and Applied Analysis 3 whereQ is a fixed positive definite matrix and k ≥ 3. Then, one has the following Lyapunov functions V n (x) = R2 (x) + R3 (x) + ⋅ ⋅ ⋅ + Rn (x) 1 + Q 1 (x) + ⋅ ⋅ ⋅ + Q n−2 (x) . (3) Proof. Rewrite V n (x) = R2 (x) + R3 (x) + ⋅ ⋅ ⋅ + Rn (x) 1 + Q 1 (x) + ⋅ ⋅ ⋅ + Q n−2 (x) = ∑ ∞

Ages of schoolmaster (n = 136) from the southeastern Florida coast from 1981-2015 were determined using sectioned sagittal otoliths. Opaque zones were annular, forming March-July (peaking in May-June). Schoolmaster ranged in age from 1-42 years; the largest fish measured 505 mm total length (TL) and was 19 years old. The oldest fish measured 440 mm TL. Estimated body size relationships for scho...

NMR relaxation rates were related to the composition of the nucleus pulposus from 11 and anulus fibrosus from six human intervertebral disks. Tissue water was proportional to glycosaminoglycan (GAG) and residue, the noncollagen, non-GAG portion of the dry weight (R2 = 0.74). The solid signal fraction depended on collagen and residue protons (R2 = 0.89). 1/T1 was proportional to collagen and res...

We adopt the following convention: a, b, r, s, x0, x are real numbers, f , g are partial functions from R to R, and X, Y are sets. The following propositions are true: (1) max(a, b) ­ min(a, b). (2) Let n be a natural number, R1, R2 be elements of R , and r1, r2 be real numbers. Then R1 a 〈r1〉 • R2 a 〈r2〉 = (R1 • R2) a 〈r1 · r2〉. (3) Let n be a natural number and R be an element of R. Suppose ∑...

Neuroferritinopathy is an autosomal dominant extrapyramidal movement disorder, caused by FTL gene mutations. Iron decreases the MR T2* decay time, therefore increasing the R2* (R2* = 1 /T2*), which correlates with brain tissue iron content. 3T structural and quantitative MR imaging assessment of R2* in 10 patients with neuroferritinopathy demonstrated a unique pattern of basal ganglia cavitatio...

After injection of Gd-EOB-DTPA, T1 and T2 were determined on a clinical MR scanner (1.5 T) in the liver and kidneys of sacrificed but intact guinea pigs with normal and obstructed biliary systems and in bile, urine, and blood collected postmortem. Tissue [Gd] was determined by radioassay of 153Gd and relaxivities (R1 and R2; units of s(-1) x mmol(-1) x kg) of Gd-EOB-DTPA calculated. Compared wi...

In atrial fibrillation, cardiac performance is dependent on both preceding RR (RR-1) and prepreceding RR (RR-2) intervals. However, relative contributions were not well defined. Left ventricular outflow peak ejection velocity (Vpe) was measured by echocardiography from 21 patients. The relation between RR-1 and Vpe could be divided into two zones; steep slope in short RR-1 intervals (< or =0.5 ...

Experimental and clinical studies strongly suggest that gelatinase A plays a central role in the evolution of glomerular injury and sclerosis. The sequences of the 5' flanking regions of the human and rat gelatinase A genes do not share similarities with other members of the matrix metalloproteinase gene family and are regulated in a distinctive manner. The human and rat gelatinase A genes incl...

and Applied Analysis 3 linear equation, which requires special tricks and techniques. In the convergence process, a compensated compactness principle is applied. The rest of this paper is organized as follows. Section 2 is devoted to studying 1.5 . More precisely, we first study the penalized equation. In order to do this, we consider the corresponding linear equation and get its formal solutio...