Search Results (20)

The aim of this paper is to reflect on the changing relationship between Catholic education and the State in the eighty years that have elapsed since the passing of the 1944 ‘Butler’ Education Act. The 1944 Education Act proved to be a transformational landmark in the development of Catholic education in this country. Whilst in many other countries in the world Catholic education is provided by independent fee-paying schools, in England it has benefited from support and funding provided by the government. The legislation consolidated the relationship between Church and State that had been established with the introduction of compulsory education in 1870. However, integrated within the national educational system, Catholic schools are also subject to political currents that can pose challenges. When I began teaching in a Catholic secondary school in the mid-1970s, compulsory education was framed within the legislation set out in the 1944 Education Act. However, the post-war consensus that informed that legislation was swept away in the wake of the passing of the 1988 Education Reform Act (ERA). This latter development marked a watershed in schooling in England and Wales. Whilst extending greater autonomy to schools through Local Management of Schools (LMS) and Grant Maintained (GM) status, central government designed and regulated education for all maintained schools through a National Curriculum and it intervened more rigorously in their management by scrutinising their performance through the establishment of regular Ofsted inspections. Arguably, the introduction of Grant Maintained schools contributed to ‘a survival of the fittest approach’, which challenged concerns for the common good. Moreover, with the passing of the 2010 Academies Act, it became possible for all maintained schools in England and Wales to convert to academies and multi-academy trusts (MATs) by direct agreement with the Secretary of State for Education. Within the Catholic sector, where the bishop has traditionally been primarily responsible for the regulation of education provided by Catholic schools across his diocese, the question of academisation has provoked divisions of opinion about the governance of Catholic schools. Drawing on a reflection on my own experience as a teacher and leader in Catholic secondary education, the paper examines opportunities and challenges that have faced Catholic education in this country over the past eighty years and considers implications for its future. Autobiographical narrative is a method of research located within the qualitative field of enquiry. Whilst its subjective and introspective nature has been open to question, it provides a researcher with the opportunity to examine personal experience to provide illuminative insight. Full article

In this paper, we deal with the canal hypersurfaces that are formed as the envelope of a family of pseudo-hyperspheres or pseudo-hyperbolic hyperspheres with centers lying on a pseudo-null curve with Bishop vector fields in four-dimensional Lorentz–Minkowski space. We give main theorems which contain the parametric expressions of these canal hypersurfaces along with their Gaussian, mean, and principal curvatures and important geometric characterizations. We also provide these characterizations for tubular hypersurfaces. Finally, we construct an example to allow for better understanding and comprehension of the results. Full article

In this study, using the generalized Bishop frame field of type-C in four-dimensional Euclidean space, we investigate the geometric properties of a soliton surface $M=M(x,y)$ associated with the Betchov–Da Rios equation. Using a unit speed x-parameter curve $M=M(x,y),$ for all y, we derive the derivative formulas for the generalized Bishop frame field of type-C. We obtain two fundamental geometric invariants of the soliton surface, k and h, characterizing the points of the surface, as well as a few other significant invariants, including Gaussian curvature, a mean curvature vector and Gaussian torsion. We use these surface invariants to prove a collection of theorems that describe the circumstances in which the soliton surface is flat, minimal, semi-umbilic or Wintgen ideal (superconformal). Additionally, we provide a theorem that describes the B-DR soliton surface’s curvature ellipse in relation to the generalized Bishop frame field of type-C in ${\mathbb{E}}^4$. Finally, we construct a foundational example to support our theoretical results and demonstrate the construction of the generalized Bishop frame field of type-C. Full article

In Minkowski 3-space, we establish a geometric framework to osculate Type-II ruled surfaces by utilizing the Type-II Bishop frame in $\left({\mathbb{E}}_1^3\right)$. Our analysis extends to higher-order singularities such as butterflies and pyramids, including explicit singularity loci. We also compare Type-II Bishop frames with rotation-minimizing frames using timelike base curves and spacelike normals. With RK4 integration, we develop a robust computational model for Weingarten surfaces and subclasses with constant curvature. The theoretical foundation for Type-II Bishop frames is extended to higher-dimensional Minkowski spaces ${\mathbb{E}}_1^n$ for $n>3$ through generalized Frenet-type equations and curvature functions. We determine exact stability conditions under perturbations of Bishop curvature using advanced singularity theory. The numerical implementations of our methods, including geometric modeling and relativistic geometry, demonstrate their effectiveness in both theoretical and applied contexts. Full article

The concept of “magnetic lines of force,” or “magnetic curves”, was introduced in the 1830s by Michael Faraday (1791–1867); his work provided the foundation for the modern understanding of magnetic fields. In differential geometry, a magnetic curve is a concept that arises from the intersection of geometry and physics. These curves represent the trajectories of a charged particle experiencing the Lorentz force as it travels through a magnetic field. These curves have garnered significant interest due to their intricate geometric properties and diverse applications. This paper provides a comprehensive exploration of magnetic curves, delving into their fundamental characteristics and classification. Full article

This study investigates the geometry of osculating type-2 ruled surfaces in Minkowski 3-space ${\mathbf{E}}_1^3$, formulated through the Type-2 Bishop frame associated with a spacelike curve whose principal normal is timelike and binormal is spacelike. Using the hyperbolic transformation linking the Frenet–Serret and Bishop frames, we analyze how the Bishop curvatures ${\zeta }_1$ and ${\zeta }_2$ affect the geometric behavior and formation of such surfaces. Explicit criteria are derived for cylindrical, developable, and minimal configurations, together with analytical expressions for Gaussian and mean curvatures. We also determine the conditions under which the base curve behaves as a geodesic, asymptotic line, or line of curvature. Several illustrative examples in Minkowski 3-space are provided to visualize the geometric influence of ${\zeta }_1$ and ${\zeta }_2$ on flatness, minimality, and developability. Overall, the Type-2 Bishop frame offers a smooth and effective framework for characterizing Lorentzian geometry and symmetry of osculating ruled surfaces, extending classical Euclidean results to the Minkowski setting. Full article

Elasticity is a well-established field within mathematical physics, yet new formulations can provide deeper insight and computational advantages. This study explores the geometry of two- and three-dimensional elastic curves using the formalism of geometric algebra, offering a unified and coordinate-free approach. This work systematically derives the Frenet, Darboux, and Bishop frames within the three-dimensional geometric algebra and employs them to integrate the elastica equation. A concise Lagrangian formulation of the problem is introduced, enabling the identification of Noetherian, conserved, multi-vector moments associated with the elastic system. A particularly compact form of the elastica equation emerges when expressed in the Bishop frame, revealing structural simplifications and making the equations more amenable to analysis. Ultimately, the geometric algebra perspective uncovers a natural correspondence between the theory of free elastic curves and classical beam models, showing how constrained theories, such as Euler–Bernoulli and Kirchhoff beam formulations, arise as special cases. These results not only clarify foundational aspects of elasticity theory but also provide a framework for future applications in continuum mechanics and geometric modeling. Full article

In this study, ruled surfaces are examined where the direction vectors are unit vectors derived from Smarandache curves, and the base curve is taken as an adjoint curve constructed using the integral curve of a Smarandache-type curve generated from the first and second Bishop normal vectors. The newly generated ruled surfaces will be referred to as Bishop adjoint ruled surfaces. Explicit expressions for the Gaussian and mean curvatures of these surfaces have been obtained, and their fundamental geometric properties have been analyzed in detail. Additionally, the conditions for developability, minimality, and singularities have been investigated. The asymptotic and geodesic behaviors of parametric curves have been examined, and the necessary and sufficient conditions for their characterization have been derived. Furthermore, the geometric properties of the surface generated by the Bishop adjoint curve and its relationship with the choice of the original curve have been established. The constructed ruled surfaces exhibit a notable degree of geometric regularity and symmetry, which naturally arise from the structural behavior of the associated adjoint curves and direction fields. This underlying symmetry plays a central role in their formulation and classification within the broader context of differential geometry. Finally, the obtained surfaces are illustrated with figures. Full article

This article aims to show how episcopal authority was built in the eastern part of the Kingdom of León (county of Castile), where a new kingdom and a vast diocese emerged in the mid-11th century. The monarchs of Castile empowered the strategic pre-urban town of Burgos in the northern Iberian Plateau as a single episcopal see, rather than the four that had existed in the area until then. The bishops were the agents that the monarchs needed in the long process that, from the destabilisation of the Visigothic diocese organisation caused by the Islamic invasion of the Iberian peninsula in the 8th century, led to the consolidation of episcopal power in the mid-13th century. The function and actions of the Burgalese bishops have been analysed in the three dimensions of their ecclesiastic authority and social significance: the patrimonial dimension (the bishop as the lord and owner of properties individually), the jurisdictional dimension and the pastoral dimension. This analysis has been able to establish three periods in the struggle of the Burgalese prelates: to define the territorial frame of their authority (the delimitation of the diocese boundaries), to recover the churches and jurisdictional rights (episcopal third and other ecclesiastic taxes) that were in the hands of the powerful Benedictine monasteries and lay people, and to affirm their hierarchical superiority over other diocese potestates. The study has identified the main strategies used by the bishops to reach those objectives: the signing of agreements to resolve disputes, the addition of abbots of collegial churches to the cathedral chapter to control key areas in the diocese, and the acquisition of properties in those areas. Full article

The aim of this work is to investigate osculating type ruled surfaces with a type 2-Bishop frame in $E^3$. We accomplish this by employing the symmetry of osculating curves. We examine osculating type ruled surfaces by taking into account the curvatures of the base curve. We investigate the geometric properties of these surfaces, focusing on their cylindrical and developable characteristics. Moreover, we calculate the Gaussian and mean curvatures and provide the requirements for the surface to be flat and minimal. We determine the requirements for the curves lying on this surface to be geodesic, asymptotic curves, or lines of curvature. Furthermore, relations between osculating type ruled surfaces with central tangent and central normal vectors are given. Finally, some examples of these surfaces are presented. Full article

In this work, we present a new Bishop frame for the conjugate curve of a curve in the 3-dimensional Lie group ${\mathbb{G}}_3$. With the help of this frame, we derive a parametric representation for a sweeping surface and show that the parametric curves on this surface are curvature lines. We then examine the local singularities and convexity of this sweeping surface and establish the sufficient and necessary conditions for it to be a developable ruled surface. Additionally, we provide detailed explanations and examples of its applications. Full article

The aim of this work is to investigate sweeping surfaces and their local singularities due to type-3 Bishop frames in Euclidean 3-space, ${\mathbb{E}}^3$. A sweeping surface a is surface that is designed from a section curve positioned along a path, which acts as the vertebral column or spine curve, and it has symmetrical characteristics. In this work, we have specified a sweeping surface and have examined its geometry and singularity. Thereafter, we deduced the circumstances required for this surface to be a developable surface. In great detail, we concentrated on the fundamental discussion on whether the resulting developable surface is a cylindrical, cone or tangent surface. Meanwhile, examples are detailed to explain the applications of the notional outcomes. Full article

Because of the current situation where the stability research of filled-slope reinforced by a frame with prestressed anchor-plates lags behind the actual engineering application, based on the ultimate balance theory, the calculation formulas of stability factor under the four arc slip surface of filled-slopes reinforced by a frame with prestressed anchor-plates are derived by using the improved Bishop method; the corresponding search method of the most dangerous slip surface is given and the calculation formulas of the pullout force of anchor-plates are improved. Based on two examples, the stability results calculated by the proposed algorithm are compared with those calculated by PLAXIS 3D and GeoStudio 2012 finite element software, and the following conclusions are drawn. (1) The improved pullout force of anchor-plates takes into account the friction of the front and rear surface of the anchor-plate and the effect of cohesion of fill soil in the passive earth pressure on the front end of the anchor-plate, which makes the force of the anchor-plate more complete. (2) The stability factor of example 1 calculated by this method differs from the results simulated by PLAXIS 3D and GeoStudio 2012 by 4.6% and 7.1%, respectively; the stability factor of example 2 calculated by this method differs from the results simulated by PLAXIS3D and GeoStudio 2012 by 3.2% and 4.5%, respectively, which can meet the engineering requirements. (3) The stability analysis method of filled-slope reinforced by a frame with prestressed anchor-plates that is proposed is reasonable and suitable for any arc slip surface in the filled-slope reinforced by a frame with prestressed anchor-plates, and it provides some guiding values for the design of practical engineering. Full article

In this paper, we consider the singularities and geometrical properties of timelike developable surfaces with Bishop frame in Minkowski 3-space. Taking advantage of the singularity theory, we give the classification of generic singularities of these developable surfaces. Furthermore, an example of application is given to illustrate the applications of the results. Full article

This article examines reparations advocacy by a vanguard of African American faith leader proponents, from Bishop Henry McNeil Turner’s late-19th century demands for federal payments toward emigrationism and Black Atlantic linkages, to 21st century Black clergy involvements in national level, local level, and sector-specific reparations policy activism. Attention is paid to evolving theoretical and operational framings of this reparations advocacy and to variances in levels of American religious and political receptivity to reparations proposals. The conclusion drawn from available evidence here is that reparations advocacy by Black religious leaders has proven more pragmatic than purist, as concerns increasingly have shifted toward maximizing public support and prospects for reparations deliverables. Full article