1. Suppose that the matrix Mk is of dimension nk × nk+1, k ∈ {1, . . . , h}. We wish to compute the product M1M2 · · ·Mh. Notice that the order of multiplication makes a difference. For example, if (n1, n2, n3, n4) = (1, 10, 1, 10), the calculation (M1M2)M3 requires 20 scalar multiplications, but the calculation M1(M2M3) requires 200 scalar multiplications. Indeed, multiplying a m × n matrix by...

Suppose we are given a particular fixed-length hash function h : 256-bits→ 128-bits. How can we use h to compute a 128-bit strong collision-free hash of a 512-bit input block? We consider several possible ways to extend h to a hash function H : 512-bits→ 128-bits. In the following, we suppose that m is 512-bits long, and we write m = m1m2, where m1 and m2 are 256 bits each. Method 1 Define H(m)...

The vasopressin-regulated urea transporter (UT)-A1 is a transmembrane protein with two glycosylated forms of 97 and 117 kDa; both are derived from a single 88-kDa core protein. However, the precise molecular sites and the function for UT-A1 N-glycosylation are not known. In this study, we compared Madin-Darby canine kidney cells stably expressing wild-type (WT) UT-A1 to Madin-Darby canine kidne...

The objective of the study in this paper is to define M₁,M₂-direction curves and M₁,M₂-donor curves of the spacelike curve γ via the Bishop frame of type-2 in E₁³. We obtained the necessary and sufficient conditions when the associated curves to be slant helices and general helices via the Bishop frame of type-2 in E₁³. After defining the spherical indicatrices of the associated curves, we obta...

A(B → M1M2) ∝ φB ⊗H ⊗ φM1 ⊗ φM2 , (2) where ⊗ denotes the convolution over parton kinematic variables, the hard kernel H absorbs perturbative dynamics, and the B (M1, M2) meson distribution amplitude φB (φM1, φM2) absorbs nonperturbative dynamics in the B → M1M2 decay. A piece of contribution to B meson decays is factorizable, if it respects either Eq. (1) in the sense of the factorization assu...

a1 Antitrypsin phenotypes were determined by isoelectric focusing in 225 adult white patients with rheumatoid arthritis (RA), 60 patients with severe rheumatoid arthritis (grade III and IV), 17 sibling pairs-HLA identical but discordant for rheumatoid arthritis, and 122 random patients with Sjogren's syndrome. No significant increase in non-M phenotypes was found in either of the groups of pati...

Biomedical Imaging Research Institute, Cedars-Sinai Medical Center, Los Angeles, California, USA Full list of author information is available at the end of the article Figure 1 Typical T2 prepared (b = 0 s/mm) images, DWI (b = 450 s/mm) in three orthogonal directions, and ADC maps for M1M2 (top) and TRSE (bottom) diffusion preparation. M1M2 diffusion-prepared images clearly depict myocardium wi...