In this paper, we determine the structure of the space of multipliers of the range of a composition operator Cj that induces by the conditional expectation between two Lp(å) spaces.

Let p be prime and , : x 7! xgx, the Discrete Lambert Map. For k ,1, let V = {0, 1, 2, . . ., pk −,1}. The iteration digraph is a directed graph with V as the vertex set and there is a unique directed edge from u to , (u) for each u 2 V. We denote this digraph by G(g, pk), where g 2 (Z/pkZ), . In this piece of work, we investigate the structural properties and find new results modulo higher powers of primes. We show that if g is of order pd, 1 ,d ,k −,1 then G(g, pk) has pk−, d d 2 e loops. If g = tp + 1, 1 ,t ,pk−, 1 −,1 then the digraph contains p k +1 2 cycles. Further, if g has order pk−, 1 then G(g, pk) has p −,1 cycles of length pk−, 1 and the digraph is cyclic. We also propose explicit formulas for the enumeration of components.

Introduction: Myasthenia gravis disease (MGD) and inflammatory myopathy (IM) are commonly reported in the literature and usually appear with thymic pathology. Lambert-Eton myasthenic syndrome (LEMS) associated with IM is extremely rare. Case Presentation: We report a 42-year-old female patient who presented with proximal muscle weakness of the upper and lower limbs, normal creatinine kinase (CK) level, and positive acetylcholine and voltage-gated calcium channel receptor antibodies. There were no oculobulbar symptoms and no history of thymoma, and the electrophysiological tests were unremarkable. Muscle biopsy revealed focal perimysial and perivascular inflammation, predominantly B-cell lymphocytes, in a non-necrotizing muscle. Conclusions: LEMS associated with IM, particularly B-cell inflammation, has never been reported in the absence of cancer history. Clinical investigations and myopathological features can help establish the diagnosis.

This paper presents an innovative approach integrating the Image Processing Method and the Beer-Lambert Law to study Sea Water Intrusion in riverine systems. Using an experimental setup, this study characterizes the dynamics between saline and fresh waters, providing detailed spatial and temporal salinity distribution analyses. The Beer-Lambert Law using Python is employed to convert image pixel data to accurate concentration measurements, essential for understanding the diffusion and mixing processes of saline wedges. Results showed that the concentration of the salt water is decreased when it flows upstream horizontally. also, the concentration of the saline water is decreased vertically, due to the salt and fresh water mixing. The "Max Concentration Profile" graph shows a sharp peak at the start, reaching approximately 1000 g/cm³, marking the point of saline introduction. Within the first 10 cm, the concentration rapidly decreases to about 600 g/cm³, indicating swift diffusion and dilution. Beyond this point, the concentration gradually decreases, stabilizing around 200 g/cm³ past the 100 cm mark, reflecting ongoing diffusion and mixing processes. The "Concentration Contour" graph shows high salinity concentrations near the bottom 10 cm of the flume. Salinity concentration decreases moving vertically up the flume, indicated by cooler colors. From approximately 30 cm in height, salinity remains low and uniform, suggesting that denser saline water settles at the bottom due to gravity.