In assessing the efficiency of the proposed recursion, we will use the following assumptions: Matrix-vector product: the computation of Ax where and requires operations.
Under this assumption, the matrix D is a diagonal matrix with only L non-zero entries given by diag (D) = [p1 ⋯ p G ] T = p, being p ∈ ℝ + G × 1.
Secondly, under the block fading assumption, the matrix Ξ = X ̄ N U ̄ N is the product of a block diagonal matrix and a block matrix with diagonal sub-matrices.
Conventional analysis methods for building structures with added viscoelastic dampers, such as direct integration, complex mode superposition, and modal strain energy method, were compared, and a procedure based on rigid diaphragm assumption and matrix condensation technique was proposed for application in the preliminary analysis and design stages.
By the positive definite assumption of matrix A, we conclude ((z^{ k)})^=0), and the proof is then complete.
For later use, we define h:,k[m] = [h1,k[m],..., hN,k[m]]T and similarily for hn,:[m] and hn,k.a Note that due to the block-fading assumption, the matrix H[m] does not depend on s.
The main result shows that under natural assumptions a matrix function that admits a thematic factorization also admits a monotone thematic factorization and the indices of a monotone thematic factorization are uniquely determined by the matrix function itself.
In 2002, Hiai and Zhan [2] proved that under the same assumption for matrices A, B, and X, the function t ↦ ∥ | A t X B 1 − t | r ∥ ⋅ ∥ | A 1 − t X B t | r ∥ is convex on [ 0, 1 ] for each r > 0. Among other things, this convexity result interpolated the above matrix Cauchy-Schwarz inequality showed by Bhatia and Davis.
Under these assumptions, the matrices R kT) and C kT), defined in section II-B, can be written as R ( k T ) = ∑ p = 1 P π p ( k T ) j p j p H + η 2 I (50) C ( k T ) = ∑ p = 1 P c p ( k T ) e 2 j ϕ p j p j p T (51).
Under some discreteness assumptions, this matrix-vector min-plus multiplication algorithm applies to several problems from the domains of context-free grammar parsing and RNA folding and, in particular, implies the asymptotically fastest O n 3 log 2 n time algorithm for single-strand RNA folding with discrete cost functions.
