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Combinatorial
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 gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 finite se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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'' +&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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 abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 countin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analysis Of Algorithms
In computer science, the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to execute them. Usually, this involves determining a function that relates the size of an algorithm's input to the number of steps it takes (its time complexity) or the number of storage locations it uses (its space complexity). An algorithm is said to be efficient when this function's values are small, or grow slowly compared to a growth in the size of the input. Different inputs of the same size may cause the algorithm to have different behavior, so best, worst and average case descriptions might all be of practical interest. When not otherwise specified, the function describing the performance of an algorithm is usually an upper bound, determined from the worst case inputs to the algorithm. The term "analysis of algorithms" was coined by Donald Knuth. Algorithm analysis is an important part of a broader ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...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] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sushruta Samhita
The ''Sushruta Samhita'' (सुश्रुतसंहिता, IAST: ''Suśrutasaṃhitā'', literally "Suśruta's Compendium") is an ancient Sanskrit text on medicine and surgery, and one of the most important such treatises on this subject to survive from the ancient world. The ''Compendium of Suśruta'' is one of the foundational texts of Ayurveda (Indian traditional medicine), alongside the '' Charaka-Saṃhitā'', ''the Bheḷa-Saṃhitā'', and the medical portions of the Bower Manuscript. It is one of the two foundational Hindu texts on the medical profession that have survived from ancient India. The ''Suśrutasaṃhitā'' is of great historical importance because it includes historically unique chapters describing surgical training, instruments and procedures which is still followed by modern science of surgery. One of the oldest ''Sushruta Samhita'' palm-leaf manuscripts is preserved at the Kaiser Library, Nepal. History Date Over a century ago, the scholar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of culturally and linguistically related city-states and other territories. Most of these regions were officially unified only once, for 13 years, under Alexander the Great's empire from 336 to 323 BC (though this excludes a number of Greek city-states free from Alexander's jurisdiction in the western Mediterranean, around the Black Sea, Cyprus, and Cyrenaica). In Western history, the era of classical antiquity was immediately followed by the Early Middle Ages and the Byzantine period. Roughly three centuries after the Late Bronze Age collapse of Mycenaean Greece, Greek urban poleis began to form in the 8th century BC, ushering in the Archaic period and the colonization of the Mediterranean Basin. This was followed by the age of Classical G ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Historian
A historian is a person who studies and writes about the past and is regarded as an authority on it. Historians are concerned with the continuous, methodical narrative and research of past events as relating to the human race; as well as the study of all history in time. Some historians are recognized by publications or training and experience.Herman, A. M. (1998). Occupational outlook handbook: 1998–99 edition. Indianapolis: JIST Works. Page 525. "Historian" became a professional occupation in the late nineteenth century as research universities were emerging in Germany and elsewhere. Objectivity During the ''Irving v Penguin Books and Lipstadt'' trial, people became aware that the court needed to identify what was an "objective historian" in the same vein as the reasonable person, and reminiscent of the standard traditionally used in English law of "the man on the Clapham omnibus". This was necessary so that there would be a legal benchmark to compare and contrast the scholar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sushruta
Sushruta, or ''Suśruta'' (Sanskrit: सुश्रुत, IAST: , ) was an ancient Indian physician. The ''Sushruta Samhita'' (''Sushruta's Compendium''), a treatise ascribed to him, is one of the most important surviving ancient treatises on medicine and is considered a foundational text of Ayurveda. The treatise addresses all aspects of general medicine, but the impressive chapters on surgery have led to the false impression that this is its main topic. The translator G. D. Singhal dubbed Suśruta "the father of surgery" on account of these detailed accounts of surgery. The ''Compendium of Suśruta'' locates its author in Varanasi, India. Date The early scholar Rudolf Hoernle proposed that some concepts from the ''Suśruta-saṃhitā'' could be found in the '' Śatapatha-Brāhmaṇa'', which he dates to the 600 BCE, and this dating is still often repeated. However, during the last century, scholarship on the history of Indian medical literature has advanced substantially ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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