In this paper we characterize the radical of an arbitrary‎ ‎submodule $N$ of a finitely generated free module $F$ over a‎ ‎commutatitve ring $R$ with identity‎. ‎Also we study submodules of‎ ‎$F$ which satisfy the radical formula‎. ‎Finally we derive‎ ‎necessary and sufficient conditions for $R$ to be a‎ ‎Pr$ddot{mbox{u}}$fer domain‎, ‎in terms of the radical of a‎ ‎cyclic submodule in $Rbigopl...

Introduction: Infants born with congenital anomalies demand individualized nutritional evaluations and recommendations. The anatomical changes of neonatal surgical diseases create specific physiological constraints.Patients with different congenital anomalies have different nutritional support needs. It is essential to know the exact physiology of these anomalies in order to be able to manage a...

By Thomas Olsen, MD, PhD The calculation of IOL power in an eye that has undergone corneal refractive surgery is not a straightforward task. However, if you understand why the calculation may go wrong, you will have a fair chance of getting a good result using modern technology. It is important to understand that the calculation errors of post-LASIK eyes are the sum of the traditional errors pl...

The contamination of melamine was evaluated in 69 infants along with follow up formula samples collected from the market for the first time in Iran using HPLC method. Since there are no previous data concerning the contamination level of melamine in all brands of infant formula samples consumed using the HPLC method in Iran, this study is the first investigation in this regard. Our results show...

Theorem 2.1. For all n ≥ 1, n log n− n < log(n!) < n log n, so log(n!) ∼ n log n. Proof. The inequality log(n!) < n log n is a consequence of the trivial inequality n! < nn. Here are three methods of showing n log n− n < log(n!). Method 1: A Riemann sum approximation for ∫ n 1 log x dx using right endpoints is log 2+ · · ·+ log n = log(n!), which overestimates, so log(n!) > ∫ n 1 log x dx = n l...

The article presents the results of a comparative analysis complexity texts on material linguistic subcorpus English textbooks “English”, “Starlight” and “Rainbow English” for 11th grade. total volume sub-corpus acting as empirical research amounted to 36 (3955 word uses). One characteristics text is its readability. It depends degree words structures used in sentences, well such quantitative p...