In mathematics and theoretical computer science, a type theory is the formal presentation of a specific type system. Type theory is the academic study of type systems.
In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value.
In quantum field theory, the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula is a method to calculate S-matrix elements from the time-ordered correlation functions of a quantum field theory. It is a step of the path that starts from the Lagrangian of some quantum field theory and leads to prediction of measurable quantities. It is named after the three German physicists Harry Lehmann, Kurt Symanzik and Wolfhart Zimmermann.
In statistics and information theory, a maximum entropy probability distribution has entropy that is at least as great as that of all other members of a specified class of probability distributions. According to the principle of maximum entropy, if nothing is known about a distribution except that it belongs to a certain class, then the distribution with the largest entropy should be chosen as the least-informative default. The motivation is twofold: first, maximizing entropy minimizes the amount of prior information built into the distribution; second, many physical systems tend to move towards maximal entropy configurations over time.
String diagrams are a formal graphical language for representing morphisms in monoidal categories, or more generally 2-cells in 2-categories. They are a prominent tool in applied category theory. When interpreted in the monoidal category of vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose graphical notation. This has led to the development of categorical quantum mechanics where the axioms of quantum theory are expressed in the language of monoidal categories.
In quantum information theory, quantum relative entropy is a measure of distinguishability between two quantum states. It is the quantum mechanical analog of relative entropy.
Mixing patterns refer to systematic tendencies of one type of nodes in a network to connect to another type. For instance, nodes might tend to link to others that are very similar or very different. This feature is common in many social networks, although it also appears sometimes in non-social networks. Mixing patterns are closely related to assortativity; however, for the purposes of this article, the term is used to refer to assortative or disassortative mixing based on real-world factors, either topological or sociological.
Assortativity, or assortative mixing, is a preference for a network's nodes to attach to others that are similar in some way. Though the specific measure of similarity may vary, network theorists often examine assortativity in terms of a node's degree. The addition of this characteristic to network models more closely approximates the behaviors of many real world networks.
In linear algebra, the computation of the permanent of a matrix is a problem that is thought to be more difficult than the computation of the determinant of a matrix despite the apparent similarity of the definitions.
In computer science, more specifically in automata and formal language theory, nested words are a concept proposed by Alur and Madhusudan as a joint generalization of words, as traditionally used for modelling linearly ordered structures, and of ordered unranked trees, as traditionally used for modelling hierarchical structures. Finite-state acceptors for nested words, so-called nested word automata, then give a more expressive generalization of finite automata on words. The linear encodings of languages accepted by finite nested word automata gives the class of visibly pushdown languages. The latter language class lies properly between the regular languages and the deterministic context-free languages. Since their introduction in 2004, these concepts have triggered much research in that area.
Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls or simply the curl theorem, is a theorem in vector calculus on . Given a vector field, the theorem relates the integral of the curl of the vector field over some surface, to the line integral of the vector field around the boundary of the surface. The classical theorem of Stokes can be stated in one sentence: The line integral of a vector field over a loop is equal to the surface integral of its curl over the enclosed surface. It is illustrated in the figure, where the direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule. For the right hand the fingers circulate along ∂Σ and the thumb is directed along n.
In the geometry of numbers, the Klein polyhedron, named after Felix Klein, is used to generalize the concept of continued fractions to higher dimensions.
Network coding has been shown to optimally use bandwidth in a network, maximizing information flow but the scheme is very inherently vulnerable to pollution attacks by malicious nodes in the network. A node injecting garbage can quickly affect many receivers. The pollution of network packets spreads quickly since the output of honest node is corrupted if at least one of the incoming packets is corrupted.
In mathematics, the Lindström–Gessel–Viennot lemma provides a way to count the number of tuples of non-intersecting lattice paths, or, more generally, paths on a directed graph. It was proved by Gessel–Viennot in 1985, based on previous work of Lindström published in 1973.
The generalized distributive law (GDL) is a generalization of the distributive property which gives rise to a general message passing algorithm. It is a synthesis of the work of many authors in the information theory, digital communications, signal processing, statistics, and artificial intelligence communities. The law and algorithm were introduced in a semi-tutorial by Srinivas M. Aji and Robert J. McEliece with the same title.
In the mathematical field of group theory, an Artin transfer is a certain homomorphism from an arbitrary finite or infinite group to the commutator quotient group of a subgroup of finite index. Originally, such mappings arose as group theoretic counterparts of class extension homomorphisms of abelian extensions of algebraic number fields by applying Artin's reciprocity maps to ideal class groups and analyzing the resulting homomorphisms between quotients of Galois groups. However, independently of number theoretic applications, a partial order on the kernels and targets of Artin transfers has recently turned out to be compatible with parent-descendant relations between finite p-groups, which can be visualized in descendant trees. Therefore, Artin transfers provide a valuable tool for the classification of finite p-groups and for searching and identifying particular groups in descendant trees by looking for patterns defined by the kernels and targets of Artin transfers. These strategies of pattern recognition are useful in purely group theoretic context, as well as for applications in algebraic number theory concerning Galois groups of higher p-class fields and Hilbert p-class field towers.
In discrete mathematics, the Bregman–Minc inequality, or Bregman's theorem, allows one to estimate the permanent of a binary matrix via its row or column sums. The inequality was conjectured in 1963 by Henryk Minc and first proved in 1973 by Lev M. Bregman. Further entropy-based proofs have been given by Alexander Schrijver and Jaikumar Radhakrishnan. The Bregman–Minc inequality is used, for example, in graph theory to obtain upper bounds for the number of perfect matchings in a bipartite graph.
The unique homomorphic extension theorem is a result in mathematical logic which formalizes the intuition that the truth or falsity of a statement can be deduced from the truth values of its parts.
The ratio of uniforms is a method initially proposed by Kinderman and Monahan in 1977 for pseudo-random number sampling, that is, for drawing random samples from a statistical distribution. Like rejection sampling and inverse transform sampling, it is an exact simulation method. The basic idea of the method is to use a change of variables to create a bounded set, which can then be sampled uniformly to generate random variables following the original distribution. One feature of this method is that the distribution to sample is only required to be known up to an unknown multiplicative factor, a common situation in computational statistics and statistical physics.
Flag algebras are an important computational tool in the field of graph theory which have a wide range of applications in homomorphism density and related topics. Roughly, they formalize the notion of adding and multiplying homomorphism densities and set up a framework to solve graph homomorphism inequalities with computers by reducing them to semidefinite programming problems. Originally introduced by Alexander Razborov in a 2007 paper, the method has since come to solve numerous difficult, previously unresolved graph theoretic questions. These include the question regarding the region of feasible edge density, triangle density pairs and the maximum number of pentagons in triangle free graphs.
