Search

We analyze the local conservation laws, auxiliary (potential) systems, potential symmetries and a class of new exact solutions for the Black-Scholes model time-dependent parameters (BST model). First, we utilize the computer package GeM to construct local conservation laws of the BST model for three different forms of multipliers. We obtain two conserved vectors for the second-order multipliers of form \begin{document}$ \Lambda(x,u,u_x,u_{xx}) $\end{document} . We define two potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document} corresponding to the conserved vectors. We construct two singlet potential systems involving a single potential variable \begin{document}$ v $\end{document} or \begin{document}$ w $\end{document} and one couplet potential system involving both potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document} . Moreover, a spectral potential system is constructed by introducing a new potential variable \begin{document}$ p_\alpha $\end{document} which is a linear combination of potential variables \begin{document}$ v $\end{document} and \begin{document}$ w $\end{document} . The potential symmetries of BST model are derived by computing the point symmetries of its potential systems. Both singlet potential systems provide three potential symmetries. The couplet potential system yields three potential symmetries and no potential symmetries exist for the spectral potential system. We utilize the potential symmetries of singlet potential systems to construct three new solutions of BST model.

Access to essential surgical care is vital for reduction in mortality and morbidity as a result of surgical conditions. These account for 28–32% of the overall global burden of disease, yet billions of people lack access to safe, affordable surgical and anesthesia care when needed. The purpose of this study was to assess the capacity for surgical care in rural hospitals across four provinces of Pakistan. This was a cross-sectional study undertaken in 10 rural hospitals across four provinces of the country. Of these, six were district and four sub-district hospitals that were purposively selected in consultation with the government. Data were gathered using the WHO-PGSSC Surgical Assessment Tool. This study estimated 3 of the 6 indicators proposed by the Lancet Commission on Global Surgery. While most hospitals had basic provisions of infrastructure and equipment, severe shortage of specialists was observed with 0.56 specialists (surgeons, gynecologists and anesthetists) present per 100,000 population. Two-hour access was possible for the catchment population of 7 out of the 10 hospitals. Of the 43 essential surgical procedures assessed, 13 or 30% procedures were available per hospital. The three Bellwether procedures were provided by only 1 hospital. Mean number of surgeries performed was 753 ± 979 per 100,000 population. Our study has demonstrated major gaps in the provision of surgical care in rural hospitals in Pakistan. While developing a strategy and national action plan is necessary, implementation can immediately begin at the local level to address the gaps that need urgent attention.

We provide the Hamiltonian version of the approximate Noether theorem developed for the perturbed ordinary differential equations (ODEs) (Govinder et al. in Phys Lett 240(3):127–131, 1998) for the approximate Hamiltonian systems. We follow the procedure adopted by Dorodnitsyn and Kozlov (J Eng Math 66(1–3):253–270, 2010) for the Hamiltonian systems of unperturbed ODEs. The approximate Legendre transformation connects the approximate Hamiltonian and approximate Lagrangian. The approximate Noether symmetries determining equation for the approximate Hamiltonian systems is defined explicitly. We provide a formula to establish an approximate first integral associated with an approximate Noether symmetry of the approximate Hamiltonian systems. We analyzed several physical models to elaborate the approach developed here.

The notions of artificial Hamiltonian (partial Hamiltonian) and partial Hamiltonian operators are used to derive the first integrals for the first order systems of ordinary differential equations (ODEs) in epidemiology, which need not be derived from standard Hamiltonian approaches. We show that every system of first order ODEs can be cast into artificial Hamiltonian system q ˙ = ∂ H ∂ p $\dot{q}=\frac{{\partial H}}{{\partial p}}$ , p ˙ = − ∂ H ∂ q + Γ ( t , q , p ) $\dot{p}=-\frac{{\partial H}}{{\partial q}}+\Gamma(t,\;q,\;p)$ (see [1]). Moreover, the second order equations and the system of second order ODEs can be written in the form of artificial Hamiltonian system. Then, the partial Hamiltonian approach is employed to derive the first integrals for systems under consideration. These first integrals are then utilized to find the exact solutions of models from the epidemiology for a distinct class of population. For physical insights, the solution curves of the closed-form expressions obtained are interpreted in order for readers understand the disease dynamics in a much deeper way. The effects of various pertinent parameters on the prognosis of the disease are observed and discussed briefly.

In this article, we present the general form of conservation laws for the nonlinear Rosenau–Hyman compacton K(m, n) equations using multiplier’s approach. General formulas for some new conservation laws are established for the K(m, n) equations. We describe three different cases where Lie symmetries are associated with these generalised conservation laws. The double reduction theory is utilised to construct some new reductions and exact solutions for different values of m and n.

Background: Accessibility and utilization of health care play a significant role in preventing complications during pregnancy, labor, and the early postnatal period. However, multiple barriers can prevent women from accessing pregnancy and neonatal care. The aim of this study was to explore the multifaceted barriers that inhibit women to seek maternal and newborn health care in Thatta, Sindh, Pakistan.Methods: This study employed an interpretive research design using purposive sampling approach. Pre-tested semi-structured interview guides were used for data collection. The data were collected through eight focus group discussions with men and women, and six in-depth interviews with lady health workers and analyzed through thematic analysis. Findings: The study identified individual, socio-cultural, and structural level barriers that inhibit women from seeking maternal and newborn care. Individual barriers included: mistrust towards public health facilities, and inadequate symptom recognition. The two identified socio-cultural barriers were: aversion to biomedical interventions, and gendered imbalances in decision making. The structural barriers included: prohibitively expensive transportation services, and ineffective referral systems. Conclusion and Suggestions: Increasing the coverage of healthcare service without addressing the multifaceted barriers that influence service utilization will not reduce the burden of maternal and neonatal mortality. As this study reveals, care seeking is influenced by a diverse array of barriers that are individual, sociocultural, and structural in nature. A combination of capacity development, health awareness, and structural interventions can address many if not all of these barriers.

Background Contracting out of health services to non-state providers has been widely used in developing countries including Pakistan. Based on three years’ experience of contracting out primary and secondary health services, this paper presents findings of third party evaluation of health services from two rural districts of Sindh Pakistan. Methods This was a baseline vs end-line cross sectional assessment of thirteen primary and secondary healthcare facilities from two rural districts (Thatta and Sujawal) of Sindh province that were contracted out in 2016 to a national non-governmental organization. Healthcare facilities included: 8 rural health centres, 4 taluka headquarter hospitals and 1 district headquarter hospital. District health information system was used to extract three years (2016 – 2019) data on key performance indicators (KPIs) as agreed in contract. We conducted record review related to human resource and budget and in-depth interviews with health managers. Health facility assessment survey and client satisfaction exit interviews were also conducted. Results KPIs showed significant improvement in service provision from baseline. General outpatient department (OPD) (33%), specialist OPD (91%), accident/emergency consultations (106%), in-patient admissions (≥100%) alongside diagnostic/laboratory service utilization (86%) increased substantially. Facility based deliveries (37%), and major (99%) and minor (172%) surgeries also showed significant increase. Preventive services (maternal tetanus toxoid, child vaccination) showed modest improvement (4-19%) in the district (overall) but slight decline in some healthcare facilities. Slight improvement in specialist workforce care was noticed, however challenges related to staff retention persisted. Adequate supplies and equipment were available except radiology services. Most healthcare facilities were also in need of repair. Delayed and partial release of funds by government was a major barrier. Lack of coordination among different stakeholders delivering healthcare in the district was also noted. Majority of clients (60%) were satisfied with service delivery but unavailability of medicine was their main concern. Conclusions Contracting out has the potential to improve service utilization. Autonomy over budget allocation and utilization, appointment of all cadre of staff, and improved coordination among all stakeholders is required to improve service delivery. Quality of care and the longer term health outcomes need further evaluation.

This article investigates the first integrals and closed- form solutions of some nonlinear first-order dynamical systems from diverse areas of applied mathematics. We use the notion of artificial Hamiltonian, and we show that every first-order system of ordinary differential equations (ODEs) can be written in the form of an artificial Hamiltonian system [see Naz and Naeem (ZNA 73(4):323–330, 2018)]. One can also express the second-order ODE or system of second-order ODEs in the form of system of first-order artificial Hamiltonian system. Then the partial Hamiltonian approach is employed to compute the partial Hamiltonian operators and the corresponding first integrals. The first integrals are utilized to construct the closed-form solutions of two-stream model for tuberculosis and dengue fever, Duffing–van der Pol oscillator, nonlinear optical oscillators under parameter restrictions, nonlinear convection model and the two- dimensional galaxy model. We show that how one can apply the existing “partial Hamiltonian approach” for nonstandard Hamiltonian systems. This study provides a new way of solving the dynamical systems of first-order ODEs, second-order ODE and second-order systems of ODEs which are expressed into the artificial Hamiltonian system.

The Lie symmetries, conservation laws, and exact solutions of a generalized nonlinear system and a (2+1)-dimensional generalized Nizhink-Novikov-Veselov (NNV) equation, arising in the study of hydrodynamics, are investigated. The multiplier approach is employed to compute the conservation laws for systems under consideration. The Lie point symmetries are derived and the association between symmetries and conserved vectors are established using symmetries conservation laws relationship. The double reduction theory is utilized which results in the reduction and exact solutions of models under investigation. All cases are discussed in detail and new solutions are determined.

The non-standard Hamiltonian system, also referred to as a partial Hamiltonian system in the literature, of the form q ˙ i = ∂ H ∂ p i , p ˙ i = − ∂ H ∂ q i + Γ i ( t , q i , p i ) ${\dot q^i} = \frac{{\partial H}}{{\partial {p_i}}},{\text{ }}{\dot p^i} = - \frac{{\partial H}}{{\partial {q_i}}} + {\Gamma ^i}(t,{\text{ }}{q^i},{\text{ }}{p_i})$ appears widely in economics, physics, mechanics, and other fields. The non-standard (partial) Hamiltonian systems arise from physical Hamiltonian structures as well as from artificial Hamiltonian structures. We introduce the term ‘artificial Hamiltonian’ for the Hamiltonian of a model having no physical structure. We provide here explicitly the notion of an artificial Hamiltonian for dynamical systems of ordinary differential equations (ODEs). Also, we show that every system of second-order ODEs can be expressed as a non-standard (partial) Hamiltonian system of first-order ODEs by introducing an artificial Hamiltonian. This notion of an artificial Hamiltonian gives a new way to solve dynamical systems of first-order ODEs and systems of second-order ODEs that can be expressed as a non-standard (partial) Hamiltonian system by using the known techniques applicable to the non-standard Hamiltonian systems. We employ the proposed notion to solve dynamical systems of first-order ODEs arising in epidemics.

Accessibility and utilization of healthcare plays a significant role in preventing complications during pregnancy, labor, and the early postnatal period. However, multiple barriers can prevent women from accessing services. The aim of this study was to explore the multifaceted barriers that inhibit women from seeking maternal and newborn health care in Thatta, Sindh, Pakistan. This study employed an interpretive research design using a purposive sampling approach. Pre-tested, semi-structured interview guides were used for data collection. The data were collected through eight focus group discussions with men and women, and six in-depth interviews with lady health workers and analyzed through thematic analysis. The study identified individual, sociocultural, and structural-level barriers that inhibit women from seeking maternal and newborn care. Individual barriers included mistrust towards public health facilities and inadequate symptom recognition. The three identified sociocultural barriers were aversion to biomedical interventions, gendered imbalances in decision making, and women’s restricted mobility. The structural barriers included ineffective referral systems and prohibitively expensive transportation services. Increasing the coverage of healthcare service without addressing the multifaceted barriers that influence service utilization will not reduce the burden of maternal and neonatal mortality. As this study reveals, care seeking is influenced by a diverse array of barriers that are individual, sociocultural, and structural in nature. A combination of capacity development, health awareness, and structural interventions can address many if not all of these barriers.

Conversely, to metallic based electromagnetic interference (EMI) shielding materials; composite polymer-based foams are normally lightweight, cheaper than metals, and less sensitive to types of environmental degradation. In this work polyurethane - Graphite composite (PU-G) foam materials were prepared with different filler concentrations. Different characterization tools such as Fourier Transform Infra-Red (FTIR) and scanning electron microscope (SEM), were used to identify the structural and topological construction of the prepared composites. Further mechanical properties for the prepared samples were studied to elucidate the opportunity of utilizing these composites in applied applications, specifically for electromagnetic interference (EMI) shielding efficiency (SE) for aerospace applications. Also, in order to adjust this research in the area of aerospace EMI SE, the evaluations were executed in the X-band at (8-12) GHz. The obtained data indicated that the moreover in filler concentration enhanced the compressive strength and compressive modulus of the prepared samples. Moreover EMI SE reached -44 dB with 30 wt % graphite concentration. Finally polyurethane – graphite composite foam material can be taken into consideration a gratifying material to be utilised in EMI SE.

In this article, a new approach is proposed to construct the symmetry groups for a class of fractional differential equations which are expressed in the modified Riemann-Liouville fractional derivative. We perform a complete group classification of a nonlinear fractional diffusion equation which arises in fractals, acoustics, control theory, signal processing and many other applications. Introducing the suitable transformations, the fractional derivatives are converted to integer order derivatives and in consequence the nonlinear fractional diffusion equation transforms to a partial differential equation (PDE). Then the Lie symmetries are computed for resulting PDE and using inverse transformations, we derive the symmetries for fractional diffusion equation. All cases are discussed in detail and results for symmetry properties are compared for different values of α . This study provides a new way of computing symmetries for a class of fractional differential equations.

The approximate partial Noether theorem proposed earlier for the ordinary differential equations (ODEs) (Naeem and Mahomed in Nonlinear Dyn 57(1–2):303–311, 2009) is generalized in phase space for the perturbed Hamiltonian-type systems. The notion of approximate partial Hamiltonian is developed. An approximate partial Hamiltonian gives rise to an approximate Hamiltonian-type perturbed dynamical system of first-order ODEs. An approximate Legendre-type transformation connects the approximate partial Lagrangian and what we term as approximate partial Hamiltonian. The formulas for approximate partial Hamiltonian operators determining equations and first integrals are provided explicitly. We have explained our approach with the help of simple illustrative example. Then, it is applied to establish the approximate first integrals, reductions and exact solutions of two perturbed cubically coupled Duffing–Van der Pol oscillators. Both resonant and nonresonant cases are considered.

The first integrals and exact solutions of mathematical models of epidemiology: a susceptible-infected-recovered-infected (SIRI) model and a tuberculosis model with demographic growth are analyzed. These models are represented by systems of first-order nonlinear ordinary differential equations, and this system is replaced by one which contains a second-order ordinary differential equation. The partial Lagrangian approach is then utilized to derive the first integrals of these models. Several cases arise. Then, we utilize the derived first integrals to construct exact solutions for the models under investigation and determine new solutions. The dynamic properties of these models are studied too. Copyright © 2016 John Wiley & Sons, Ltd.

This article analyses the nonclassical symmetries and group invariant solution of boundary layer equations for two-dimensional heated flows. First, we derive the nonclassical symmetry determining equations with the aid of the computer package SADE. We solve these equations directly to obtain nonclassical symmetries. We follow standard procedure of computing nonclassical symmetries and consider two different scenarios, ξ 1≠0 and ξ 1=0, ξ 2≠0. Several nonclassical symmetries are reported for both scenarios. Furthermore, numerous group invariant solutions for nonclassical symmetries are derived. The similarity variables associated with each nonclassical symmetry are computed. The similarity variables reduce the system of partial differential equations (PDEs) to a system of ordinary differential equations (ODEs) in terms of similarity variables. The reduced system of ODEs are solved to obtain group invariant solution for governing boundary layer equations for two-dimensional heated flow problems. We successfully formulate a physical problem of heat transfer analysis for fluid flow over a linearly stretching porous plat and, with suitable boundary conditions, we solve this problem.

Background Where access to facilities for childhood diarrhea and pneumonia is inadequate, community case management (CCM) is an effective way of improving access to care. In Pakistan, utilization of CCM for these diseases through the Lady Health Worker Program remains low. Challenges of access to facilities persist leading to delayed care and poor outcomes. Estimating caregiver knowledge, understanding their perceptions and practices, and recognizing how these are related to care seeking decisions about childhood diarrhea and pneumonia is crucial to bring about coherence between supply and demand-side practices. Methods Data was collected from family caregivers to explore their knowledge, perceptions and practices regarding childhood diarrhea and pneumonia. Data from a household survey with 7025 caregivers, seven focus group discussion (FGDs), seven in-depth interviews (IDIs), and 20 detailed narrative interviews are used to explore caregiver knowledge, perceptions and practices. Results Household survey shows that most family caregivers recognize main signs and symptoms of diarrhea such as loose stools (76%). Fewer recognize signs and symptoms of pneumonia such as breathing problems (21%). Few caregivers (18%) have confidence in lady health workers’ (LHWs) ability to treat childhood diarrhea and pneumonia. Care seeking from LHWs remains negligible (

In vitro antifungal potential of Trichoderma isolates, selective botanical extracts and fungicides against A. solani was evaluated. Trichoderma isolates, i.e. T. harzianum, T. viride and T. hamatum, were tested for their antifungal effect by dual culture technique at 48, 96, 144 and 172 hrs. T. hamatum produced the highest inhibition of A. solani in vitro, followed by T. hazianum and T. viride after 172 hrs. Methanolic leaf extracts of Elettaria cardamomum, Syzygium aromaticum, Curcuma longa and root extract of Parthenium hysterophorus showed up to 100% inhibition of A. solani, compared to control, while methanolic stem and leaf extracts of P. hysterophorus produced up to 90% inhibition of the pathogen. In vitro, six different systemic fungicides Triger 25% EC (Tebuconazole), Solex (Carbendazim 40% + Triadimefon 10%), Dew (Difenoconazole), Amistor Top SC (Azoxystrobin + Difenoconazole), Corel 25% EC (Difenoconazole), Reflex (Difenoconazole + Propiconazole) were tested against A. solani at 5, 10 and 15 ppm concentrations after 48, 96, 144 and 172 hrs. Corel and reflex at all concentrations produced best growth inhibition of A. solani. The inhibition was maximum by all fungicides at 15 ppm after 172 hrs. All fungicides had a promising inhibitory effect on A. solani, except Solex. It can be concluded from the present investigation that a combination of these strategies can be used in integrated disease management of A. solani on potato.

In this article, we study time–space fractional advection–dispersion (FADE) equation and time–space fractional Whitham–Broer–Kaup (FWBK) equation that have significant roles in hydrology. We introduce suitable transformations to convert fractional-order derivatives to integer-order derivatives and as a result these equations transform into partial differential equations (PDEs). Then the Lie symmetries and the corresponding optimal systems of the resulting PDEs are derived. The symmetry reductions and exact independent solutions based on optimal system are investigated which constitute the exact solutions of original fractional differential equations.

We present symmetry classification and exact solutions of generalized modified Boussinesq (GMB) equation. The direct method of group classification is utilized to determine four different functional forms of . The GMB equation admits two-dimensional principle algebra for arbitrary and the algebra extends to three-dimensional for other forms of . Similarity reductions are made in each case and exact solutions are derived.

The conservation laws for (1+1)-dimensional non-linear generalized regularized long wave (GRLW) equation are derived via partial Noether approach after increasing its order. The GRLW equation is a third-order partial differential equation. We convert GRLW equation to fourth order equation by assuming new dependent variable v to be the derivative of original dependent variable u by setting either u = v x or u = v t . The partial Noether’s approach is then used to derive the conservation laws. The derived conserved vectors are adjusted to satisfy the divergence relationship. Finally, the conservation laws are expressed in the variable u and they constitute the conservation laws for the third-order GRLW equation. The Lie point symmetries for GRLW equation are computed. The double reduction theory based on symmetry and its associated conserved vector is utilized and two independent exact solutions are obtained. Moreover, the Lie symmetry method is used to derive an invariant solution. One of the solutions obtained by the double reduction method is the same as derived by Lie symmetry method. The second solution constructed by the double reduction theory is not obtained by the Lie symmetry method. A similar analysis is performed for regularized long wave (RLW) and modified Benjamin–Bona–Mahoney (MBBM) equations.

A special type of (1 + n )-dimensional linear evolution equation is considered. A class of the equations generated by the Fokker–Planck equation becomes the subcase of the considered equation. Conserved vectors using the partial Lagrangian approach is derived in terms of the coefficients of the discussed equation. Derived results are used for the different models from different sciences. We also discuss the conservation laws of the heat equation on curved manifolds and in different coordinate systems. Potential systems are also obtained for some models. At last conclusion is given.

SETTING Six towns of Karachi, Pakistan. OBJECTIVES 1) To strengthen the capacity of general practitioners (GPs) in providing tuberculosis (TB) treatment through DOTS; and 2) to enhance collaboration between the public and private sectors in TB management and case reporting. DESIGN A quasi-experimental study design was adopted to ensure enrolment of TB patients through trained GPs with the support of laboratory networks and to improve the case detection rate. RESULTS The following challenges were faced during implementation of the model in urban settings: no systematic list of GPs was available; the majority of the GPs were untrained health practitioners working in squatter settlements, where formally trained GPs are most needed; the motivation of GPs with high patient loads is very low; and access to a laboratory is difficult. Of 35 patients enrolled in the first quarter (third quarter 2009), 87% completed their treatment successfully. CONCLUSION Public-private mix (PPM) DOTS is feasible in the cities of Pakistan. However, the cost, time and effort required to establish the programme is higher than in many other developing countries.

The problem of symmetry classification for the heat equation on torus is studied by means of classical Lie group theory. The Lie point symmetries are constructed and Lie algebra is formed for equation under consideration. Then these algebras are used to classify its subalgebras up to conjugacy classes. In general the heat equation on torus admits one-, two-, three- and four-dimensional algebras. For one-dimensional algebra £1 and £2 the heat equation on torus is reduced in independent variables whereas in two-dimensional algebras £3 and £4 the considered heat equation is investigated by quadrature. While three- and four-dimensional algebras lead to a trivial solution.

The conservation laws for a ( 1 + n ) -dimensional heat equation on curved surfaces are constructed by using a partial Noether’s approach associated with the partial Lagrangian. The partial Noether determining equation for the general case n ≥ 2 gives some extra conditions on gauge terms and conserved vectors. The analysis is then applied for heat equations on cone, sphere, torus and plane. Moreover, we use the symmetry conservation law relation and some new conservation laws are deduced.

We provide linearizability criteria for a class of systems of two third-order ordinary differential equations that is cubically nonlinear in the first derivative, by differentiating a system of second-order quadratically nonlinear ordinary differential equations and using the original system to replace the second derivatives. The procedure developed splits into two cases: those for which the coefficients are constant and those for which they are variables. Both cases are discussed and examples given.

We present here the exact solutions for the equations of magnetohydrodynamic (MHD) aligned flow of a second grade fluid. The exact solutions are constructed for steady and unsteady equations by employing inverse method.

We study the construction of approximate first integrals of approximate partial Euler–Lagrange equations via approximate partial Noether operators corresponding to a partial Lagrangian. An approximate Noether-like theorem which gives the approximate first integrals for the perturbed equations without regard to a standard Lagrangian is deduced. These approximate partial Noether operators, in general, do not form an approximate Lie algebra. The results are applied to a system of two coupled nonlinear oscillators. The approximate first integrals are obtained for both the resonant and nonresonant cases with the help of approximate partial Noether operators associated with a partial Lagrangian. This approach can give rise to further studies in the construction of approximate first integrals for perturbed equations without a variational principle.

We obtain Noether and partial Noether operators corresponding to a Lagrangian and a partial Lagrangian for a system of two linear second-order ordinary differential equations (ODEs) with variable coefficients. The canonical form for a system of two second-order ordinary differential equations is invoked and a special case of this system is studied for both Noether and partial Noether operators. Then the first integrals with respect to Noether and partial Noether operators are obtained for the linear system under consideration. We show that the first integrals for both the Noether and partial Noether operators are the same. This can give rise to further studies on systems from a partial Lagrangian viewpoint as systems in general do not admit Lagrangians.

The notions of partial Lagrangians, partial Noether operators and partial Euler–Lagrange equations are used in the construction of first integrals for ordinary differential equations that need not be derivable from variational principles. We obtain a Noether-like theorem that provides the first integral by means of a formula which has the same structure as the Noether integral. However, the invariance condition for the determination of the partial Noether operators is different as we have a partial Lagrangian and as a result partial Euler–Lagrange equations. Applications given include those that admit a standard Lagrangian such as the harmonic oscillator, modified Emden and Ermakov–Pinney equations and systems of two second-order equations that do not have standard Lagrangians. Copyright © 2007 John Wiley & Sons, Ltd.

Antagonism amongst mixtures of inoculant strains of Rhizobiaceae on the basis of bacteriocin production was assessed. A total number of 10 strains of Rhizobium , Bradyrhizobium and Agrobacterium were screened for their bacteriocin production ability. It was observed that Rhizobium leguminosarum bv. viciae strain LC-31 produced a medium typed bacteriocin that was found to be highly effective in growth inhibition of some strains of R. leguminosarum bv. viciae and Agrobacterium sp. It was isolated and partially purified to homogeneity by chloroform extraction followed by ammonium sulfate fractionation. The bacteriocin fraction consistently migrated as a 50 kDa polypeptide on SDS-PAGE. Ex vivo assays were carried out using the partially purified bacteriocin protein fraction of R. leguminosarum bv. viciae strain LC-31 against related strains of the same species. It showed an activity pattern similar to that exhibited by strain LC-31.

This paper analyzes the first integrals and exact solutions of mathematical models of epidemiology via the partial Lagrangian approach by replacing the three first-order nonlinear ordinary differential equations by an equivalent system containing one second-order equation and a first-order equation. The partial Lagrangian approach is then utilized for the second-order ODE to construct the first integrals of the underlying system. We investigate the SIR and HIV models. We obtain two first integrals for the SIR model with and without demographic growth. For the HIV model without demography, five first integrals are established and two first integrals are deduced for the HIV model with demography. Then we utilize the derived first integrals to construct exact solutions to the models under investigation. The dynamic properties of these models are studied too. Numerical solutions are derived for SIR models by finite difference method and are compared with exact solutions.

Introduction The surgical Apgar score (SAS) can predict 30-day major complications or death after surgery. Studies have validated the score in different patient populations and suggest it should be used to objectively guide postoperative care. We aimed to see whether using the SAS in a decisive approach in a future randomised controlled trial (RCT) would be likely to demonstrate an effect on postoperative care and clinical outcome. Methods A total of 143 adults undergoing general/vascular surgery in 9 National Health Service hospitals were recruited to a pilot single blinded RCT and the data for 139 of these were analysed. Participants were randomised to a control group with standard postoperative care or to an intervention group with care influenced (but not mandated) by the SAS (decisive approach). The notional primary outcome was 30-day major complications or death. Results Incidence of major complications was similar in both groups (control: 20/69 [29%], intervention: 23/70 [33%], p=0.622). Immediate admissions to the critical care unit was higher in the intervention group, especially in the SAS 0–4 subgroup (4/6 vs 2/7) although this was not statistically significant (p=0.310). Validity was also confirmed in area under the curve (AUC) analysis (AUC: 0.77). Conclusions This pilot study found that a future RCT to investigate the effect of using the SAS in a decisive approach may demonstrate a difference in postoperative care. However, significant changes to the design are needed if differences in clinical outcome are to be achieved reliably. These would include a wider array of postoperative interventions implemented using a quality improvement approach in a stepped wedge cluster design with blinded collection of outcome data.

Different approaches to construct first integrals for ordinary differential equations and systems of ordinary differential equations are studied here. These approaches can be grouped into three categories: direct methods, Lagrangian or partial Lagrangian formulations, and characteristic (multipliers) approaches. The direct method and symmetry conditions on the first integrals correspond to first category. The Lagrangian and partial Lagrangian include three approaches: Noether’s theorem, the partial Noether approach, and the Noether approach for the equation and its adjoint as a system. The characteristic method, the multiplier approaches, and the direct construction formula approach require the integrating factors or characteristics or multipliers. The Hamiltonian version of Noether’s theorem is presented to derive first integrals. We apply these different approaches to derive the first integrals of the harmonic oscillator equation. We also study first integrals for some physical models. The first integrals for nonlinear jerk equation and the free oscillations of a two-degree-of-freedom gyroscopic system with quadratic nonlinearities are derived. Moreover, solutions via first integrals are also constructed.

1 Centre for Mathematics and Statistical Sciences, Lahore School of Economics, Lahore 53200, Pakistan 2 Centro de Matematica, Computacao e Cognicao, Universidade Federal do ABC, Rua Santa Adelia 166, Bairro Bangu, 09.210-170 Santo Andre, SP, Brazil 3 Department of Mathematics, School of Science and Engineering, Lahore University of Management Sciences, Lahore Cantt, Lahore 54792, Pakistan 4Dipartimento di Matematica ed Informatica, Universita di Catania Viale A. Doria 6, 95125 Catania, Italy

We study the conservation laws of evolution equation, lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equation. The symbolic software GeM (Cheviakov (2007) and (2010)) is used to derive the multipliers and conservation law fluxes. Software GeM is Maple-based package, and it computes conservation laws by direct method and first homotopy and second homotopy formulas.

The nonclassical symmetries of boundary layer equations for two-dimensional and radial flows are considered. A number of exact solutions for problems under consideration were found in the literature, and here we find new similarity solution by implementing the SADE package for finding nonclassical symmetries.

The group invariant solution for Prandtl-s boundary layer equations for an incompressible fluid governing the flow in radial free, wall and liquid jets having finite fluid velocity at the orifice are investigated. For each jet a symmetry is associated with the conserved vector that was used to derive the conserved quantity for the jet elsewhere. This symmetry is then used to construct the group invariant solution for the third-order partial differential equation for the stream function. The general form of the group invariant solution for radial jet flows is derived. The general form of group invariant solution and the general form of the similarity solution which was obtained elsewhere are the same., {"references":["W. H. Schwarz, The radial free jet, Chem. Eng. Sci. 18 (1963) 779-786.","H. Schlichting, Laminare Strahlausbreitung, Z. Angew. Math. Mech. 13\n(1933) 260-263.","W. G. Bickley, The plane jet, Phil. Mag. 23 (1937) 727-731.","D. P. Mason, Group invariant solution and conservation law for free\nlaminar two-dimensional jet, Journal Nonlinear Math. Phys. 9 (2002) 92-\n101.","R. Naz, F. M. Mahomed, D. P. Mason, Conservation laws via the\npartial Lagrangian and group invariant solutions for radial and twodimensional\nfree jets, Submitted to Journal of Nonlinear Analysis: Real\nworld applications.","M. B. Glauert, The wall jet, J. Fluid Mech. 1 (1956) 625-643.","N. Riley, Asymptotic expansions in radial jets, J.Math. and Phys. 41\n(1962) 132-146.","N. Riley, An Application of Mangler-s transformation, Quart. J. Mech.\nApplied. Math. 14 (1961) 197-202 .","N. Riley, Radial jets with swirl, Part I. Incompressible Flow, Quart. Journ.\nmech. and Applied Math. 15, (1962) 435-458.\n[10] E. J. Watson, The radial spread of a liquid jet over a horizontal plane,\nJ. Fluid Mech. 20 (1964) 481-499.\n[11] R. Naz, F. M. Mahomed and D. P. Mason, Symmetry solutions of a thirdorder\nordinary differential equation which arises from Prandtl boundary\nlayer equations, Journal Nonlinear Math. Phys. Accepted for publication.\n[12] R. Naz, D. P. Mason and F. M. Mahomed, Conservation laws and\nphysical conserved quantities for laminar two-dimensional and radial\njets, Accepted for publication in journal of Nonlinear Analysis: Real\nWorld Applications.\n[13] A. H. Kara and F. M. Mahomed, Relationship between symmetries and\nconservation laws, Int. J. Theor. Phys. 39 (2000) 23-40.\n[14] M. B. Glauert, symposium uber Grenzschichtforschung, edited by H.\nGortler (Springerverlag 1957)."]}

Residual metallic fragments after intra-abdominal penetrating injuries due to fragments of artillery shells are generally inert, although they may rarely re-activate. The authors present a case where such a fragment presented as a renal stone 17 years after injury. The literature was reviewed and the initial difficulties with the radiological diagnosis, as well as treatment approaches and post-operative findings are discussed in the light of existing reports. An additional challenge arose as the authors working in a tertiary endourology centre attempted to remain as minimally invasive as possible, but were limited by the very nature of the foreign body.

The partial Noether operators and first integrals of a general system of two linear second-order ordinary differential equations (ODEs) with variable coefficients are studied by means of a partial Lagrangian. The canonical form for the general system of two second-order ordinary differential equations is invoked and all cases of this system are discussed with respect to partial Noether operators. We also tabulate the results for the special case b(x) = c(x) of the system which was considered elsewhere using a Lagrangian and a partial Lagrangian. The first integrals are obtained explicitly by exploiting a Noether-like theorem with the help of partial Noether operators. This study gives a new way to construct first integrals for systems without a variational principle as not all linear equations have a Lagrangian. Physical applications to conservative and oscillator mechanical systems are given.

We construct all partial Noether operators corresponding to a partial Lagrangian for a system with two degrees of freedom. Then all the first integrals are obtained explicitly by utilizing a Noether-like theorem with the help of the partial Noether operators. We show how the first integrals can be constructed for the system without the need of a variational principle although the Lagrangian L = y ′2 /2 + z ′2 /2 − v(y,z) does exist for the system. Our objective is twofold: one is to see the effectiveness of the partial Noether approach and the other to determine all the first integrals of the system under study which have not been reported before. Thus, we deduce a complete classification of the potentials v(y,z) for which first integrals exist. This can give rise to further studies on systems which are not Hamiltonian via partial Noether operators and the construction of first integrals from a partial Lagrangian viewpoint.