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Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is ...
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Schröder–Hipparchus Number
In combinatorics, the Schröder–Hipparchus numbers form an integer sequence that can be used to count the number of plane trees with a given set of leaves, the number of ways of inserting parentheses into a sequence, and the number of ways of dissecting a convex polygon into smaller polygons by inserting diagonals. These numbers begin :1, 1, 3, 11, 45, 197, 903, 4279, 20793, 103049, ... . They are also called the super-Catalan numbers, the little Schröder numbers, or the Hipparchus numbers, after Eugène Charles Catalan and his Catalan numbers, Ernst Schröder and the closely related Schröder numbers, and the ancient Greek mathematician Hipparchus who appears from evidence in Plutarch to have known of these numbers. Combinatorial enumeration applications The Schröder–Hipparchus numbers may be used to count several closely related combinatorial objects:.. *The ''n''th number in the sequence counts the number of different ways of subdividing of a polygon with ''n'' +& ...
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Discrete Mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with ...
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Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ...
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Counting
Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term '' enumeration'' refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...). There is archaeological evidence suggesting that humans have been count ...
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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 ...
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Leon Mirsky
Leonid Mirsky (19 December 1918 – 1 December 1983) was a Russian-British mathematician who worked in number theory, linear algebra, and combinatorics.... Mirsky's theorem is named after him. Biography Mirsky was born in Russia on 19 December 1918 to a medical family, but his parents sent him to live with his aunt and uncle, a wool merchant in Germany, when he was eight. His uncle's family moved to Bradford, England in 1933, bringing Mirsky with them. He studied at Herne Bay High School and King's College, London, graduating in 1940. Because of the evacuation of London during the Blitz, students at King's College were moved to Bristol University, where Mirsky earned a master's degree. He took a short-term faculty position at Sheffield University in 1942, and then a similar position in Manchester; he returned to Sheffield in 1945, where (except for a term as visiting faculty at Bristol) he would stay for the rest of his career. He became a lecturer in 1947, earned a Ph.D. from Sh ...
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Statistical Physics
Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic nature. Its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion. Statistical mechanics develop the phenomenological results of thermodynamics from a probabilistic examination of the underlying microscopic systems. Historically, one of the first topics in physics where statistical methods were applied was the field of classical mechanics, which is concerned with the motion of particles or objects when subjected to a force. Scope Statistical physics explains and quanti ...
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Evolutionary Biology
Evolutionary biology is the subfield of biology that studies the evolutionary processes (natural selection, common descent, speciation) that produced the diversity of life on Earth. It is also defined as the study of the history of life forms on Earth. Evolution is based on the theory that all species are related and they gradually change over time. In a population, the genetic variations affect the physical characteristics i.e. phenotypes of an organism. These changes in the phenotypes will be an advantage to some organisms, which will then be passed onto their offspring. Some examples of evolution in species over many generations are the Peppered Moth and Flightless birds. In the 1930s, the discipline of evolutionary biology emerged through what Julian Huxley called the modern synthesis of understanding, from previously unrelated fields of biological research, such as genetics and ecology, systematics, and paleontology. The importance of studying Evolutionary bio ...
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Ancient History
Ancient history is a time period from the beginning of writing and recorded human history to as far as late antiquity. The span of recorded history is roughly 5,000 years, beginning with the Sumerian cuneiform script. Ancient history covers all continents inhabited by humans in the period 3000 BCAD 500. The three-age system periodizes ancient history into the Stone Age, the Bronze Age, and the Iron Age, with recorded history generally considered to begin with the Bronze Age. The start and end of the three ages varies between world regions. In many regions the Bronze Age is generally considered to begin a few centuries prior to 3000 BC, while the end of the Iron Age varies from the early first millennium BC in some regions to the late first millennium AD in others. During the time period of ancient history, the world population was already exponentially increasing due to the Neolithic Revolution, which was in full progress. While in 10,000 BC, the world population stood at ...
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Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: :\. Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set is called finite if there exists a bijection :f\colon S\to\ for some natural number . The number is the set's cardinality, denoted as . The empty set o ...
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Hipparchus
Hipparchus (; el, Ἵππαρχος, ''Hipparkhos'';  BC) was a Greek astronomer, geographer, and mathematician. He is considered the founder of trigonometry, but is most famous for his incidental discovery of the precession of the equinoxes. Hipparchus was born in Nicaea, Bithynia, and probably died on the island of Rhodes, Greece. He is known to have been a working astronomer between 162 and 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity. He was the first whose quantitative and accurate models for the motion of the Sun and Moon survive. For this he certainly made use of the observations and perhaps the mathematical techniques accumulated over centuries by the Babylonians and by Meton of Athens (fifth century BC), Timocharis, Aristyllus, Aristarchus of Samos, and Eratosthenes, among others. He developed trigonometry and constructed trigonometric tables, and he solved ...
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Chrysippus
Chrysippus of Soli (; grc-gre, Χρύσιππος ὁ Σολεύς, ; ) was a Greek Stoic philosopher. He was a native of Soli, Cilicia, but moved to Athens as a young man, where he became a pupil of the Stoic philosopher Cleanthes. When Cleanthes died, around 230 BC, Chrysippus became the third head of the Stoic school. A prolific writer, Chrysippus expanded the fundamental doctrines of Cleanthes' mentor Zeno of Citium, the founder and first head of the school, which earned him the title of the Second Founder of Stoicism. Chrysippus excelled in logic, the theory of knowledge, ethics, and physics. He created an original system of propositional logic in order to better understand the workings of the universe and role of humanity within it. He adhered to a deterministic view of fate, but nevertheless sought a role for personal freedom in thought and action. Ethics, he thought, depended on understanding the nature of the universe, and he taught a therapy of extirpating th ...
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