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In the mathematical field of graph theory, a spanning tree T of a connected, undirected graph G is a tree composed of all the vertices and some ( or perhaps all ) of the edges of G. Informally, a spanning tree of G is a selection of edges of G that form a tree spanning every vertex.
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mathematical and field
In the nineteenth century the major figures of mathematical acoustics were Helmholtz in Germany, who consolidated the field of physiological acoustics, and Lord Rayleigh in England, who combined the previous knowledge with his own copious contributions to the field in his monumental work The Theory of Sound ( 1877 ).
Acoustic theory is the field relating to mathematical description of sound waves.
Theoretical models have also been developed to study the physics of phase transitions, such as the Landau-Ginzburg theory, Critical exponents and the use of mathematical techniques of quantum field theory and the renormalization group.
Higher-dimensional categories are part of the broader mathematical field of higher-dimensional algebra, a concept introduced by Ronald Brown.
He developed a method of measuring the horizontal intensity of the magnetic field which was in use well into the second half of the 20th century, and worked out the mathematical theory for separating the inner and outer ( magnetospheric ) sources of Earth's magnetic field.
The computable numbers form a real closed field and can be used in the place of real numbers for many, but not all, mathematical purposes.
What methods, what new facts will the new century reveal in the vast and rich field of mathematical thought?
In particular, this field contains all the numbers named in the mathematical constants article, and all algebraic numbers ( and therefore all rational numbers ).
The process is algorithmic and based on dependencies ( mathematical relations ) that exist among relations ' field types.
volume ) and its position in time ( perceived as a scalar dimension along the t-axis ), as well as the spatial constitution of objects within — structures that correlate with both particle and field conceptions, interact according to relative properties of mass — and are fundamentally mathematical in description.
The Copenhagen interpretation is a consensus among some of the pioneers in the field of quantum mechanics that it is undesirable to posit anything that goes beyond the mathematical formulae and the kinds of physical apparatus and reactions that enable us to gain some knowledge of what goes on at the atomic scale.
There are different mathematical ways of representing the electromagnetic field.
In terms of practice, mathematical finance also overlaps heavily with the field of computational finance ( also known as financial engineering ).
Some modification of the Feynman rules of calculation may well outlive the elaborate mathematical structure of local canonical quantum field theory ...” So far there are no opposing opinions.
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces ; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.
In his exposition, he acknowledged the existence of what are now called imaginary numbers, although he did not understand their properties ( described for the first time by his Italian contemporary Rafael Bombelli, although mathematical field theory was developed centuries later ).
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function.
The IHÉS, founded in 1958 by businessman and mathematical physicist Léon Motchane with the help of Robert Oppenheimer and Jean Dieudonné, aims to bring together top researchers in the field.
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points.
Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory.
The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis — family friends of Euler — were responsible for much of the early progress in the field.
Where objective measurement is desired, light levels can be quantified by field measurement or mathematical modeling, with results typically displayed as an isophote map or light contour map.
Category theory, another field within " foundational mathematics ", is rooted on the abstract axiomatization of the definition of a " class of mathematical structures ", referred to as a " category ".
mathematical and graph
Graph ( mathematics ) | Graphs like this are among the objects studied by discrete mathematics, for their interesting graph property | mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithm s.
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects from a certain collection.
* Node ( graph theory ), a vertex in a mathematical graph
The items may be stored individually as records in a database ; or may be elements of a search space defined by a mathematical formula or procedure, such as the roots of an equation with integer variables ; or a combination of the two, such as the Hamiltonian circuits of a graph.
The musical staff is analogous to a mathematical graph of pitch with respect to time.
Integration is a mathematical operation that corresponds to the informal idea of finding the area under the graph of a function.
( A mathematical " lattice " is a partially ordered set, characterizable as a directed acyclic graph, in which the relationship between any two vertices is either " dominates ," " is dominated by ," or neither.
The problem is part of the mathematical field of topological graph theory which studies the embedding of graphs on surfaces.
After enrolling in a Computer Science Ph. D. program at Stanford University, Page was in search of a dissertation theme and considered exploring the mathematical properties of the World Wide Web, understanding its link structure as a huge graph.
Petersen's interests in mathematics were manifold ( geometry, function theory, number theory, mathematical physics, mathematical economics, cryptography and graph theory ).
For general functions, the graphic representation cannot be applied and the formal definition of the graph of a function suits the need of mathematical statements, e. g., the closed graph theorem in functional analysis.
For example, it can be almost synonymous with mathematical function ( as in learning curve ), or graph of a function ( as in Phillips curve ).
The term has been generalized to include measurements in much more general mathematical spaces ; for example, in graph theory, one might consider a geodesic between two vertices / nodes of a graph.
In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph ( whether directed or undirected ).
The resulting mathematical structure is called a graph.
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges.
Flooding algorithms are also useful for solving many mathematical problems, including maze problems and many problems in graph theory.
Structural induction is a proof method that is used in mathematical logic ( e. g., in the proof of Łoś ' theorem ), computer science, graph theory, and some other mathematical fields.
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