|
Mark Kac
Mark Kac ( ; Polish: ''Marek Kac''; August 3, 1914 – October 26, 1984) was a Polish American mathematician. His main interest was probability theory. His question, " Can one hear the shape of a drum?" set off research into spectral theory, the idea of understanding the extent to which the spectrum allows one to read back the geometry. (In the end, the answer was "no", in general.) Biography He was born to a Polish-Jewish family; their town, Kremenets (Polish: "Krzemieniec"), changed hands from the Russian Empire (by then Soviet Ukraine) to Poland after the Peace of Riga, when Kac was a child.Obituary in ''Rochester Democrat & Chronicle'', 11 November 1984 Kac completed his Ph.D. in mathematics at the Polish |
Krzemieniec
Kremenets ( uk, Крем'янець, Кременець, translit. ''Kremianets'', ''Kremenets''; pl, Krzemieniec; yi, קרעמעניץ, Kremenits) is a city in Ternopil Oblast (province) of western Ukraine. It is the administrative center of the Kremenets Raion (district), and lies 18 km north-east of the great Pochayiv Monastery. The city is situated in the historic region of Volhynia. It hosts the administration of Kremenets urban hromada, one of the hromadas of Ukraine. Population: History According to some sources the Kremenets fortress was built in the 8th or 9th century, and later became a part of Kievan Rus'. The first documented reference to the fortress is given in a Polish encyclopedic dictionary written in 1064. The first reference to Kremenets in Old Slavic literature dates from 1226 when the city's ruler, Mstislav the Bold, defeated the Hungarian army of King Andrew II nearby. During the Mongol invasion of Rus in 1240–1241, Kremenets was one of few c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman–Kac Formula
The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. In 1947, when Kac and Feynman were both Cornell faculty, Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions. The Feynman–Kac formula resulted, which proves rigorously the real case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still unproven. It offers a method of solving certain partial differential equations by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods. Theorem Consider the partial differential equation :\frac(x,t) + \mu(x,t) \frac(x,t) + \tfrac \sigma^2(x,t) \frac(x,t) -V(x,t) u(x,t) + f(x,t) = 0, defined for all x \in \mathbb and t \in , T/math>, subject to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish-Jewish
The history of the Jews in Poland dates back at least 1,000 years. For centuries, Poland was home to the largest and most significant Ashkenazi Jewish community in the world. Poland was a principal center of Jewish culture, because of the long period of statutory toleration, religious tolerance and Qahal, social autonomy which ended after the Partitions of Poland in the 18th century. During World War II there was a nearly complete genocide, genocidal destruction of the Polish Jewish community by Nazi Germany and its collaborators of various nationalities, during the German occupation of Poland between 1939 and 1945, called the Holocaust. Since the fall of communism in Poland, there has been a renewed interest in Jewish culture, featuring an annual Jewish Culture Festival, new study programs at Polish secondary schools and universities, and the opening of Warsaw's Museum of the History of Polish Jews. From the founding of the Kingdom of Poland (1025–1385), Kingdom of Poland i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was therefore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hearing The Shape Of A Drum
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of overtones, via the use of mathematical theory. "Can One Hear the Shape of a Drum?" is the title of a 1966 article by Mark Kac in the '' American Mathematical Monthly'' which made the question famous, though this particular phrasing originates with Lipman Bers. Similar questions can be traced back all the way to physicist Arthur Schuster in 1882. For his paper, Kac was given the Lester R. Ford Award in 1967 and the Chauvenet Prize in 1968. The frequencies at which a drumhead can vibrate depend on its shape. The Helmholtz equation calculates the frequencies if the shape is known. These frequencies are the eigenvalues of the Laplacian in the space. A central question is whether the shape can be predicted if the frequencies are known; for example, whether a Reuleaux triangle can be recognized in this way. Kac admitted that he did not know whethe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of the sample space is called an event. Central subjects in probability theory include discrete and continuous random variables, probability distributions, and stochastic processes (which provide mathematical abstractions of non-deterministic or uncertain processes or measured quantities that may either be single occurrences or evolve over time in a random fashion). Although it is not possible to perfectly predict random events, much can be said about their behavior. Two major results in probability ... [...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, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); 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' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) 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 mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish American
Polish Americans ( pl, Polonia amerykańska) are Americans who either have total or partial Polish ancestry, or are citizens of the Republic of Poland. There are an estimated 9.15 million self-identified Polish Americans, representing about 2.83% of the U.S. population. Polish Americans are the second-largest Central European ethnic group after German Americans, and the eighth largest ethnic group overall in the United States. The first Polish immigrants came to the Jamestown colony in 1608, twelve years before the Pilgrims arrived in Massachusetts. Two Polish volunteers, Casimir Pulaski and Tadeusz Kościuszko, led armies in the Revolutionary War and are remembered as American heroes. Overall, around 2.2 million Poles and Polish subjects immigrated into the United States, between 1820 and 1914, chiefly after national insurgencies and famine. They included former Polish citizens of Roman Catholic, Protestant, Jewish or other minority descent. Exact immigration figures are unk ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Language
Polish (Polish: ''język polski'', , ''polszczyzna'' or simply ''polski'', ) is a West Slavic language of the Lechitic group written in the Latin script. It is spoken primarily in Poland and serves as the native language of the Poles. In addition to being the official language of Poland, it is also used by the Polish diaspora. There are over 50 million Polish speakers around the world. It ranks as the sixth most-spoken among languages of the European Union. Polish is subdivided into regional dialects and maintains strict T–V distinction pronouns, honorifics, and various forms of formalities when addressing individuals. The traditional 32-letter Polish alphabet has nine additions (''ą'', ''ć'', ''ę'', ''ł'', ''ń'', ''ó'', ''ś'', ''ź'', ''ż'') to the letters of the basic 26-letter Latin alphabet, while removing three (x, q, v). Those three letters are at times included in an extended 35-letter alphabet, although they are not used in native words. The traditional ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birkhoff Prize
The George David Birkhoff Prize in applied mathematics is awarded – jointly by the American Mathematical Society (AMS) and the Society for Industrial and Applied Mathematics (SIAM) – in honour of George David Birkhoff (1884–1944). It is currently awarded every three years for an outstanding contribution to: "applied mathematics in the highest and broadest sense". The recipient of the prize has to be a member of one of the awarding societies, as well as a resident of the United States of America, Canada or Mexico. The prize was established in 1967 and currently (2020) amounts to US$5,000. Recipients See also * List of mathematics awards * Prizes named after people A prize is an award to be given to a person or a group of people (such as sporting teams and organizations) to recognize and reward their actions and achievements. Notes {{DEFAULTSOR ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chauvenet Prize
The Chauvenet Prize is the highest award for mathematical expository writing. It consists of a prize of $1,000 and a certificate, and is awarded yearly by the Mathematical Association of America in recognition of an outstanding expository article on a mathematical topic. The prize is named in honor of William Chauvenet and was established through a gift from J. L. Coolidge in 1925. The Chauvenet Prize was the first award established by the Mathematical Association of America. A gift from MAA president Walter B. Ford in 1928 allowed the award to be given every 3 years instead of the originally planned 5 years. Winners *1925 G. A. Bliss *1929 T. H. Hildebrandt *1932 G. H. Hardy *1935 Dunham Jackson *1938 G. T. Whyburn *1941 Saunders Mac Lane *1944 R. H. Cameron *1947 Paul Halmos *1950 Mark Kac *1953 E. J. McShane *1956 Richard H. Bruck *1960 Cornelius Lanczos *1963 Philip J. Davis *1964 Leon Henkin *1965 Jack K. Hale & Joseph P. LaSalle *1967 Guido Weiss *1968 Mark ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telegraph Process
In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise (also called popcorn noise or random telegraph signal). If the two possible values that a random variable can take are ''c_1'' and ''c_2'', then the process can be described by the following master equations: :\partial_t P(c_1, t, x, t_0)=-\lambda_1 P(c_1, t, x, t_0)+\lambda_2 P(c_2, t, x, t_0) and :\partial_t P(c_2, t, x, t_0)=\lambda_1 P(c_1, t, x, t_0)-\lambda_2 P(c_2, t, x, t_0). where \lambda_1 is the transition rate for going from state c_1 to state c_2 and \lambda_2 is the transition rate for going from going from state c_2 to state c_1. The process is also known under the names Kac process (after mathematician Mark Kac), and dichotomous random process. Solution The master equation is compactly written in a matrix form by introducing a vector \mathbf= x, t_0),P(c_2, t, x, t_0)/math>, :\frac=W\mathbf P where :W=\begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Probability Theory")
