Separation of variables in PDEs using nonlinear transformations: Applications to reaction–diffusion type equations

The paper describes a new approach to constructing exact solutions of nonlinear partial differential equations that employs separation of variables using special (nonlinear integral) transformations and the splitting principle. To illustrate its effectiveness, the method is applied to nonlinear reaction–diffusion type equations that involve variable coefficients and arbitrary functions. New exact functional separable solutions as well as generalized traveling wave solutions are obtained

Genetic and environmental contributions to facial morphological variation: a 3D population-based twin study

Introduction Facial phenotype is influenced by genes and environment; however, little is known about their relative contributions to normal facial morphology. The aim of this study was to assess the relative genetic and environmental contributions to facial morphological variation using a three-dimensional (3D) population-based approach and the classical twin study design. Materials and Methods 3D facial images of 1380 female twins from the TwinsUK Registry database were used. All faces were landmarked, by manually placing 37 landmark points, and Procrustes registered. Three groups of traits were extracted and analysed: 19 principal components (uPC) and 23 principal components (sPC), derived from the unscaled and scaled landmark configurations respectively, and 1275 linear distances measured between 51 landmarks (37 manually identified and 14 automatically calculated). The intraclass correlation coefficients, rMZ and rDZ, broad-sense heritability (h2), common (c2) and unique (e2) environment contributions were calculated for all traits for the monozygotic (MZ) and dizygotic (DZ) twins. Results Heritability of 13 uPC and 17 sPC reached statistical significance, with h2 ranging from 38.8% to 78.5% in the former and 30.5% to 84.8% in the latter group. Also, 1222 distances showed evidence of genetic control. Common environment contributed to one PC in both groups and 53 linear distances (4.3%). Unique environment contributed to 17 uPC and 20 sPC and 1245 distances. Conclusions Genetic factors can explain more than 70% of the phenotypic facial variation in facial size, nose (width, prominence and height), lips prominence and inter-ocular distance. A few traits have shown potential dominant genetic influence: the prominence and height of the nose, the lower lip prominence in relation to the chin and upper lip philtrum length. Environmental contribution to facial variation seems to be the greatest for the mandibular ramus height and horizontal facial asymmetr

Symmetry reductions and new functional separable solutions of nonlinear Klein–Gordon and telegraph type equations

The paper is concerned with different classes of nonlinear Klein–Gordon and telegraph type equations with variable coefficient

Multi-parameter reaction–diffusion systems with quadratic nonlinearity and delays: new exact solutions in elementary functions

The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays

Facial genetics: A brief overview

Historically, craniofacial genetic research has understandably focused on identifying the causes of craniofacial anomalies and it has only been within the last 10 years, that there has been a drive to detail the biological basis of normal-range facial variation. This initiative has been facilitated by the availability of low-cost hi-resolution three-dimensional systems which have the ability to capture the facial details of thousands of individuals quickly and accurately. Simultaneous advances in genotyping technology have enabled the exploration of genetic influences on facial phenotypes, both in the present day and across human history. There are several important reasons for exploring the genetics of normal-range variation in facial morphology.     - Disentangling the environmental factors and relative parental biological contributions to heritable traits can help to answer the age-old question "why we look the way that we do?"     - Understanding the etiology of craniofacial anomalies; e.g., unaffected family members of individuals with non-syndromic cleft lip/palate (nsCL/P) have been shown to differ in terms of normal-range facial variation to the general population suggesting an etiological link between facial morphology and nsCL/P.     - Many factors such as ancestry, sex, eye/hair color as well as distinctive facial features (such as, shape of the chin, cheeks, eyes, forehead, lips, and nose) can be identified or estimated using an individual's genetic data, with potential applications in healthcare and forensics.     - Improved understanding of historical selection and adaptation relating to facial phenotypes, for example, skin pigmentation and geographical latitude.     - Highlighting what is known about shared facial traits, medical conditions and genes

The heat and mass transfer modeling with time delay

Nonlinear hyperbolic reaction-diffusion equations with a delay in time are investigated. All equations considered here contain one arbitrary function. Exact solutions are also presented for more complex nonlinear equations in which delay arbitrarily depends on time. Exact solutions with a generalized separation of variables are found. For special cases, new exact solutions in the form of a traveling waves are obtained, some of which can be represented in terms of elementary functions. All of these solutions contain free (arbitrary) parameters, so that one can use them to solve modeling problems of heat and mass transfer with relaxation phenomena

Multilevel analysis of the influence of maternal smoking and alcohol consumption on the facial shape of English adolescents

This cross-sectional study aims to assess the influence of maternal smoking and alcohol consumption during pregnancy on the facial shape of non-syndromic English adolescents and demonstrate the potential benefits of using multilevel principal component analysis (mPCA). A cohort of 3755 non-syndromic 15-year-olds from the Avon Longitudinal Study of Parents and Children (ALSPAC), England, were included. Maternal smoking and alcohol consumption during the 1st and 2nd trimesters of pregnancy were determined via questionnaire at 18 weeks gestation. 21 facial landmarks, used as a proxy for the main facial features, were manually plotted onto 3D facial scans of the participants. The effect of maternal smoking and maternal alcohol consumption (average 1–2 glasses per week) was minimal, with 0.66% and 0.48% of the variation in the 21 landmarks of non-syndromic offspring explained, respectively. This study provides a further example of mPCA being used effectively as a descriptive analysis in facial shape research. This is the first example of mPCA being extended to four levels to assess the influence of environmental factors. Further work on the influence of high/low levels of smoking and alcohol and providing inferential evidence is required

A 3D morphometric perspective for facial gender analysis and classification using geodesic path curvature features

The relationship between the shape and gender of a face, with particular application to automatic gender classification, has been the subject of significant research in recent years. Determining the gender of a face, especially when dealing with unseen examples, presents a major challenge. This is especially true for certain age groups, such as teenagers, due to their rapid development at this phase of life. This study proposes a new set of facial morphological descriptors, based on 3D geodesic path curvatures, and uses them for gender analysis. Their goal is to discern key facial areas related to gender, specifically suited to the task of gender classification. These new curvature-based features are extracted along the geodesic path between two biological landmarks located in key facial areas. Classification performance based on the new features is compared with that achieved using the Euclidean and geodesic distance measures traditionally used in gender analysis and classification. Five different experiments were conducted on a large teenage face database (4745 faces from the Avon Longitudinal Study of Parents and Children) to investigate and justify the use of the proposed curvature features. Our experiments show that the combination of the new features with geodesic distances provides a classification accuracy of 89%. They also show that nose-related traits provide the most discriminative facial feature for gender classification, with the most discriminative features lying along the 3D face profile curve