Related Research Articles
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 can be considered "discrete" rather than "continuous". 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. However, there is no exact definition of the term "discrete mathematics".
In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.
In category theory, Met is a category that has metric spaces as its objects and metric maps as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by Isbell (1964).
In combinatorics, a Helly family of order k is a family of sets in which every minimal subfamily with an empty intersection has k or fewer sets in it. Equivalently, every finite subfamily such that every k-fold intersection is non-empty has non-empty total intersection. The k-Helly property is the property of being a Helly family of order k.
In metric geometry, the metric envelope or tight span of a metric space M is an injective metric space into which M can be embedded. In some sense it consists of all points "between" the points of M, analogous to the convex hull of a point set in a Euclidean space. The tight span is also sometimes known as the injective envelope or hyperconvex hull of M. It has also been called the injective hull, but should not be confused with the injective hull of a module in algebra, a concept with a similar description relative to the category of R-modules rather than metric spaces.
In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of Aronszajn & Panitchpakdi (1956) that these two different types of definitions are equivalent.
In mathematics, a complex line is a one-dimensional affine subspace of a vector space over the complex numbers. A common point of confusion is that while a complex line has dimension one over C, it has dimension two over the real numbers R, and is topologically equivalent to a real plane, not a real line.
Nachman Aronszajn was a Polish American mathematician. Aronszajn's main field of study was mathematical analysis, where he systematically developed the concept of reproducing kernel Hilbert space. He also contributed to mathematical logic.
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra.
Algebraic statistics is the use of algebra to advance statistics. Algebra has been useful for experimental design, parameter estimation, and hypothesis testing.
In graph theory, a starSk is the complete bipartite graph K1,k : a tree with one internal node and k leaves. Alternatively, some authors define Sk to be the tree of order k with maximum diameter 2; in which case a star of k > 2 has k − 1 leaves.
Louis Joseph Billera is a Professor of Mathematics at Cornell University.
John Rolfe Isbell was an American mathematician, for many years a professor of mathematics at the University at Buffalo (SUNY).
Michael Lee Develin is an American mathematician known for his work in combinatorics and discrete geometry.
Mathematics is a broad subject that is commonly divided in many areas that may be defined by their objects of study, by the used methods, or by both. For example, analytic number theory is a subarea of number theory devoted to the use of methods of analysis for the study of natural numbers.
Richard M. Pollack was an American geometer who spent most of his career at the Courant Institute of Mathematical Sciences at New York University, where he was Professor Emeritus until his death.
Paul Allen Catlin was a mathematician, professor of mathematics who worked in graph theory and number theory. He wrote a significant paper on the series of chromatic numbers and Brooks' theorem, titled Hajós graph coloring conjecture: variations and counterexamples.
In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets are several closely related definitions of well-spaced sets of points, and the packing radius and covering radius of these sets measure how well-spaced they are. These sets have applications in coding theory, approximation algorithms, and the theory of quasicrystals.
In mathematics, a diversity is a generalization of the concept of metric space. The concept was introduced in 2012 by Bryant and Tupper, who call diversities "a form of multi-way metric". The concept finds application in nonlinear analysis.
References
- Hans-Jurgen Bandelt; Andreas Dress (1992). "A canonical decomposition theory for metrics on a finite set". Advances in Mathematics . 92 (1): 47–105. doi: 10.1016/0001-8708(92)90061-O .
- Andreas Dress; Vincent Moulton; Werner Terhalle (1996). "T-theory: An Overview". European Journal of Combinatorics . 17 (2–3): 161–175. doi: 10.1006/eujc.1996.0015 .
- John Isbell (1964). "Six theorems about metric spaces". Commentarii Mathematici Helvetici . 39: 65–74. doi:10.1007/BF02566944.
- Bernd Sturmfels; Josephine Yu (2004). "Classification of Six-Point Metrics". The Electronic Journal of Combinatorics. 11: R44. doi: 10.37236/1797 .
 | This combinatorics-related article is a stub. You can help Wikipedia by expanding it. |