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The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
If M has a normal form, the Böhm tree is finite and has a simple correspondence to the normal form. If M does not have a normal form, normalization may "grow" some subtrees infinitely, or it may get "stuck in a loop" attempting to produce a result for part of the tree, which produce infinitary trees and meaningless terms respectively. Since ...
A forest diagram is one where all the internal lines have momentum that is completely determined by the external lines and the condition that the incoming and outgoing momentum are equal at each vertex. The contribution of these diagrams is a product of propagators, without any integration. A tree diagram is a connected forest diagram.
In mathematics, the Kirby calculus in geometric topology, named after Robion Kirby, is a method for modifying framed links in the 3-sphere using a finite set of moves, the Kirby moves. Using four-dimensional Cerf theory , he proved that if M and N are 3-manifolds , resulting from Dehn surgery on framed links L and J respectively, then they are ...
Such a diagram can be obtained from a connected tree diagram by taking two external lines of the same type and joining them together into an edge. Diagrams with loops (in graph theory, these kinds of loops are called cycles , while the word loop is an edge connecting a vertex with itself) correspond to the quantum corrections to the classical ...
The ZX-calculus is a rigorous graphical language for reasoning about linear maps between qubits, which are represented as string diagrams called ZX-diagrams. A ZX-diagram consists of a set of generators called spiders that represent specific tensors. These are connected together to form a tensor network similar to Penrose graphical notation.
In mathematics, Schubert calculus [1] is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert in order to solve various counting problems of projective geometry and, as such, is viewed as part of enumerative geometry. Giving it a more rigorous foundation was the aim of Hilbert's 15th problem.
A Jordan curve or a simple closed curve in the plane is the image of an injective continuous map of a circle into the plane, :.A Jordan arc in the plane is the image of an injective continuous map of a closed and bounded interval [,] into the plane.