A theoretical relation is presented between the seismological FOURIER AMPLITUDE spectrum and the mean squared value of the elastic response, which is defined by Gaussian distribution. By shifting a general process to its mean value, spectrum of the mean squared value of the displacements computed from the FOURIER AMPLITUDE spectrum and the real part of the relative displacement transfer function of the single-degree-of-freedom elastic oscillator. It is shown that the relation presented in this work opens the door for a better understanding of the relationship between time invariant mean squared value of linear response of a single degree freedom system and seismological variables, such as magnitude, focal distance, and path and soil conditions. For illustrating the proposed theoretical relation, the mean squared values of a drift have been calculated for earthquake ground motions with different magnitude, focal distance and soil.

Let G be a locally compact group and w be a symmetric weight function on G. We define a co-product Gw on the weighted algebra L¥ (G, w-1) of essentially w-bounded Borel measurable functions on G and show that L¥ (G, w-1) becomes a Kac algebra with natural co-inverse kw and Haar weight  fw. We use the machinery of Kac algebras to introduce the weighted FOURIER and FOURIER-Stieltjes algebra A(G, w-1) and B(G, w-1) of G.

In this paper, we introduce the decomposition property on locally compact groups and we give a multi-norm structure based on the FOURIER and FOURIER-Stieltjes algebras on locally compact groups with this property. We show that compact groups have the decomposition property. This construction generalizes the known multi-norm structure of L1-algebras on compact abelian groups. Also, we study some special multi-bounded maps on the FOURIER algebras and improve some results in this theory.

In this paper we shall examine the quadratic FOURIER transform which is introduced by the generalized quadratic function for one order parameter in the ordinary FOURIER transform. This will be done by analyzing corresponding six subcases of the quadratic FOURIER transform within a reproducing kernel Hilbert spaces framework.

Introduction: Clinical performance of light cured resin composites is related to their degree of polymerization. The purpose of this study was to compare the degree of conversion of packable and hybrid composites by FTIR (FOURIER Transform Infrared Spectroscopy).Materials & Methods: In this experimental study, 40 composite disks were prepared in two groups from Z250 and P60 composites. Each group was divided into four subgroups (2mm thickness cured with QTH unit, 2mm thickness cured with LED unit, 5mm thickness cured with QTH unit, and 5mm thickness cured with LED unit). Then samples were evaluated by FTIR to determine the degree of conversion (DC). Data were analyzed by Kolmogorov-Smirnov and three way ANOVA.Results: There was not a significant difference among the DC of the materials tested. LED significantly increased the degree of conversion of materials tested (P<0.001). DC values were significantly greater in 2mm diameter samples vs 5mm ones (P<0.001).Conclusion: The use of incremental technique in deep cavities restored with these composites is suggested. Also using LED is advised for curing composites because of a better result.

Extended AbstractIntroductionRainfall plays an important role in maintaining life on Earth and maintaining the balance of ecosystems. It is essential to understand its importance for various environmental, agricultural, and hydrological aspects. Agriculture relies heavily on rainfall for crop growth. Adequate rainfall ensures that the soil remains fertile and productive. Forecasted rainfall is critical for various sectors including disaster management and urban planning. Accurate forecasts enable individuals and organizations to make informed decisions that can reduce risks and increase productivity. This information allows them to effectively plan planting and harvesting schedules and ensure optimal crop performance. Time series models are statistical tools used to analyze and forecast data points collected over time. Common types of models include autoregressive (AR), moving average (MA), and autoregressive integrated moving average (ARIMA) models. These models use a chronological sequence of observations and allow analysts to identify patterns, trends, and seasonal changes that can provide future predictions. Therefore, the purpose of this research is to develop an integrated system to investigate and evaluate the precipitation trend and its changes in the statistical period of 24 years and use it to predict precipitation in the next 5 years. For this purpose, the rainfall data of three stations, Tabriz, Amol and Yazd, which have different climates, were used. On the other hand, for the evaluation of time series and forecasting operations, two models of FOURIER series and auto-regression were used, and then the obtained results were analyzed with evaluation criteria and graphic diagrams.Materials and MethodsIn this study, the monthly rainfall time series of three stations in Tabriz, Amol, and Yazd during 24 years (2023-2000) was taken from the database of Mathematica software. First, the data were examined and by applying the outlier data processing, the data that had a significant difference were removed and the data that were not measured were completed with the interpolation method. After the data was completed and evaluated, it was divided into two parts, training and testing. Normally, the ratio of this division in the fields of hydrology and hydraulics is 70 to 30. For this purpose, 70% of the data (2016-2000) was assigned to training and 30% (2023-2017) to the test. Then, to predict monthly rainfall, two models, FOURIER and autoregressive, which are based on time series, were used. These methods are described below. FOURIER series is the mathematical representation of a periodic function as an infinite sum of sine and cosine functions. This concept is fundamental in various fields such as signal processing, physics, and engineering, and allows complex periodic signals to be analyzed. Auto-Regressive (AR) models work on the principle that the current value of a time series can be expressed as a linear combination of its past values with a random error. The performance comparison of the two models was evaluated using four criteria: root mean square error (RMSE), correlation coefficient (r), Nash Sutcliffe coefficient (NSE), and Wilmot coefficient (WI). The results showed.Results and Discussion The results showed that the FOURIER model has an average error of 1.21, correlation coefficient of 0.87, Nash Sutcliffe coefficient of 0.74, and Wilmot coefficient of 0.91. The predictions made with the FOURIER model were more reliable than the autocorrelation model. The graph related to the FOURIER model in all three stations has almost the same trend as the real values and has less difference. But in some places, there are few fits. On the other hand, the graph of the AR model has a big difference from the real values, and in most of the points, it has a poor fit. Considering the scatter diagrams of FOURIER model and AR, the scatter diagrams of FOURIER model have less dispersion. Therefore, the coefficient of determination for the FOURIER model in predicting the monthly rainfall of Tabriz is equal to 0.86, Amol is equal to 0.73, and Yazd is equal to 0.68. For the AR model, it is equal to 0.18 for Tabriz, 0.34 for Amol, and 0.48 for Yazd. These results show that the FOURIER model has identified changes in precipitation trends better than the AR model and has provided more reliable predictions. Also, the predicted five-year trends by the FOURIER model have more natural changes than the AR model, but on the contrary, the AR model has fewer fluctuations and has changes close to a straight line.Conclusion Precipitation forecasting is critical for various sectors including agriculture, natural disaster management, and climate adaptation. Accurate forecasts can significantly impact food security, infrastructure planning, and environmental conservation. Complete and abundant data has a significant impact on the accuracy of predictions and improves it. The FOURIER model with the lowest error and the highest degree of correlation estimated acceptable forecasts for the precipitation of three stations, and on the other hand, the forecasts of the trend of the FOURIER model for five years have acceptable changes and are somewhat similar to the trend of real precipitation. FOURIER series assume that the data are periodic, which may not be true for all precipitation patterns. Many regions experience irregular rainfall distributions that do not fit well into periodic models, leading to inaccurate forecasts. Therefore, in this research, by integrating the probability distribution with the FOURIER model, this problem was solved as much as possible, and accurate predictions of precipitation were made. While FOURIER models can be effective for short-term forecasting, their performance degrades over longer periods. This limitation is due to the fact that the basic assumptions and periodic nature of the model are not maintained with the extension of the forecasting horizon, making it challenging to accurately predict precipitation beyond a certain time frame. Also, the effectiveness of the FOURIER model depends on the quality and temporal resolution of the input data.

Background: S1 radiculopathy is a high prevalence illness and its early diagnosis is of crucial importance. Several reports have revealed the efficacy of magnetic stimulation in diagnosis of this condition. We conducted this study to compare MMEP latency and MMEP AMPLITUDE in patients with S1 radiculopathy and healthy controls in Shohadaye Tajreash Hospital in 2007.Materials and Methods: This historical cohort study investigated 31 patients with S1 radiculopathy (according to clinical history and physical examination) and 33 healthy controls. H-reflex, MMEP latency and MMEP AMPLITUDE were evaluated in both groups. Patients and controls were not informed about the type of study and performers were unaware of the participants condition. Results were analyzed using statistical tests.Results: There was not significant difference between cases and controls regarding age and sex. Mean MMEP latency was 15.23±2.47ms and 13.42±1.6ms in patients and controls, respectively (P<0.05) and mean MMEP AMPLITUDE was 0.55±0.53µv and 1.29±1.22µv in patients and controls, respectively (P<0.06). We found abnormal MMEP latency in 12.2% of controls and 48.4% of patients (P<0.005) and abnormal MMEP AMPLITUDE in 87.9% of controls and 100% of patients (P<0.06).Conclusion: MMEP latency seems to be efficient in diagnosis of S1 radiculopathy. A study with a precise diagnostic design is recommended.