Exact solutions of sixth and fifth degree equations

The real problematic with algebraic polynomial equations is how to exactly solve any sixth and fifth degree polynomial equations. In this study, we give a new absolute method that presents a new decomposition to exactly solve a sixth degree polynomial equation, while the corresponding fifth degree equation can be easily transformed into a sixth degree equation of this kind (sixth degree equation solvable by this method), then the sextic equation (sixth degree equation) obtained will be solved by applying the principles of this method; moreover, the solutions of the quintic equation (fifth degree equation) will be easily deduced.

Forecasting COVID-19 infection trends and new hospital admissions in England due to SARS-CoV-2 Variant of Concern Omicron

Objectives: On November 26, 2021, WHO designated the variant B.1.1.529 as a new SARS-CoV-2 variant of concern (VoC), named Omicron, originally identified in South Africa. Several mutations in Omicron indicate that it may have an impact on how it spreads, resistance to vaccination, or the severity of illness it causes. We used our previous modelling algorithms to forecast the spread of Omicron in England. Design: We followed EQUATOR TRIPOD guidance for multivariable prediction models. Setting: England. Participants: Not applicable. Interventions: Non-interventional, observational study with a predicted forecast of outcomes. Main outcome measures: Trends in daily COVID-19 cases with a 7-day moving average and of new hospital admissions. Methods: Modelling included a third-degree polynomial curve in existing epidemiological trends on the spread of Omicron and a new Gaussian curve to estimate a downward trend after a peak in England. Results: Up to February 15, 2022, we estimated a projection of 250,000 COVID-19 daily cases of Omicron spread in the worse scenario, and 170,000 in the best scenario. Omicron might represent a relative increase from the background daily rates of COVID-19 infection in England of mid December 2021 of 1.9 to 2.8-fold. With a 5-day lag-time, daily new hospital admissions would peak at around 5,063 on January 23, 2022 in the worse scenario. Conclusion: This warning of pandemic surge of COVID-19 due to Omicron is calling for further reinforcing in England and elsewhere of universal hygiene interventions (indoor ventilation, social distance, and face masks), and anticipating the need of new total or partial lockdowns in England.

A note on the polynomial surface fitting method for estimation of winds at different isobaric levels using the observed wind at 850 hPa level

A third degree polynomial surface fitting technique is adopted to compute the winds at different isobaric levels for the limited area domain from 35°E to 140°E and 30° S to 40oN for the month of June. The polynomial surfaces are fitted to the ratios of the monthly mean winds at different isobaric levels with the 850 hPa winds. The surfaces are fitted to the II and v components of the winds separately. The winds for the level are then reconstructed using the computed coefficients and observed wind at 850 hPa. The results of the technique are presented and discussed in the paper.

Best Regression for Eye Recognition

Human eye offers a number of opportunities for biometric recognition. The essential parts of the eye like cornea, iris, veins and retina can determine different characteristics. Systems using eyes’ features are widely deployed for identification in government requirement levels and laws; but also beginning to have more space in portable validation world. The first image was prepared to be used and monitored using CLAHE which means (Contrast Limited Adaptive Histogram Equalization) to improve the contrast of the image, after that the 3D surface plot was created for this image then different types of regression were used and the better one was chosen. The results showed that power regression is better, and fitter than other fitting methods (8th, 7th, 6th, 5th, 4th, 3rd, 2nd) degree polynomial, and straight line respectively, when depending on the sum of residual squared. The estimations of R-square demonstrated that (5th, 6th, 7th, 8th) have a great proportion of variance in the model followed by (power, 4th, 3rd, 2nd, straight line) respectively. The conclusion from these results is that the power regression has a better fitting than other types of fitting functions for this study and similar ones.

Analytical determination of DNA melting characteristic parameters using the optimal degree polynomial regression model

The article describes a new technique for determining two main parameters of DNA melting: the melting temperature Tm and the temperature melting range ΔT, based on the plotting of an approximating polynomial function for the DNA melting curve. An algorithm is proposed for reducing the melting curve to approximation by the fourth degree polynomial function in accordance with the physical aspect of the DNA melting process. The correctness of the optimal degree choice from the condition of minimizing the value of the Akaike’s information criterion corrected has been confirmed. Analytical expressions for calculating the values of Tm and ΔT are given oriented to a polynomial function of the fourth degree. Results comparison of applying the proposed and well-known techniques based on the experimental data is performed. The advantages of the new technique are revealed.

The Object According State Prediction to Diagnostic Data

The predicting methodology the state of the object based on diagnostic data is considered. With the selected parameter that determines the state of the object, it is measured in real time at a fixed sampling step. According to the measurement data, the value of this parameter is predicted in the future. This operation is implemented by an extrapolator of the l order - a l degree polynomial, built using the least squares method based on the previous measurements results. The changing process model of the diagnosed parameter is a random time function described by the stationary centered random component sum and a mathematical expectation deterministic change. The estimating prediction error method and the extrapolator parameters influence on its value are presented.

Fetal cerebellar development: 3D morphometric analysis of fetal postmortem MRI among Chinese fetuses.

The development of the cerebellum starts from early gestational period and extends postnatal. Because of its protracted development, the cerebellum is susceptible to developmental anomalies such as Dandy-Walker malformations, Blakes pouch cysts and vermin hypoplasia. Measurements of fetal cerebellar parameters of a normal growing fetus in each week of gestation is important for setting up morphometric standards and hence used as clinical reference data. Any deviation from the normal cerebellar parameters alerts the clinicians for the possibility of presence of cerebellar malformations. Study objective: The objective of this study was to assess the fetal cerebellar growth by quantifying the following parameters; fetal cerebellar volume, anterior-posterior diameter and superior-inferior diameter. Methods: We used 3T and 7T MRI to scan the postmortem fetal brains at different stages of development and subsequently analyze the images using ITK-SNAP software. Results: The mean superior-inferior cerebellar diameter was found to be 19.12+2. 70mm.The linear(y=bo+b1t) model was the best fit (r2=0.996, F=32022.961) to describe the relationship between the gestational age and the superior-inferior diameter(y=5.89+0.49t). There was significant correlation between the superior-inferior cerebellar diameter and the gestation age, Pearson correlation coefficient of 0.999, r=0.001. The median cerebellar volume was 8607.7mm, the mean rank high among males(78.12) as compared to female(68.25). There was no statistically significant difference of the cerebellar volume between males and females (u=2193.5,p=0.16). The quadratic(y=bo+b1t+b2t2) model was the best fit regression equation (r2=0.994,F=10791.157) describing the relationship between the cerebellar volume and the gestational age. The median anterior-posterior diameter was 12.45 mm. There was significant correlation between anterior-posterior diameter and the gestational age with Spearmans rho of (0.997, p=0.01). The linear model was the best fit the best fit model (y=bo+b1t) describing the relationship between anterior-posterior diameter and the gestational age(y=3.31+0.5t) r2=0.998, F=70646.838. Conclusion: Significant correlation between the superior-inferior cerebellar diameters, the anterior-posterior cerebellar diameter and the gestational age was found. These two linear parameters follow the first-degree polynomial in relation to the gestational age. The cerebellar volume follows the second-degree polynomial as it increases with the gestational age and correlate significantly with the gestational age. This study has provided new insight to the development of the cerebellum, and setup a benchmark data of which the deviation from it will alert the clinicians for the possibility of presence of cerebellar malformations. Key words: Cerebellar Development, Cranial Magnetic Resonance imaging, Cerebellar Malformations