Reduction Operators of Linear Second-Order Parabolic Equations

The reduction operators, i.e., the operators of nonclassical (conditional) symmetry, of (1+1)-dimensional second order linear parabolic partial differential equations and all the possible reductions of these equations to ordinary differential ones are exhaustively described. This problem proves to be equivalent, in some sense, to solving the initial equations. The ``no-go'' result is extended to the investigation of point transformations (admissible transformations, equivalence transformations, Lie symmetries) and Lie reductions of the determining equations for the nonclassical symmetries. Transformations linearizing the determining equations are obtained in the general case and under different additional constraints. A nontrivial example illustrating applications of reduction operators to finding exact solutions of equations from the class under consideration is presented. An observed connection between reduction operators and Darboux transformations is discussed.Comment: 31 pages, minor misprints are correcte

Oral-health-related background factors and dental service utilisation among Sudanese children with and without a congenital heart defects

Background: Sudanese children with congenital heart defects (CHDs) were found to have poorer oral health than those without CHDs. The aims of this study were to: describe the patterns of oral-health-related background factors in children with and without CHD and explore any differences, and to evaluate the effects of background factors on caries and gingivitis prevalence and dental services utilisation. Methods: In this analytical cross-sectional study, caregivers of children aged 3–12 years with (CHD cases n = 111) and without CHDs (Controls n = 182), underwent face-to-face interviews using a structured questionnaire. The questionnaire items covered several oral health background factors (independent variables) including: child’s health status, oral hygiene practices, dental services utilization, mother’s level of education, and caregiver’s perception and awareness of their child’s oral health. The relationship between these factors and occurrence of ‘caries’ and ‘gingivitis’ as well as ‘child’s dental services utilisation’ (dependent variables) were explored using multiple adjusted and hierarchal logistic regression analyses. Results: Compared with controls, CHD cases had lower frequencies of brushing and use of fluoridated toothpaste, and their caregivers were less knowledgeable about caries. Among CHD cases, the variables (brushing and fluoridated toothpaste use) had significant impacts on caries prevalence (odd ratio (OR) =5.6, 95% confidence interval (CI): 1.4–22.8 and OR = 0.3, 95% CI: 0.1–0.8 for infrequent compared to frequent ones, respectively) as well as the mother’s level of education (OR = 2.6, 95% CI: 1.0–6.4). When differences in background factors were controlled for, the adjusted ORs for caries and gingivitis prevalence in CHD cases compared with controls were 1.8, (95% CI: 1.1–3.2) and 5.3 (95% CI: 2.9–9.4), respectively. Among CHD cases, the child’s age (8–12 years: OR = 11.9, 95% CI: 1.9–71.6), and the mother’s level of education (lower education: OR = 0.2, 95% CI: 0.03–0.9) were significantly associated with the child’s dental services utilisation. Conclusions: Lower frequencies of brushing and use of fluoride tooth paste were reported among CHD cases, and brushing had the predominant significant impact on caries prevalence. The child’s age and the mother’s level of education were the main factors affecting the child’s (CHD cases) dental services utilisation

Complete group classification of a class of nonlinear wave equations

Preliminary group classification became prominent as an approach to symmetry analysis of differential equations due to the paper by Ibragimov, Torrisi and Valenti [J. Math. Phys. 32, 2988-2995] in which partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the infinite-dimensional equivalence algebra of the class under consideration. In the present paper we implement, up to both usual and general point equivalence, the complete group classification of the same class using the algebraic method of group classification. This includes the complete preliminary group classification of the class and finding Lie symmetry extensions which are not associated with subalgebras of the equivalence algebra. The complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel algebra. The set of admissible point transformations of the class is exhaustively described in terms of the partition of the class into normalized subclasses. A version of the algebraic method for finding the complete equivalence groups of a general class of differential equations is proposed.Comment: 39 page

Realizations of Real Low-Dimensional Lie Algebras

Using a new powerful technique based on the notion of megaideal, we construct a complete set of inequivalent realizations of real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. Our classification amends and essentially generalizes earlier works on the subject. Known results on classification of low-dimensional real Lie algebras, their automorphisms, differentiations, ideals, subalgebras and realizations are reviewed.Comment: LaTeX2e, 39 pages. Essentially exetended version. Misprints in Appendix are correcte

Group classification of (1+1)-Dimensional Schr\"odinger Equations with Potentials and Power Nonlinearities

We perform the complete group classification in the class of nonlinear Schr\"odinger equations of the form $i\psi_t+\psi_{xx}+|\psi|^\gamma\psi+V(t,x)\psi=0$ where $V$ is an arbitrary complex-valued potential depending on $t$ and $x,$ $\gamma$ is a real non-zero constant. We construct all the possible inequivalent potentials for which these equations have non-trivial Lie symmetries using a combination of algebraic and compatibility methods. The proposed approach can be applied to solving group classification problems for a number of important classes of differential equations arising in mathematical physics.Comment: 10 page

Multidimensional Cosmology: Spatially Homogeneous models of dimension 4+1

In this paper we classify all 4+1 cosmological models where the spatial hypersurfaces are connected and simply connected homogeneous Riemannian manifolds. These models come in two categories, multiply transitive and simply transitive models. There are in all five different multiply transitive models which cannot be considered as a special case of a simply transitive model. The classification of simply transitive models, relies heavily upon the classification of the four dimensional (real) Lie algebras. For the orthogonal case, we derive all the equations of motion and give some examples of exact solutions. Also the problem of how these models can be compactified in context with the Kaluza-Klein mechanism, is addressed.Comment: 24 pages, no figures; Refs added, typos corrected. To appear in CQ

Metal-Insulator Transitions in Degenerate Hubbard Models and Ax_xC60_{60}

Mott-Hubbard metal-insulator transitions in $N$-fold degenerate Hubbard models are studied within the Gutzwiller approximation. For any rational filling with $x$ (integer) electrons per site it is found that metal-insulator transition occurs at a critical correlation energy $U_c(N,x)=U_c(N,2N-x)=\gamma(N,x)|\bar{\epsilon}(N,x)|$, where $\bar{\epsilon}$ is the band energy per particle for the uncorrelated Fermi-liquid state and $\gamma(N,x)$ is a geometric factor which increases linearly with $x$. We propose that the alkali metal doped fullerides $A_xC_{60}$ can be described by a 3-fold degenerate Hubbard model. Using the current estimate of band width and correlation energy this implies that most of ${\rm A_xC_{60}}$, at integer $x$, are Mott-Hubbard insulators and ${\rm A_3C_{60}}$ is a strongly correlated metal.Comment: 10 pages, Revte

Continuous Symmetries of Difference Equations

Lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations. In this article we review the results of a much more recent program: the use of Lie groups to study difference equations. We show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners. One is to use generalized symmetries acting on solutions of difference equations, but leaving the lattice invariant. The other is to restrict to point symmetries, but to allow them to also transform the lattice.Comment: Review articl

Homogeneity in prediction of survival probabilities for subcategories of hipprosthesis data : the Nordic Arthroplasty Register Association, 2000–2013

Introduction: The four countries in the Nordic Arthroplasty Register Association (NARA) share geographic proximity, culture, and ethnicity. Pooling data from different sources in order to obtain higher precision and accuracy of survival-probability estimates is appealing. Nevertheless, survival probabilities of hip replacements vary between the countries. As such, risk prediction for individual patients within countries may be problematic if data are merged. In this study, our primary question was to address when data merging for estimating prosthesis survival in subcategories of patients is advantageous for survival prediction of individual patients, and at what sample sizes this may be advised. Methods: Patients undergoing total hip replacements for osteoarthritis between January 1, 2000 and December 31, 2013 in the four Nordic countries were studied. A total of 184,507 patients were stratified into 360 patient subcategories based on country, age-group, sex, fixation, head size, and articulation. For each patient category, we determined the sample size needed from a single country to obtain a more accurate and precise estimate of prosthesis-survival probability at 5 and 10 years compared to an estimate using data from all countries. The comparison was done using mean-square error. Results: We found large variations in the sample size needed, ranging from 40 to 2,060 hips, before an estimate from a single Nordic country was more accurate and precise than estimates based on the NARA data. Conclusion: Using pooled survival-probability estimates for individual risk prediction may be imprecise if there is heterogeneity in the pooled data sources. By applying mean-square error, we demonstrate that for small sample sizes, applying the larger NARA database may provide a more accurate and precise estimate; however, this effect is not consistent and varies with the characteristics of the subcategory