The theory of basic sets (bases) of polynomials has a significant role in mathematics and its applications, e.g., approximation theory, mathematical physics, Geometry, and partial differential equations. The interest of the present work focused on the expansion of analytic functions into BPs. Given a sequence of a base of polynomialsThe expansion of an analytic function as a basic series ∑ began with the papers by Whittaker and Cannon [14,15,30,31] about 90 years ago. Basic series generalize Taylor series, where can be Legendre, Laguerre, Chebyshev, Hermite, Bessel, Bernoulli and Euler polynomials (see [1,4,5,10,11,21]). This theory found a lot of applications, mainly in the theory of functions depending on one or several complex variables as well as in the approximation of solutions of differential equations or matrix functions.The topic of derivative BPs in one complex variable has been studied early (see [26,27,28]), the searchers considered the disks in the complex plane C. For several complex variables (see [16,17,19,24,25]), the representation domains are polycyclinderical, hyperspherical and hyperelliptical regions. Recently,in [12,35] the
Background: Postmenopausal osteoporosis (PMO) is a debilitating disease induced by estrogen deficiency characterized by bone loss and bone micro-architecture degradation contributing to fragility fractures. Osteopontin (OPN) has been found to be involved in bone turnover by activating the resorption process. Objectives: This study aim to evaluate the relationship between serum OPN levels and bone mineral density (BMD) for identification of female individuals at high risk of osteoporotic fractures. Methods: A total of eighty eight postmenopausal women (PMW) were recruited in this study. Densitometry results addressed cases of the study into 3 groups: thirty with osteoporosis, thirty with osteopenia and twenty eight PMW without complications as a control group. Serum OPN was measured by using Enzyme-Linked immunosorbent assay (ELISA) kits and BMD was obtained at lumbar spine and femoral neck by dualenergy X-ray absorptiometry (DEXA). Results: Osteopontin levels were significantly higher in osteoporotic group compared to osteopenic and control groups (P ˂ 0.001). Negative correlation was found between BMD and OPN in osteoporotic group. By using the Receiver Operating Characteristic (ROC) curve, the cutoff level of ≥ 10.294 ng/mL for OPN was chosen for suggesting osteoporosis which yield a sensitivity of 66.67 % and specificity of 96.43 %. Also, there were a significant increase in serum ALP levels in osteoporotic and osteopenic groups compared to control group while serum Ca failed to show a significant difference between the studied groups. In addition, serum TAC showed a significant decrease in osteoporotic group compared to control group. Conclusion: These findings suggest that the circulating OPN might play role in the pathophysiology of PMO. Its blood concentration can be used as a sensitive monitoring indicator for the early detection of osteoporosis independently of BMD.
Although increased awareness of morbidity and costs related to osteoporotic fractures, real progress achieved only through early detection of osteoporosis before any fractures occur. Dual Energy X-ray Absorptiometry (DEXA) is commonly used for diagnosis of osteoporosis by measuring bone mineral density (BMD
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