By the compact embedding E ↪ L 2 × L 2 , for a given A ∈ M 2 , there is a compact self-adjoint operator T A : E → E associated with A such that 〈 T A z, w 〉 = ∫ Ω ( A ( x ) z, w ) d x, z, w ∈ E. The operator T A possesses the property that λ ( A ) is an eigenvalue of (LA) if and only if there is nonzero z ∈ E such that λ ( A ) T A z = z.
For a given a matrix A ∈ C m × n, the symbols A* and r(A) will stand for the conjugate transpose and the rank of the matrix A, respectively.
Given a network G = (V, E), and denote its vertex set as V, edge set as E and adjacency matrix as A. Given a m-partition P m, define a corresponding n × m assignment matrix X = [ h1, h2, …, h m] with h ic = 1 if v i ∊ V c, and h ic = 0 otherwise, for 1 ≤ c ≤ m.
One can construct the open symplectic mapping torus \(T_\phi \) for a given a Weinstein manifold \(M\) and a compactly supported symplectomorphism \ \phi \).
Given an m × n haplotype matrix A, a block B i, j) of matrix A is viewed as m haplotype strings; they are partitioned into groups by merging identical haplotype strings into the same group.
For a given (yin A(M)) the mapping (F cdot,y)) is a (vert cdot vert _{rho} -contraction for each (rhoinLambda), so by Cain-Nashed theorem ([13], Theorem 2.2) it has a unique fixed point in X.
A lesion M is a result of a random process with a given distribution P m (M), which specifies the probability of occurrence of the lesion M due to mutation.
It is well-known that given a module M and a submodule M ′, then the lattice of submodules of the factor module M / M ′ is canonically isomorphic to the lattice of the submodules U of M satisfying M ′ ⊆ U.
Given a message m, an attribute set ω, and labe.params, lABE.Encrypt(m, ω, labe.params) outputs the cipertext CM as follows: 1. Choose (k in Z_{q}^) and compute the key K=k·PK.
Alternatively, Bafna and Bansal [ 11] introduced an algorithm for computing the minimum number of recombination events, I j[ m −j], needed to obtain a recombinant j given a set, m −j, of its possible ancestors.
Given a marking m and a place s, we say that the place s contains m(s) tokens.
