Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Americans are nationals and citizens of the United States of America. Although nationals and citizens make up the majority of Americans, some dual citizens and permanent residents may claim American nationality; the United States is home to people of many different ethnic origins. As a result, American culture and law does not equate nationality with race or ethnicity, but with citizenship and permanent allegiance. English-speakers, speakers of many other languages use the term "American" to mean people of the United States; the word "American" can refer to people from the Americas in general. The majority of Americans or their ancestors immigrated to America or are descended from people who were brought as slaves within the past five centuries, with the exception of the Native American population and people from Hawaii, Puerto Rico and the Philippine Islands, who became American through expansion of the country in the 19th century, additionally America expanded into American Samoa, the U. S. Virgin Islands and Northern Mariana Islands in the 20th century.
Despite its multi-ethnic composition, the culture of the United States held in common by most Americans can be referred to as mainstream American culture, a Western culture derived from the traditions of Northern and Western European colonists and immigrants. It includes influences of African-American culture. Westward expansion integrated the Creoles and Cajuns of Louisiana and the Hispanos of the Southwest and brought close contact with the culture of Mexico. Large-scale immigration in the late 19th and early 20th centuries from Southern and Eastern Europe introduced a variety of elements. Immigration from Asia and Latin America has had impact. A cultural melting pot, or pluralistic salad bowl, describes the way in which generations of Americans have celebrated and exchanged distinctive cultural characteristics. In addition to the United States and people of American descent can be found internationally; as many as seven million Americans are estimated to be living abroad, make up the American diaspora.
The United States of America is a diverse country and ethnically. Six races are recognized by the U. S. Census Bureau for statistical purposes: White, American Indian and Alaska Native, Black or African American, Native Hawaiian and Other Pacific Islander, people of two or more races. "Some other race" is an option in the census and other surveys. The United States Census Bureau classifies Americans as "Hispanic or Latino" and "Not Hispanic or Latino", which identifies Hispanic and Latino Americans as a racially diverse ethnicity that comprises the largest minority group in the nation. People of European descent, or White Americans, constitute the majority of the 308 million people living in the United States, with 72.4% of the population in the 2010 United States Census. They are considered people who trace their ancestry to the original peoples of Europe, the Middle East, North Africa. Of those reporting to be White American, 7,487,133 reported to be Multiracial. Additionally, there are Latinos.
Non-Hispanic Whites are the majority in 46 states. There are four minority-majority states: California, New Mexico, Hawaii. In addition, the District of Columbia has a non-white majority; the state with the highest percentage of non-Hispanic White Americans is Maine. The largest continental ancestral group of Americans are that of Europeans who have origins in any of the original peoples of Europe; this includes people via African, North American, Central American or South American and Oceanian nations that have a large European descended population. The Spanish were some of the first Europeans to establish a continuous presence in what is now the United States in 1565. Martín de Argüelles born 1566, San Agustín, La Florida a part of New Spain, was the first person of European descent born in what is now the United States. Twenty-one years Virginia Dare born 1587 Roanoke Island in present-day North Carolina, was the first child born in the original Thirteen Colonies to English parents. In the 2017 American Community Survey, German Americans, Irish Americans, English Americans and Italian Americans were the four largest self-reported European ancestry groups in the United States forming 35.1% of the total population.
However, the English Americans and British Americans demography is considered a serious under-count as they tend to self-report and identify as "Americans" due to the length of time they have inhabited America. This is over-represented in the Upland South, a region, settled by the British. Overall, as the largest group, European Americans have the lowest poverty rate and the second highest educational attainment levels, median household income, median personal income of any racial demographic in the nation. According to the American Jewish Archives and the Arab American National Museum, some of the first Middle Easterners and North Africans arrived in the Americas between the late 15th and mid-16th centuries. Many were fleeing ethnic or ethnoreligious persecution during the Spanish Inquisition, a few were taken to the Americas as slaves. In 2014, The United States Census Bureau began finalizing the ethnic classification of MENA populations. According to the Arab American Institute, Arab
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 and social sciences, chemistry, engineering 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; the National Medal of Science was established on August 25, 1959, by an act of the Congress of the United States under Pub. L. 86–209. The medal was to honor scientists in the fields of the "physical, mathematical, or engineering sciences"; the Committee on the National Medal of Science was established on August 23, 1961, by executive order 10961 of President John F. Kennedy. On January 7, 1979, the American Association for the Advancement of Science passed a resolution proposing that the medal be expanded to include the social and behavioral sciences.
In response, Senator Ted Kennedy introduced the Science and Technology Equal Opportunities Act into the Senate on March 7, 1979, expanding the medal to include these scientific disciplines as well. President Jimmy Carter's signature enacted this change as Public Law 96-516 on December 12, 1980. In 1992, the National Science Foundation signed a letter of agreement with the National Science and Technology Medals Foundation that made the National Science and Technology Medals Foundation the metaorganization over both the National Medal of Science and the similar National Medal of Technology; the first National Medal of Science was awarded on February 18, 1963, for the year 1962 by President John F. Kennedy to Theodore von Kármán for his work at the Caltech Jet Propulsion Laboratory; the citation accompanying von Kármán's award reads: For his leadership in the science and engineering basic to aeronautics. The first woman to receive a National Medal of Science was Barbara McClintock, awarded for her work on plant genetics in 1970.
Although Public Law 86-209 provides for 20 recipients of the medal per year, it is typical for 8–15 accomplished scientists and engineers to receive this distinction each year. There have been a number of years; those years include: 1985, 1984, 1980, 1978, 1977, 1972 and 1971. The awards ceremony is organized by the Office of Technology Policy, it is presided by the sitting United States president. Each year the National Science Foundation sends out a call to the scientific community for the nomination of new candidates for the National Medal of Science. Individuals are nominated by their peers with each nomination requiring three letters of support from individuals in science and technology. Nominations are sent to the Committee of the National Medal of Science, a board composed of fourteen presidential appointees comprising twelve scientists, two ex officio members—the director of the Office of Science and Technology Policy and the president of the National Academy of Sciences. According to the Committee, successful candidates must be U.
S. citizens or permanent residents who are applying for U. S. citizenship, who have done work of outstanding merit or that has had a major impact on scientific thought in their field. The Committee values those who promote the general advancement of science and individuals who have influenced science education, although these traits are less important than groundbreaking or thought-provoking research; the nomination of a candidate is effective for three years. The Committee makes their recommendations to the President for the final awarding decision; the National Medal of Science depicts Man, surrounded by earth and sky, contemplating and struggling to understand Nature. The crystal in his hand represents the universal order and suggests the basic unit of living things; the formula being outlined in the sand symbolizes scientific abstraction. National Medal of Arts National Medal of Technology and Innovation National Science Foundation Searchable Database of National Medal of Science Recipients National Science & Technology Medals Foundation Using the National Medal of Science to recognize advances in psychology
Ashkenazi Jews known as Ashkenazic Jews or Ashkenazim, are a Jewish diaspora population who coalesced in the Holy Roman Empire around the end of the first millennium. The traditional diaspora language of Ashkenazi Jews is Yiddish, developed after they had moved into northern Europe: beginning with Germany and France in the Middle Ages. For centuries they used Hebrew only as a sacred language, until the revival of Hebrew as a common language in Israel. Throughout their time in Europe, Ashkenazim have made many important contributions to its philosophy, literature, art and science; the term "Ashkenazi" refers to Jewish settlers who established communities along the Rhine river in Western Germany and in Northern France dating to the Middle Ages. Once there, they adapted traditions carried from Babylon, the Holy Land, the Western Mediterranean to their new environment; the Ashkenazi religious rite developed in cities such as Mainz and Troyes. The eminent French Rishon rabbi Shlomo Itzhaki would have a significant influence on the Jewish religion.
In the late Middle Ages, due to religious persecution, the majority of the Ashkenazi population shifted eastward, moving out of the Holy Roman Empire into the areas part of the Polish–Lithuanian Commonwealth. In the course of the late 18th and 19th centuries, those Jews who remained in or returned to the German lands generated a cultural reorientation; the Holocaust of the Second World War decimated the Ashkenazim, affecting every Jewish family. It is estimated that in the 11th century Ashkenazi Jews composed three percent of the world's total Jewish population, while an estimate made in 1930 had them as 92 percent of the world's Jews. Prior to the Holocaust, the number of Jews in the world stood at 16.7 million. Statistical figures vary for the contemporary demography of Ashkenazi Jews, ranging from 10 million to 11.2 million. Sergio Della Pergola, in a rough calculation of Sephardic and Mizrahi Jews, implies that Ashkenazi Jews make up less than 74% of Jews worldwide. Other estimates place Ashkenazi Jews as making up about 75% of Jews worldwide.
Genetic studies on Ashkenazim—researching both their paternal and maternal lineages—suggest a predominant amount of shared Middle Eastern ancestry, complemented by varying percentages of European admixture. These studies have arrived at diverging conclusions regarding both the degree and the sources of their European ancestry, have focused on the extent of the European genetic origin observed in Ashkenazi maternal lineages. Ashkenazi Jews are popularly contrasted with Sephardi Jews, who descend from Jews who settled in the Iberian Peninsula, Mizrahi Jews, who descend from Jews who remained in the Middle East; the name Ashkenazi derives from the biblical figure of Ashkenaz, the first son of Gomer, son of Japhet, son of Noah, a Japhetic patriarch in the Table of Nations. The name of Gomer has been linked to the ethnonym Cimmerians. Biblical Ashkenaz is derived from Assyrian Aškūza, a people who expelled the Cimmerians from the Armenian area of the Upper Euphrates, whose name is associated with the name of the Scythians.
The intrusive n in the Biblical name is due to a scribal error confusing a vav ו with a nun נ. In Jeremiah 51:27, Ashkenaz figures as one of three kingdoms in the far north, the others being Minni and Ararat corresponding to Urartu, called on by God to resist Babylon. In the Yoma tractate of the Babylonian Talmud the name Gomer is rendered as Germania, which elsewhere in rabbinical literature was identified with Germanikia in northwestern Syria, but became associated with Germania. Ashkenaz is linked to Scandza/Scanzia, viewed as the cradle of Germanic tribes, as early as a 6th-century gloss to the Historia Ecclesiastica of Eusebius. In the 10th-century History of Armenia of Yovhannes Drasxanakertc'i Ashkenaz was associated with Armenia, as it was in Jewish usage, where its denotation extended at times to Adiabene, Khazaria and areas to the east, his contemporary Saadia Gaon identified Ashkenaz with the Saquliba or Slavic territories, such usage covered the lands of tribes neighboring the Slavs, Eastern and Central Europe.
In modern times, Samuel Krauss identified the Biblical "Ashkenaz" with Khazaria. Sometime in the Early Medieval period, the Jews of central and eastern Europe came to be called by this term. Conforming to the custom of designating areas of Jewish settlement with biblical names, Spain was denominated Sefarad, France was called Tsarefat, Bohemia was called the Land of Canaan. By the high medieval period, Talmudic commentators like Rashi began to use Ashkenaz/Eretz Ashkenaz to designate Germany, earlier known as Loter, where in the Rhineland communities of Speyer and Mainz, the most important Jewish communities arose. Rashi uses leshon Ashkenaz to describe German speech, Byzantium and Syrian Jewish letters referred to the Crusaders as Ashkenazim. Given the close links between the Jewish communities of France a
A mathematician is someone who uses an extensive knowledge of mathematics in his or her work to solve mathematical problems. Mathematics is concerned with numbers, quantity, space and change. One of the earliest known mathematicians was Thales of Miletus, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' 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", with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypatia of Alexandria, she succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells.
Science and mathematics in the Islamic world during the Middle Ages followed various models and modes of funding varied based on scholars. It was extensive patronage and strong intellectual policies implemented by specific rulers that allowed scientific knowledge to develop in many areas. Funding for translation of scientific texts in other languages was ongoing throughout the reign of certain caliphs, it turned out that certain scholars became experts in the works they translated and in turn received further support for continuing to develop certain sciences; as these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many scholars working under Muslim rule in medieval times is that they were polymaths. Examples include the work on optics and astronomy of Ibn al-Haytham; the Renaissance brought an increased emphasis on science to Europe.
During this period of transition from a feudal and ecclesiastical culture to a predominantly secular one, many notable mathematicians had other occupations: Luca Pacioli. As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in the seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle, at Cambridge where Isaac Newton was Lucasian Professor of Mathematics & Physics. Moving into the 19th century, the objective of universities all across Europe evolved from teaching the “regurgitation of knowledge” to “encourag productive thinking.” In 1810, Humboldt convinced the King of Prussia to build a university in Berlin based on Friedrich Schleiermacher’s liberal ideas. Thus and laboratories started to evolve. British universities of this period adopted some approaches familiar to the Italian and German universities, but as they enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment, the same influences that inspired Humboldt.
The Universities of Oxford and Cambridge emphasized the importance of research, arguably more authentically implementing Humboldt’s idea of a university than German universities, which were subject to state authority. Overall, science became the focus of universities in the 20th centuries. Students could conduct research in seminars or laboratories and began to produce doctoral theses with more scientific content. According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge; the German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of the kind of research done by private and individual scholars in Great Britain and France. In fact, Rüegg asserts that the German system is responsible for the development of the modern research university because it focused on the idea of “freedom of scientific research and study.” Mathematicians cover a breadth of topics within mathematics in their undergraduate education, proceed to specialize in topics of their own choice at the graduate level.
In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics. Mathematicians involved with solving problems with applications in real life are called applied mathematicians. Applied mathematicians are mathematical scientists who, with their specialized knowledge and professional methodology, approach many of the imposing problems presented in related scientific fields. With professional focus on a wide variety of problems, theoretical systems, localized constructs, applied mathematicians work in the study and formulation of mathematical models. Mathematicians and applied mathematicians are considered to be two of the STEM careers; the discipline of applied mathematics concerns
In mathematics convolution is a mathematical operation on two functions to produce a third function that expresses how the shape of one is modified by the other. The term convolution refers to the process of computing it; some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, it differs from cross-correlation only in that either f or g is reflected about the y-axis. For continuous functions, the cross-correlation operator is the adjoint of the convolution operator. Convolution has applications that include probability, computer vision, natural language processing and signal processing and differential equations; the convolution can be defined for functions on Euclidean space, other groups. For example, periodic functions, such as the discrete-time Fourier transform, can be defined on a circle and convolved by periodic convolution. A discrete convolution can be defined for functions on the set of integers. Generalizations of convolution have applications in the field of numerical analysis and numerical linear algebra, in the design and implementation of finite impulse response filters in signal processing.
Computing the inverse of the convolution operation is known as deconvolution. The convolution of f and g is written f ∗ g, using an star, it is defined as the integral of the product of the two functions after one is shifted. As such, it is a particular kind of integral transform: An equivalent definition is: ≜ ∫ − ∞ ∞ f g d τ. While the symbol t is used above, it need not represent the time domain, but in that context, the convolution formula can be described as a weighted average of the function f at the moment t where the weighting is given by g shifted by amount t. As t changes, the weighting function emphasizes different parts of the input function. For functions f, g supported on only [0, ∞), the integration limits can be truncated, resulting in: = ∫ 0 t f g d τ for f, g: [ 0, ∞ ) → R. For the multi-dimensional formulation of convolution, see domain of definition. A common engineering convention is: f ∗ g ≜ ∫ − ∞ ∞ f g d τ ⏟, which has to be interpreted to avoid confusion. For instance, f ∗ g is equivalent to.
Convolution describes the output of an important class of operations known as linear time-invariant. See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created; the existing ones are only modified. In other words, the output transform is the pointwise product of the input transform with a third transform. See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms. One of the earliest uses of the convolution integral appeared in D'Alembert's derivation of Taylor's theorem in Recherches sur différents points importants du système du monde, published in 1754. An expression of the type: ∫ f ⋅ g d u is used by Sylvestre François Lacroix on page 505 of his book entitled Treatise on differences and series, the last of 3 volumes of the encyclopedic series: Traité du calcul différentiel et du calcul intégral, Chez Courcier, Paris, 1797–1800.
Soon thereafter, convolution operations appear in the works of Pierre Simon Laplace, Jean-Baptiste Joseph Fourier, Siméon Denis Poisson, others. The term itself did not come into wide use until the 60s. Prior to that it was sometimes known as Faltung, composition product, superposition integral, Carson's integral, yet it appears as early as 1903. The o
Charles Louis Fefferman is an American mathematician at Princeton University. His primary field of research is mathematical analysis. A child prodigy, Fefferman entered the University of Maryland at age 14, had written his first scientific paper by the age of 15, he graduated with degrees in math and physics at 17, earned his PhD in mathematics three years from Princeton University, under Elias Stein. Fefferman achieved a full professorship at the University of Chicago at the age of 22, making him the youngest full professor appointed in the United States. At 24, he returned to Princeton as a full professor -- a position, he won the Alan T. Waterman Award in 1976 and the Fields Medal in 1978 for his work in mathematical analysis convergence and divergence, he was elected to the National Academy of Sciences in 1979. He was appointed the Herbert Jones Professor at Princeton in 1984. In addition to the above, his honors include the Salem Prize in 1971, the Bôcher Memorial Prize in 2008, the Bergman Prize in 1992, the Wolf Prize in Mathematics for 2017, as well as election to the American Academy of Arts and Sciences.
Fefferman contributed several innovations that revised the study of multidimensional complex analysis by finding fruitful generalisations of classical low-dimensional results. Fefferman's work on partial differential equations, Fourier analysis, in particular convergence, divergence, singular integrals and Hardy spaces earned him a Fields Medal at the International Congress of Mathematicians at Helsinki in 1978, he was a Plenary Speaker of the ICM in 1974 in Vancouver. His early work included a study of the asymptotics of the Bergman kernel off the boundaries of pseudoconvex domains in C n, he has studied mathematical physics, harmonic analysis, fluid dynamics, neural networks, mathematical finance and spectral analysis, amongst others. Charles Fefferman and his wife Julie have two daughters and Lainie. Lainie Fefferman is a composer, taught math at Saint Ann's School and holds a degree in music from Yale University as well as a Ph. D. in music composition from Princeton. She has an interest in Middle Eastern music.
Nina is a computational biologist whose research is concerned with the application of mathematical models to complex biological systems. Charles Fefferman's brother, Robert Fefferman, is a mathematician and former Dean of the Physical Sciences Division at the University of Chicago. Fefferman's most cited papers, in the order of citations, include the following. Fefferman, C.. "Weighted norm inequalities for maximal functions and singular integrals", Studia Mathematica, 51: 241–250 Fefferman, C.. "Some maximal inequalities", American Journal of Mathematics, 93: 107–115, doi:10.2307/2373450, JSTOR 2373450 Fefferman, Charles, "The Bergman kernel and biholomorphic mappings of pseudoconvex domains", Inventiones mathematicae, 26: 1–65, doi:10.1007/bf01406845 Fefferman, Charles L. "The uncertainty principle", Bulletin of the American Mathematical Society, 9: 129–206, doi:10.1090/s0273-0979-1983-15154-6 Fefferman, Charles, "Inequalities for singular convolution operators", Acta Mathematica, 124: 9–36, doi:10.1007/bf02394567 Constantin, P..
Charles Fefferman at the Mathematics Genealogy Project Charles Fefferman Curriculum Vitae "Ad Honorem Charles Fefferman". Notices of the American Mathematical Society. 64: 1254–1273. December 2017