PARALLEL IMPLEMENTATION OF RELATIONAL ALGEBRA
... • a sequential common control unit (CU), where programs and scalar constants are stored; • an associative processing unit forming a two-dimensional array of single-bit PEs; • a matrix memory for the associative processing unit. The CU passes each instruction to all PEs in one unit of time. All activ ...
... • a sequential common control unit (CU), where programs and scalar constants are stored; • an associative processing unit forming a two-dimensional array of single-bit PEs; • a matrix memory for the associative processing unit. The CU passes each instruction to all PEs in one unit of time. All activ ...
Strict Monotonicity of Sum of Squares Error and Normalized Cut in
... C 00 ⊂ C 0 and C 0 ⊂ C implies C 00 ⊂ C, clearly holds. The join C ∨C 0 is the clustering with the maximum number of clusters such that C ⊆ C ∨ C 0 and C 0 ⊆ C ∨ C 0 . The meet C ∧C 0 is the clustering with the minimum number of clusters such that C ∧ C 0 ⊆ C and C ∧ C 0 ⊆ C 0 . For every pair C and ...
... C 00 ⊂ C 0 and C 0 ⊂ C implies C 00 ⊂ C, clearly holds. The join C ∨C 0 is the clustering with the maximum number of clusters such that C ⊆ C ∨ C 0 and C 0 ⊆ C ∨ C 0 . The meet C ∧C 0 is the clustering with the minimum number of clusters such that C ∧ C 0 ⊆ C and C ∧ C 0 ⊆ C 0 . For every pair C and ...
Enhanced PDF - Project Euclid
... T HEOREM 1.6 (Sparse circular law). Let 0 < α ≤ 1 be a constant, and let x be a random complex variable with mean zero and variance one. Let Xn be the sparse matrix ensemble for x with parameter α. Then the ESD for √1n Xn converges in probability to the uniform distribution on the unit disk. An illu ...
... T HEOREM 1.6 (Sparse circular law). Let 0 < α ≤ 1 be a constant, and let x be a random complex variable with mean zero and variance one. Let Xn be the sparse matrix ensemble for x with parameter α. Then the ESD for √1n Xn converges in probability to the uniform distribution on the unit disk. An illu ...
Noncommutative Ricci ow in a matrix geometry
... 0 for µ = 1, 2. Again in terms of the notation above, it follows from τ (δh)δhj ≤ 0 that τ (δh)δhj = 0, in particular 0 = τ ((δh)δh) = − hδh, δhi. So δµ a = δµ (h + λ) = 0, hence a ∈ C1 by Proposition 2.1(a). ...
... 0 for µ = 1, 2. Again in terms of the notation above, it follows from τ (δh)δhj ≤ 0 that τ (δh)δhj = 0, in particular 0 = τ ((δh)δh) = − hδh, δhi. So δµ a = δµ (h + λ) = 0, hence a ∈ C1 by Proposition 2.1(a). ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.