Suppose R is a union of two subsets Rl and R2 whose Steiner minimal trees SMT(R 1) and SMT(R2) are known. The decomposition question is when the Steiner minimal tree SMT(R) for R is just the union of two Steiner minimal trees on R 1 and R2 In this paper a special case studied, that R 1 =bcda, R2=defg are two non-overlapping rectangles with a common vertex d so that a,d,e lie on one line. We con...

INTRODUCTION Accurate R2* measurements are critical for many quantitative imaging applications. R2* is estimated via mono-exponential fitting of signal decay within a series of gradient echo (GRE) images sampled at increasing TE. Root-sum-of-square (RSS) remains the most common approach to combine surface coil array images at each TE prior to the data fitting procedure. Rapid signal decay can r...

(3) (δh[f ])(x) = (∇h 2 [f ])(x)− (∇−h2 [f ])(x). (4) (~ ∆h[r f1 + f2])(n+ 1)(x) = r · (~ ∆h[f1])(n+ 1)(x) + (~ ∆h[f2])(n+ 1)(x). (5) (~ ∆h[f1 + r f2])(n+ 1)(x) = (~ ∆h[f1])(n+ 1)(x) + r · (~ ∆h[f2])(n+ 1)(x). (6) (~ ∆h[r1 f1 − r2 f2])(n+ 1)(x) = r1 · (~ ∆h[f1])(n+ 1)(x)− r2 · (~ ∆h[f2])(n+ 1)(x). (7) (~ ∆h[f ])(1) = ∆h[f ]. (8) (~ ∇h[r f1 + f2])(n+ 1)(x) = r · (~ ∇h[f1])(n+ 1)(x) + (~ ∇h[f2])(...

Tuberculosis (TB) is the leading cause of death among individuals infected with the human immunodeficiency virus (HIV). The study of the joint dynamics of HIV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. Furthermore, although there is overlap in the populations at risk of HIV and TB infections, the magnitude of the proport...

The present study aimed to evaluate the feasibility of performing carbogen gas-challenge blood oxygen level-dependent (BOLD) magnetic resonance imaging (MRI) measurements in patients with hepatocellular carcinoma (HCC). A total of 25 patients with HCC underwent T2* mapping derived from multi-echo gradient-recalled echo imaging prior to and following breathing carbogen (95% O2 and 5% CO2) for 10...

We consider the non-selfadjoint operator H = [ −∆ + μ− V1 −V2 V2 ∆− μ+ V1 ] where μ > 0 and V1, V2 are real-valued decaying potentials. Such operators arise when linearizing a focusing NLS equation around a standing wave. Under natural spectral assumptions we obtain L(R)× L(R)→ L∞(R2)× L∞(R2) dispersive decay estimates for the evolution ePac. We also obtain the following weighted estimate ‖wePa...

The length of a vector v = ( x y ) in R2 is || ( x y ) || = √ x2 + y2, and the distance between two vectors v and w is the length of their difference: ||v − w||. A function h : R2 → R2 is called an isometry when it preserves distances: ||h(v)− h(w)|| = ||v −w|| for all v and w in R2. Examples of isometries include translation by ( 1 −1/2 ) , a rotation by 40 degrees counterclockwise around the ...

In this paper, we give a short survey on discrete geometry on red and blue points in the plane, most of whose results were obtained in the past decade. We consider balanced subdivision problems, geometric graph problems, graph embedding problems, Gallai-type problems and others. 1 Notation and Definitions In this paper, we give a short survey on discrete geometry on red and blue points in the p...