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Here, we prove that under stochastic geometry modeling, the coverage probability of cellular communications networks is independent of base station density, λ. This proof identifies a glitch in a huge number of research results, accululated in the literature over the past decade, where this probability is mistakenly expressed as a function of λ.

Background: Diastolic dysfunction and coronary artery disease (CAD) are interrelated. The complications of CAD, myocardial ischemia or infarction, are major causes of diastolic dysfunction. Objective: The aim of this study was to determine the accuracy of CMR in assessment of left ventricular diastolic function in ischemic heart disease patients by using a combination of left atrium size measurement and phase-contrast evaluation of transmitral flow. Also, to evaluate the extent of myocardial scarring in these patients and to correlate between the degree of myocardial scarring and the degree of left ventricular diastolic dysfunction. Patients and methods: In the current study a total number of 22 patients with known ischemic heart disease were enrolled for contrast enhanced CMR examination between January 2019 and October 2019. All patients underwent CMR and echocardiography. Two of the 22 patients were excluded from the study because of technical difficulties or failure of data acquisition (patient did not hear or did not obey breathing instructions). So, 20 patients were included in the present study. Results: our study showed feasibility of routinely performing evaluation of MV flow to assess diastolic function using CMR. We were able to obtain clinical variables including E:A ratios and deceleration times, which agreed with those obtained using TTE. Conclusion: In patients with CAD, extent of myocardial scarring reliably predicts the degree of diastolic dysfunction. LGE-CMR provides a powerful means to noninvasively assess the degree of myocardial scarring.

Background: stenosis or occlusion of the internal carotid artery (ICA) causes a reduction in arterial pressure distal to the stenosis or occlusion. This reduction leads to hypoperfusion which causes chemical changes in the brain and can be detected by proton magnetic resonance spectroscopy (1H-MRS). Objective: The aim of the current study was to explore the value of proton magnetic resonance spectroscopy imaging in patients with internal carotid artery stenosis or occlusion. Patients and Methods: This study included a total of 30 non infarcted patients with stenosis or occlusion of unilateral internal carotid artery, attending at the Department of Radiodiagnosis, Faculty of Medicine, Al-Azhar University (AlHussein) Hospital. Patients underwent MR spectroscopy to detect metabolic changes of the brain. Results: In 30 noninfarcted patients, there was decreasing in N-acetylaspartate (NAA) and increasing in choline in the hemisphere compared with the contralateral side. In addition, there was lactate peaks which found only in 8 patients. Conclusion: 1H-MRSI can reveal abnormal metabolic changes which occur in cerebral tissues with no infarction. However, internal carotid artery may show stenosis or occlusion at an initial stage, which in turn may help guide management decisions and preoperative assessment.

The objective of this article is to investigate the Ulam-Hyres stability and Ulam-Hyres-Rassias stability for some general integral equations f(x)=∫_E F(x,f(x))dx, x∈E, where E is a nonempty set of a Banach space. The main tool used in the analysis is a recent fixed point theory. In this way, we obtain results in 2-Banach Spaces.