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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.
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values. Discrete mathematics therefore excludes topics in "continuous mathematics" such as 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. However, there is no exact definition of the term "discrete mathematics." Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions.
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity, as it relates to astronomy and the geodesy of the Earth, and later in the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th century and the 19th century.
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.
Mathematics includes the study of such topics as numbers, formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes. There is no general consensus about its exact scope or epistemological status.
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points to Euclidean points, and vice-versa.
Synthetic geometry is the study of geometry without the use of coordinates or formulae. It relies on the axiomatic method and the tools directly related to them, that is, compass and straightedge, to draw conclusions and solve problems.
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations.
Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.
Mathematics encompasses a growing variety and depth of subjects over its history, and comprehension of it requires a system to categorize and organize these various subjects into a more general areas of mathematics. A number of different classification schemes have arisen, and though they share some similarities, there are differences due in part to the different purposes they serve.
A spatial network is a graph in which the vertices or edges are spatial elements associated with geometric objects, i.e., the nodes are located in a space equipped with a certain metric. The simplest mathematical realization of spatial network is a lattice or a random geometric graph, where nodes are distributed uniformly at random over a two-dimensional plane; a pair of nodes are connected if the Euclidean distance is smaller than a given neighborhood radius. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks and biological neural networks are all examples where the underlying space is relevant and where the graph's topology alone does not contain all the information. Characterizing and understanding the structure, resilience and the evolution of spatial networks is crucial for many different fields ranging from urbanism to epidemiology.
In mathematics, a space is a set with some added structure.
Geometry is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer.
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.
In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.
Edgar Nelson Gilbert was an American mathematician and coding theorist, a longtime researcher at Bell Laboratories whose accomplishments include the Gilbert–Varshamov bound in coding theory, the Gilbert–Elliott model of bursty errors in signal transmission, and the Erdős–Rényi model for random graphs.
In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.
In probability, statistics and related fields, a Poisson point process is a type of random mathematical object that consists of points randomly located on a mathematical space. The Poisson point process is often called simply the Poisson process, but it is also called a Poisson random measure, Poisson random point field or Poisson point field. This point process has convenient mathematical properties, which has led to it being frequently defined in Euclidean space and used as a mathematical model for seemingly random processes in numerous disciplines such as astronomy, biology, ecology, geology, seismology, physics, economics, image processing, and telecommunications.
Ilona Palásti (1924–1991) was a Hungarian mathematician who worked at the Alfréd Rényi Institute of Mathematics. She is known for her research in discrete geometry, geometric probability, and the theory of random graphs. With Alfréd Rényi and others, she was considered to be one of the members of the Hungarian School of Probability.
References
- ↑ Herbert Solomon (1978). Geometric Probability. Philadelphia, PA: Society for Industrial and Applied Mathematics.
- Daniel A. Klain, Gian-Carlo Rota - Introduction to Geometric Probability.
- Maurice G. Kendall, Patrick A. P. Moran - Geometrical Probability.
- Eugene Seneta, Karen Hunger Parshall, François Jongmans - Nineteenth-Century Developments in Geometric Probability: J. J. Sylvester, M. W. Crofton, J.-É. Barbier, and J. Bertrand
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