Search results
Results from the WOW.Com Content Network
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual.In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression.
This moduli stack has a completion consisting of the moduli stack of stable curves (for given and ), which is proper over Spec Z. For example, M 0 {\displaystyle {\mathcal {M}}_{0}} is the classifying stack B PGL ( 2 ) {\displaystyle B{\text{PGL}}(2)} of the projective general linear group.
A dyadic tensor T is an order-2 tensor formed by the tensor product ⊗ of two Cartesian vectors a and b, written T = a ⊗ b.Analogous to vectors, it can be written as a linear combination of the tensor basis e x ⊗ e x ≡ e xx, e x ⊗ e y ≡ e xy, ..., e z ⊗ e z ≡ e zz (the right-hand side of each identity is only an abbreviation, nothing more):
The first line in the action contains the Einstein–Hilbert action, the dilaton kinetic term, and the Kalb–Ramond field strength tensor =. The first term on the second line has the appropriately modified field strength tensors for the three C p {\displaystyle C_{p}} gauge fields, while the last term is a Chern–Simons term .
The first nine blocks in the solution to the single-wide block-stacking problem with the overhangs indicated. In statics, the block-stacking problem (sometimes known as The Leaning Tower of Lire (Johnson 1955), also the book-stacking problem, or a number of other similar terms) is a puzzle concerning the stacking of blocks at the edge of a table.
In two dimensions, the Levi-Civita symbol is defined by: = {+ (,) = (,) (,) = (,) = The values can be arranged into a 2 × 2 antisymmetric matrix: = (). Use of the two-dimensional symbol is common in condensed matter, and in certain specialized high-energy topics like supersymmetry [1] and twistor theory, [2] where it appears in the context of 2-spinors.
It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space. Well-formulated expressions are constrained by the rules of Einstein summation: any index may appear at most twice and furthermore a raised index must contract with a lowered index ...
The number of each upper and lower indices of a tensor gives its type: a tensor with p upper and q lower indices is said to be of type (p, q), or to be a type-(p, q) tensor. The number of indices of a tensor, regardless of variance, is called the degree of the tensor (alternatively, its valence, order or rank, although rank is ambiguous).