Introduction It is well known that MRI may provide information on trabecular bone structure as well as trabecular bone density. Two major methods are quantitative magnetic resonance (QMR) and high resolution magnetic resonance imaging (HR-MRI). QMR measures the relaxation times T2, T2* of bone marrow to assess the inhomogeneities in trabecular bone structure while HR-MRI directly look at the 2D...

For simplicity, we adopt the following convention: x, x0, y, y0, r are real numbers, z, z0 are elements of R2, f , f1, f2 are partial functions from R2 to R, R is a rest, and L is a linear function. Let us note that every rest is total. Let f be a partial function from R2 to R and let z be an element of R2. The functor pdiff1(f, z) yielding a function from R2 into R is defined as follows: (Def....

Ribosomal RNA genes are abundant repetitive sequences in most eukaryotes. Ribosomal DNA (rDNA) contains many insertions derived from mobile elements including non-long terminal repeat (non-LTR) retrotransposons. R2 is the well-characterized 28S rDNA-specific non-LTR retrotransposon family that is distributed over at least 4 bilaterian phyla. R2 is a large family sharing the same insertion speci...

2 ∥∥∥(x∗ − x̃) + σ n z̃− δ∥∥2l2 s.t. δ ∈ C − x̃. Letting R1 and R2 denote orthogonal subspaces that contain Q1 and Q2, i.e., Q1 ⊆ R1 and Q2 ⊆ R2, and letting δ = PR1(δ), δ (2) = PR2(δ), δ̂ (1) n (C) = PR1(δ̂n(C)), δ̂ (2) n (C) = PR2(δ̂n(C)) denote the projections of δ, δ̂n(C) onto R1, R2, we can rewrite the above reformulated optimization problem as: [ δ̂ (1) n (C), δ̂ (2) n (C) ] = arg min δ∈Q1,δ∈Q2 1 2...

In this paper we prove a quantitative form of Landis’ conjecture in the plane. Precisely, let W (z) be a measurable real vector-valued function and V (z) ≥ 0 be a real measurable scalar function, satisfying ‖W‖L∞(R2) ≤ 1 and ‖V ‖L∞(R2) ≤ 1. Let u be a real solution of ∆u − ∇(Wu) − V u = 0 in R2. Assume that u(0) = 1 and |u(z)| ≤ exp(C0|z|). Then u satisfies inf |z0|=R sup |z−z0|<1 |u(z)| ≥ exp(...

The set of Penrose tilings, when provided with a natural compact metric topology, becomes a strictly ergodic dynamical system under the action of R2 by translation. We show that this action is an almost 1:1 extension of a minimal R2 action by rotations on T4, i.e., it is an R2 generalization of a Sturmian dynamical system. We also show that the inflation mapping is an almost 1:1 extension of a ...

The carbon monoxide (CO) rotational transition lines are the most common tracers of molecular gas within giant molecular clouds (MCs). We study the ratio (R2−1/1−0) between CO’s first two emission lines and examine what information it provides about the physical properties of the cloud. To study R2−1/1−0 we perform smooth particle hydrodynamic simulations with time dependent chemistry (using GA...