in Starobielsk. The exact place and date of his death remain unknown, but it is believed that he died, when aged 30, murdered by the NKVD in Kharkiv. His parents, to whom he gave his manuscripts before the beginning of World War II, were transported to the Soviet Union in 1940 and later died of hunger in a camp. Their fate is described by ZygmundSee his commemoration in the volume of Marcinkiewicz's collected papers . described the last and his lost mathematical works ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematical model, models, and mathematics#Calculus and analysis, change. History One of the earliest known mathematicians was Thales of Miletus (); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales's theorem. The number of known mathematicians grew when Pythagoras of Samos () established the Pythagorean school, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
National Medal Of Science
The National Medal of Science is an honor bestowed by the President of the United States to individuals in science and engineering who have made important contributions to the advancement of knowledge in the fields of behavioral science, behavioral and social sciences, biology, chemistry, engineering, mathematics and physics. The twelve member presidential Committee on the National Medal of Science is responsible for selecting award recipients and is administered by the National Science Foundation (NSF). It is the highest science award in the United States. History The National Medal of Science was established on August 25, 1959, by an act of the Congress of the United States under . The medal was originally to honor scientists in the fields of the "physical, biological, mathematical, or engineering sciences". The Committee on the National Medal of Science was established on August 23, 1961, by Executive order (United States), executive order 10961 of President John F. Kennedy. O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calderón–Zygmund Kernel
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, whose kernel function ''K'' : R''n''×R''n'' → R is singular along the diagonal ''x'' = ''y''. Specifically, the singularity is such that , ''K''(''x'', ''y''), is of size , ''x'' − ''y'', −''n'' asymptotically as , ''x'' − ''y'', → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over , ''y'' − ''x'', > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on ''L''''p''(R''n''). The Hilbert transform The archetypal singular integral operator is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paley–Zygmund Inequality
In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its first two moments. The inequality was proved by Raymond Paley and Antoni Zygmund. Theorem: If ''Z'' ≥ 0 is a random variable with finite variance, and if 0 \le \theta \le 1, then : \operatorname( Z > \theta\operatorname ) \ge (1-\theta)^2 \frac. Proof: First, : \operatorname = \operatorname Z \, \mathbf_ + \operatorname Z \, \mathbf_ The first addend is at most \theta \operatorname /math>, while the second is at most \operatorname ^2 \operatorname( Z > \theta\operatorname ^ by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎ Related inequalities The Paley–Zygmund inequality can be written as : \operatorname( Z > \theta \operatorname ) \ge \frac. This can be improved. By the Cauchy–Schwarz inequality, : \operatorname .html" ;"title=" - \theta \operatorname[Z"> - \theta \operatorname[Z \le \operatorna ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marcinkiewicz–Zygmund Inequality
In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables. It is a generalization of the rule for the sum of variances of independent random variables to moments of arbitrary order. It is a special case of the Burkholder-Davis-Gundy inequality in the case of discrete-time martingales. Statement of the inequality Theorem J. Marcinkiewicz and A. Zygmund. Sur les fonctions indépendantes. ''Fund. Math.'', 28:60–90, 1937. Reprinted in Józef Marcinkiewicz, ''Collected papers'', edited by Antoni Zygmund, Panstwowe Wydawnictwo Naukowe, Warsaw, 1964, pp. 233–259. Yuan Shih Chow and Henry Teicher. ''Probability theory. Independence, interchangeability, martingales''. Springer-Verlag, New York, second edition, 1988. If \textstyle X_, \textstyle i=1,\ldots,n, are independent random variables such that \textstyle E\left( X_\right) =0 and \text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calderón–Zygmund Lemma
In mathematics, the Calderón–Zygmund lemma is a fundamental result in Fourier analysis, harmonic analysis, and singular integrals. It is named for the mathematicians Alberto Calderón and Antoni Zygmund. Given an integrable function , where denotes Euclidean space and denotes the complex numbers, the lemma (mathematics), lemma gives a precise way of partition of a set, partitioning into two set (mathematics), sets: one where is essentially small; the other a countable set, countable collection of cubes where is essentially large, but where some control of the function is retained. This leads to the associated Calderón–Zygmund decomposition of , wherein is written as the sum of "good" and "bad" functions, using the above sets. Covering lemma Let be integrable and be a positive constant. Then there exists an open set such that: :(1) is a disjoint union of open cubes, , such that for each , ::\alpha\le \frac \int_ , f(x), \, dx \leq 2^d \alpha. :(2) almost everyw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Integral
In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator : T(f)(x) = \int K(x,y)f(y) \, dy, whose kernel function ''K'' : R''n''×R''n'' → R is singular along the diagonal ''x'' = ''y''. Specifically, the singularity is such that , ''K''(''x'', ''y''), is of size , ''x'' − ''y'', −''n'' asymptotically as , ''x'' − ''y'', → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over , ''y'' − ''x'', > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on ''L''''p''(R''n''). The Hilbert transform The archetypal singular integral operator is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Guido Weiss
Guido Leopold Weiss (29 December 1928–25 December 2021 in St. Louis) was an American mathematician, working in analysis, especially Fourier analysis and harmonic analysis. Childhood Weiss was born in Trieste Italy into a Jewish family. His parents, Edoardo and Vonda Weiss, were both psychiatrists. Weiss was forced out of school at the age of 9, upon the passage of Italy's Italian Racial Laws, which forbade all Jewish children from attending public school. He attended a Jewish school in Rome until the end of 1939 when his father was sponsored by members of the Menninger family to emigrate to America. The family settled in Topeka, Kansas. Career Weiss studied at the University of Chicago, where he received in 1951 his master's degree and in 1956 under Antoni Zygmund his PhD with thesis ''On certain classes of function spaces and on the interpolation of sublinear operators''. At DePaul University he became an instructor in 1955, an assistant professor in 1956, and in 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias M
Elias ( ; ) is the hellenized version for the name of Elijah (; ; , or ), a prophet in the Northern Kingdom of Israel in the 9th century BC, mentioned in several holy books. Due to Elias' role in the scriptures and to many later associated traditions, the name is used as a personal name in numerous languages. Variants * Éilias Irish * Elia Italian, English * Elias Norwegian * Elías Icelandic * Éliás Hungarian * Elías Spanish * Eliáš, Elijáš Czech * Elijah, Elia, Ilyas, Elias Indonesian * Elias, Eelis, Eljas Finnish * Elias Danish, German, Swedish * Elias Portuguese * Elias, Iliya () Persian * Elias, Elis Swedish * Elias, Elyas (ኤሊያስ) Ethiopian * Elias, Elyas Philippines * Eliasz Polish * Élie French * Elija Slovene * Elijah English, Hebrew * Elis Welsh * Elisedd Welsh * Eliya (එලියා) Sinhala * Eliyas (Ілияс) Kazakh * Eliyahu, Eliya (אֵלִיָּהוּ, אליה) Biblical Hebrew, Hebrew * Elyās, Ilyās, Eliya (, ) Arabic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
