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Directed self-assembly (DSA) is a type of directed assembly which utilizes block co-polymer morphology to create lines, space and hole patterns, facilitating for a more accurate control of the feature shapes. Then it uses surface interactions as well as polymer thermodynamics to finalize the formation of the final pattern shapes. [1]
The spiral array model can be viewed as a generalized tonnetz, which maps pitches into a two-dimensional lattice (array) structure. The spiral array wraps up the two-dimensional tonnetz into a three-dimensional lattice, and models higher order structures such as chords and keys in the interior of the lattice space. This allows the spiral array ...
Diagonal matrices D and E, and unitary U and V, are not necessarily unique in general. Comment: U and V matrices are not the same as those from the SVD. Analogous scale-invariant decompositions can be derived from other matrix decompositions; for example, to obtain scale-invariant eigenvalues. [3] [4]
In other words, the matrix of the combined transformation A followed by B is simply the product of the individual matrices. When A is an invertible matrix there is a matrix A −1 that represents a transformation that "undoes" A since its composition with A is the identity matrix. In some practical applications, inversion can be computed using ...
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where the u n are the inputs, the y m are the outputs, and the g mn are the transfer functions. This may be written more succinctly in matrix operator notation as, = where Y is a column vector of the outputs, G is a matrix of the transfer functions, and U is a column vector of the inputs.
For a simple graph with vertex set U = {u 1, …, u n}, the adjacency matrix is a square n × n matrix A such that its element A ij is 1 when there is an edge from vertex u i to vertex u j, and 0 when there is no edge. [1] The diagonal elements of the matrix are all 0, since edges from a vertex to itself are not
In linear algebra, the Cholesky decomposition or Cholesky factorization (pronounced / ʃ ə ˈ l ɛ s k i / shə-LES-kee) is a decomposition of a Hermitian, positive-definite matrix into the product of a lower triangular matrix and its conjugate transpose, which is useful for efficient numerical solutions, e.g., Monte Carlo simulations.