Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}). In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

Adult Stature Estimation from Radiographic Metatarsal Length in a Contemporary Korean Population

The ability to estimate stature can be important in the identification of skeletal remains. This study aims to develop a Korean-specific equation predicting stature using radiographic measurements in the contemporary Korean population. 200 healthy Korean adults, including 102 males and 98 females, were randomly selected (age, range 20–86 years). The first and second metatarsals of the foot were measured by a standing X-ray using a digital medical image viewer. The result showed a statistically significant correlation between metatarsal length and stature in Korean populations (male, R = 0.46, p < 0.001; female, R = 0.454, p < 0.001). Values of correlation coefficients (R) of the equations were 0.431 to 0.477. Compared to equations derived from other races, the Korean-specific equation showed significantly lower error values for estimating the actual height of Koreans through cross-validation. In conclusion, this study is the first to propose a Korean-specific regression formula for estimating stature using metatarsal length and a verified formula for precise application to the Korean population. However, given the relatively low correlation coefficient, the stature estimation formula derived from this study can be utilized when other bones that allow more accurate stature estimation are not available.

Stable and self-consistent charged gravastar model within the framework of $$f(R,\,T)$$ gravity

In this work, we discuss the configuration of a gravastar (gravitational vacuum stars) in the context of $$f(R, \,T )$$ f ( R , T ) gravity by employing the Mazur–Mottola conjecture (Mazur and Mottola in Report No. LA-UR-01-5067, 2001; Mazur and Mottola, Proc Natl Acad Sci USA 101:9545, 2004). The gravastar is conceptually a substitute for a black hole theory as available in the literature and it has three regions with different equation of states. By assuming that the gravastar geometry admits a conformal Killing vector, the Einstein–Maxwell field equations have been solved in different regions of the gravastar by taking a specific equation of state as proposed by Mazur and Mottola. We match our interior spacetime to the exterior spherical region which is completely vacuum and described by the Reissner–Nordström geometry. For the particular choice of $$f(R,\,T)$$ f ( R , T ) of $$f(R, \,T )=R+2\gamma T$$ f ( R , T ) = R + 2 γ T , here we analyze various physical properties of the thin shell and also present our results graphically for these properties. The stability analysis of our present model is also studied by introducing a new parameter $$\eta $$ η and we explore the stability regions. Our proposed gravastar model in the presence of charge might be treated as a successful stable alternative of the charged black hole in the context of this version of gravity.

Age-specific definition of low anti-Mullerian hormone and associated pregnancy outcome in women undergoing IVF treatment

Background The age-specific definition of low anti-müllerian hormone (AMH) is lacking. This study aims to define an age-specific reference for low AMH and to evaluate the associated outcome in women undergoing IVF treatment. Methods A retrospective study was performed in women receiving IVF treatment at the Shenzhen maternity and child healthcare hospital between September 2016 and September 2018. We excluded cases without AMH concentration. Polynomial least-squares regression was used to estimate the age-specific reference ranges for AMH after log-transformed. The age-specific 10th centile was defined as the threshold of low AMH concentration. Results A total of 909 patients were analyzed in this study. The age-specific reference ranges for AMH were established using linear regression model and resulted in an age-specific equation for mean: mean of LnAMH = (− 0.085 × age) + 3.722 (ng/ml, in unit). Women with AMH level higher than 10th centile had favorable outcomes in ovarian stimulation compared to those with low AMH level. In patients younger than 35 years, the rates of clinical pregnancy per transfer and ongoing pregnancy per transfer in the subgroup with AMH level higher than 10th centile were significantly higher than that in the subgroup of low AMH level, 57% versus 31.3% p = 0.003 and 51.9% versus 21.9% p = 0.001, respectively. Conclusion Women with AMH lower than age-specific 10th centile had significantly unfavorable outcomes after IVF treatment. The age-specific 10th centile of AMH concentration may be useful to predict the outcome of IVF treatment.

Possible Contamination from Rainwater in Community Pool

The project is meant to create an equation that can be used to estimate the amount of organic pollutant – bacteria - that is present in a swimming pool per day from rainwater. This equation is derived through a differential equation of the rate in minus the rate out. The created differential equation is an ordinary linear differential equation and is solved using an integration factor. The general solution is then converted into a specific equation using an initial condition. The resulting equation provides an approximate number of organic contaminants x(t) present in the pool after an amount of time in days (t). The equation finds that the pool – during its closure – has been cleaned often enough. It also provides a method to estimate the amount of contamination from rain after any other extended closures.

Rotation Curve and Dark Matter Halo profile in Finsler Geometry

Galactic dark matter is an active area of research in recent time. Several researchers proposed several descriptions of radial profiles of dark matter halos by using N-body simulations. Among them, Navarro, Frenk and White (NFW) dark matter profile provides the most accurate description of dark matter halos. It is believed that dark matter is smooth and distributed uniformly throughout space. Using Finslerian geometrical background and a specific equation of state, we propose a new way to estimate the rotational velocity of galaxies based on the NFW dark matter profile. On small scales the first few distances (about 30 kpc) the velocity increases whereas in the outer region of the galaxies, the rotational velocity is found to be more or less constant which supports observed rotational velocities.

Generalised radiating fields in Einstein–Gauss–Bonnet gravity

A five-dimensional spherically symmetric generalised radiating field is studied in Einstein–Gauss–Bonnet gravity. We assume the matter distribution is an extended Vaidya-like source and the resulting Einstein–Gauss–Bonnet field equations are solved for the matter variables and mass function. The evolution of the mass, energy density and pressure are then studied within the spacetime manifold. The effects of the higher order curvature corrections of Einstein–Gauss–Bonnet gravity are prevalent in the analysis of the mass function when compared to general relativity. The effects of diffusive transport are then considered and we derive the specific equation for which diffusive behaviour is possible. Gravitational collapse is then considered and we show that collapse ends with a weak and conical singularity for the generalised source, which is not the case in Einstein gravity.

Validity of Using Velocity to Estimate Intensity in Resistance Exercises in Men and Women

This study aimed to examine the validity of using bar velocity to estimate relative load in squat and bench-press exercises for both young men and women. Twenty-five men and 25 women performed a progressive loading test up to 1-RM in the squat and bench-press exercises, which were repeated after 2-weeks. Relationships between mean propulsive velocity and%1-RM were analysed. A second-order polynomial equation for predicting the corresponding velocity of each percentage of 1-RM was developed for men (validation). This equation was then applied in women (cross-validation). Moreover, a specific equation for women was developed (validation) and was also applied in a sub-sample of women (cross-validation). Close relationships (R2: 0.91–0.95) between bar velocity and relative load were observed in both sexes for squat and bench press. Men’s equation applied to women showed a high level of agreement, although lower bias and higher level of agreement was observed when a sex-specific equation was applied in women, both validation and cross-validation samples. In conclusion, lifting velocity can be used to accurately prescribe the relative load regardless of sex in both upper-body and lower-body exercises, although when estimating load from velocity measures it will be necessary to use the sex-specific equation for each exercise.

The effect of different analytical platforms and methods on the performance of population-specific adjusted calcium equation

Background A recent attempt to improve the diagnostic value of adjusted calcium addressed a primary care-specific adjusted calcium equation, but validated the new equation for Roche Cobas, BCG and NM-BAPTA methods only. In this study, we aim to validate a population-specific equation for other methods and platforms. Method We collected retrospective patient data-sets from 15 hospital laboratories using a range of commercially available analytical platforms and methods for calcium and albumin measurements. Raw data-sets were collected and filtered according to Payne’s criteria, and separate adjusted calcium equations were derived for hospitalized and primary care patients. Results Mean albumin and calcium results were significantly higher in primary care populations ( P < 0.0001). The prevalence of hypocalcaemia using adjusted calcium ranged between 6% and 44% for inpatient data-sets and was higher in users of BCG methods. The application of community-specific adjustment equation to primary care data-sets reduced the prevalence of hypocalcaemia (mean 1.7%, range 0.8–3.7%). Conclusion We demonstrated that the use of a community-specific calcium adjustment equation to a primary care population reduces both the percentage and the variation of hypocalcaemia between different laboratories.