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Herbert Robbins
Herbert Ellis Robbins (January 12, 1915 – February 12, 2001) was an American mathematician and statistician. He did research in topology, measure theory, statistics, and a variety of other fields. He was the co-author, with Richard Courant, of '' What is Mathematics?'', a popularization that is still () in print. The Robbins lemma, used in empirical Bayes methods, is named after him. Robbins algebras are named after him because of a conjecture (since proved) that he posed concerning Boolean algebras. The Robbins theorem, in graph theory, is also named after him, as is the Whitney–Robbins synthesis, a tool he introduced to prove this theorem. The well-known unsolved problem of minimizing in sequential selection the expected rank of the selected item under full information, sometimes referred to as the fourth secretary problem, also bears his name: Robbins' problem (of optimal stopping). Biography Robbins was born in New Castle, Pennsylvania. As an undergraduate, Rob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New Castle, Pennsylvania
New Castle is a city in Lawrence County, Pennsylvania, United States, and the county seat of Lawrence County. It is northwest of Pittsburgh, and near the Pennsylvania–Ohio border, just southeast of Youngstown, Ohio. As of the 2020 U.S. Census, the city had a population of 21,926. It is the commercial center of a fertile agricultural region, officially the New Castle micropolitan area, which had a population of 86,070 in 2020. New Castle also anchors the northwestern part of the Pittsburgh-New Castle-Weirton combined area. History In 1798, John Carlysle Stewart, a civil engineer, traveled to western Pennsylvania to resurvey the "donation lands", which had been reserved for veterans of the Revolutionary War. He discovered that the original survey had neglected to stake out approximately at the confluence of the Shenango River and Neshannock Creek, at that time a part of Allegheny County. The Indian town of Kuskusky was listed on early maps in this location. Claiming the land ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real analysis, mathematical physics, the calculus of variations and partial differential equations. He wrote textbooks widely used by generations of students of physics and mathematics. He is also known for founding the institute now bearing his name. Life and career Courant was born in Lublinitz, in the Prussian Province of Silesia. His parents were Siegmund Courant and Martha Courant ''née'' Freund of Oels. Edith Stein was Richard's cousin on the paternal side. During his youth his parents moved often, including to Glatz, then to Breslau and in 1905 to Berlin. He stayed in Breslau and entered the university there, then continued his studies at the University of Zürich and the University of Göttingen. He became David Hilbert's assist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doctorate
A doctorate (from Latin ''docere'', "to teach"), doctor's degree (from Latin ''doctor'', "teacher"), or doctoral degree is an academic degree awarded by universities and some other educational institutions, derived from the ancient formalism ''licentia docendi'' ("licence to teach"). In most countries, a research degree qualifies the holder to teach at university level in the degree's field or work in a specific profession. There are a number of doctoral degrees; the most common is the Doctor of Philosophy (PhD), awarded in many different fields, ranging from the humanities to scientific disciplines. In the United States and some other countries, there are also some types of technical or professional degrees that include "doctor" in their name and are classified as a doctorate in some of those countries. Professional doctorates historically came about to meet the needs of practitioners in a variety of disciplines. Many universities also award honorary doctorates to individuals d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marston Morse
Harold Calvin Marston Morse (March 24, 1892 – June 22, 1977) was an American mathematician best known for his work on the ''calculus of variations in the large'', a subject where he introduced the technique of differential topology now known as Morse theory. The Morse–Palais lemma, one of the key results in Morse theory, is named after him, as is the Thue–Morse sequence, an infinite binary sequence with many applications. In 1933 he was awarded the Bôcher Memorial Prize for his work in mathematical analysis. Biography He was born in Waterville, Maine to Ella Phoebe Marston and Howard Calvin Morse in 1892. He received his bachelor's degree from Colby College (also in Waterville) in 1914. At Harvard University, he received both his master's degree in 1915 and his PhD in 1917. He wrote his PhD thesis, ''Certain Types of Geodesic Motion of a Surface of Negative Curvature'', under the direction of George David Birkhoff. Morse was a Benjamin Peirce Instructor at Harvard in 191 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pennsylvania
Pennsylvania (; ( Pennsylvania Dutch: )), officially the Commonwealth of Pennsylvania, is a state spanning the Mid-Atlantic, Northeastern, Appalachian, and Great Lakes regions of the United States. It borders Delaware to its southeast, Maryland to its south, West Virginia to its southwest, Ohio to its west, Lake Erie and the Canadian province of Ontario to its northwest, New York to its north, and the Delaware River and New Jersey to its east. Pennsylvania is the fifth-most populous state in the nation with over 13 million residents as of 2020. It is the 33rd-largest state by area and ranks ninth among all states in population density. The southeastern Delaware Valley metropolitan area comprises and surrounds Philadelphia, the state's largest and nation's sixth most populous city. Another 2.37 million reside in Greater Pittsburgh in the southwest, centered around Pittsburgh, the state's second-largest and Western Pennsylvania's largest city. The state's su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robbins' Problem (of Optimal Stopping)
In probability theory, Robbins' problem of optimal stopping, named after Herbert Robbins, is sometimes referred to as the fourth secretary problem or the problem of minimizing the expected rank with full information. Its statement is as follows. Let ''X''1, ... , ''X''''n'' be independent, identically distributed random variables, uniform on , 1 We observe the ''X''''k'''s sequentially and must stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation, and what is its corresponding value? The general solution to this full-information expected rank problem is unknown. The major difficulty is that the problem is fully history-dependent, that is, the optimal rule depends at every stage on all preceding values, and not only on simpler sufficient statistics of these. Only bounds are known for the limiting value ''v'' as ''n'' goes to infinity, namely 1.908 < ''v'' <&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secretary Problem
The secretary problem demonstrates a scenario involving optimal stopping theory For French translation, secover storyin the July issue of ''Pour la Science'' (2009). that is studied extensively in the fields of applied probability, statistics, and decision theory. It is also known as the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem. The basic form of the problem is the following: imagine an administrator who wants to hire the best secretary out of n rankable applicants for a position. The applicants are interviewed one by one in random order. A decision about each particular applicant is to be made immediately after the interview. Once rejected, an applicant cannot be recalled. During the interview, the administrator gains information sufficient to rank the applicant among all applicants interviewed so far, but is unaware of the quality of yet unseen applicants. The question is about the optimal strategy ( sto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ear Decomposition
In graph theory, an ear of an undirected graph ''G'' is a path ''P'' where the two endpoints of the path may coincide, but where otherwise no repetition of edges or vertices is allowed, so every internal vertex of ''P'' has degree two in ''G''. An ear decomposition of an undirected graph ''G'' is a partition of its set of edges into a sequence of ears, such that the one or two endpoints of each ear belong to earlier ears in the sequence and such that the internal vertices of each ear do not belong to any earlier ear. Additionally, in most cases the first ear in the sequence must be a cycle. An open ear decomposition or a proper ear decomposition is an ear decomposition in which the two endpoints of each ear after the first are distinct from each other. Ear decompositions may be used to characterize several important graph classes, and as part of efficient graph algorithms. They may also be generalized from graphs to matroids. Characterizing graph classes Several important classes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robbins Theorem
Robbins may refer to: People * Robbins (name), a surname Fictional characters * Al Robbins, medical doctor in ''CSI: Crime Scene Investigation'' * Arizona Robbins, surgeon in ''Grey's Anatomy'' * Ashley Mizuki Robbins, protagonist in the video games '' Another Code: Two Memories'' and '' Another Code: R – A Journey into Lost Memories'' * Jack Robbins, character on ''EastEnders'' television series * Lily Robbins, character in ''The Lily Series'' * Parker Robbins, comic book character Places Antarctica * Robbins Hill, a hill at the terminus of Blue Glacier Australia * Robbins Passage and Boullanger Bay Important Bird Area, Tasmania USA * Robbins, California, town in Sutter County * Robbins, Illinois, village in Cook County * Robbins, Michigan, an unincorporated community * Robbins, Missouri, an unincorporated community * Robbins, North Carolina, city in Moore County * Robbins, Tennessee, unincorporated community in Scott County *Robbins, Virginia, ghost town Other * Bas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boolean Algebra (structure)
In abstract algebra, a Boolean algebra or Boolean lattice is a complemented distributive lattice. This type of algebraic structure captures essential properties of both set operations and logic operations. A Boolean algebra can be seen as a generalization of a power set algebra or a field of sets, or its elements can be viewed as generalized truth values. It is also a special case of a De Morgan algebra and a Kleene algebra (with involution). Every Boolean algebra gives rise to a Boolean ring, and vice versa, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry of the theory described by the duality principle. __TOC__ History The term "Boolean algebra" honors George Boole (1815–1864), a self-educated English ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robbins Algebra
In abstract algebra, a Robbins algebra is an algebra containing a single binary operation, usually denoted by \lor, and a single unary operation usually denoted by \neg. These operations satisfy the following axioms: For all elements ''a'', ''b'', and ''c'': # Associativity: a \lor \left(b \lor c \right) = \left(a \lor b \right) \lor c # Commutativity: a \lor b = b \lor a # ''Robbins equation'': \neg \left( \neg \left(a \lor b \right) \lor \neg \left(a \lor \neg b \right) \right) = a For many years, it was conjectured, but unproven, that all Robbins algebras are Boolean algebras. This was proved in 1996, so the term "Robbins algebra" is now simply a synonym for "Boolean algebra". History In 1933, Edward Huntington proposed a new set of axioms for Boolean algebras, consisting of (1) and (2) above, plus: *''Huntington's equation'': \neg(\neg a \lor b) \lor \neg(\neg a \lor \neg b) = a. From these axioms, Huntington derived the usual axioms of Boolean algebra. Very soon thereafte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
