Exact Solution of the Hyperbolic Generalization of Burgers Equation, Describing Travelling Fronts and their Interaction

We present new analytical solutions to the hyperbolic generalization of Burgers equation, describing interaction of the wave fronts. To obtain them, we employ a modified version of the Hirota method.Comment: 12 pages, 3 figure

Hoc Est Sacrificium Laudis: The Influence of Hebrews on the Origin, Structure, and Theology of the Roman Canon Missae

One area of study that received a newfound level of attention during the twentieth century’s Liturgical Movement was the relationship between the Bible and liturgy. The Constitution on the Sacred Liturgy, Sacrosanctum concilium, highlights the importance and centrality of this relationship, declaring that “[s]acred scripture is of the greatest importance in the celebration of the liturgy” (SC 24). The broad movements of ressourcement and la nouvelle théologie, particularly figures such as Jean Daniélou and Henri de Lubac, emphasized the deep unity between Scripture and the very text of liturgical rites and argued that the liturgy is an expression of spiritual exegesis (whether it is called “typology” or “allegory”). What did not figure in these studies was a specific demonstration of these broad claims through the study of particular liturgical texts. This dissertation seeks to fill that lacuna through a study of one liturgical text—the Roman Canon Missae—and its relationship to one specific book of the Bible: the Epistle to the Hebrews. A significant motivation for this research is a concern to demonstrate how this new scriptural avenue of inquiry can provide an additional source of rich material to liturgical scholars for any liturgical text, not just the Roman Canon. My approach situates this exploration of the ways Hebrews was used as a source within the broader orbit of the emergence and development of the text of the Roman Canon in order to demonstrate that attention to the place of Scripture, or even a single biblical book, can radically enrich the search for the origin and early evolution of liturgical rites. This new methodology includes a detailed proposal for a way to categorize the ways in which a liturgical text can utilize Scripture as a source. Most of the unique features of the Roman Canon—including its unique institution narrative, emphasis on sacrifice, repeated requests for the Father’s merciful acceptance of the sacrificial offering, the use of the phrase sacrificium laudis as a way to name and describe the eucharistic sacrifice, the centrality of Melchizedek’s sacrifice in conjunction with those of Abel and Abraham, and the content of the anaphora’s doxology—are all found in the Epistle to the Hebrews

Correlated exponential functions in high precision calculations for diatomic molecules

Various properties of the general two-center two-electron integral over the explicitly correlated exponential function are analyzed for the potential use in high precision calculations for diatomic molecules. A compact one dimensional integral representation is found, which is suited for the numerical evaluation. Together with recurrence relations, it makes possible the calculation of the two-center two-electron integral with arbitrary powers of electron distances. Alternative approach via the Taylor series in the internuclear distance is also investigated. Although numerically slower, it can be used in cases when recurrences lose stability. Separate analysis is devoted to molecular integrals with integer powers of interelectronic distances $r_{12}$ and the vanishing corresponding nonlinear parameter. Several methods of their evaluation are proposed.Comment: 26 pages, includes two tables with exemplary calculation

On the Mathematical and Geometrical Structure of the Determining Equations for Shear Waves in Nonlinear Isotropic Incompressible Elastodynamics

Using the theory of $1+1$ hyperbolic systems we put in perspective the mathematical and geometrical structure of the celebrated circularly polarized waves solutions for isotropic hyperelastic materials determined by Carroll in Acta Mechanica 3 (1967) 167--181. We show that a natural generalization of this class of solutions yields an infinite family of \emph{linear} solutions for the equations of isotropic elastodynamics. Moreover, we determine a huge class of hyperbolic partial differential equations having the same property of the shear wave system. Restricting the attention to the usual first order asymptotic approximation of the equations determining transverse waves we provide the complete integration of this system using generalized symmetries.Comment: 19 page

Reactive-infiltration instabilities in rocks. Fracture dissolution

A reactive fluid dissolving the surface of a uniform fracture will trigger an instability in the dissolution front, leading to spontaneous formation of pronounced well-spaced channels in the surrounding rock matrix. Although the underlying mechanism is similar to the wormhole instability in porous rocks there are significant differences in the physics, due to the absence of a steadily propagating reaction front. In previous work we have described the geophysical implications of this instability in regard to the formation of long conduits in soluble rocks. Here we describe a more general linear stability analysis, including axial diffusion, transport limited dissolution, non-linear kinetics, and a finite length system.Comment: to be published in J. Fluid. Mec

A Kind of Affine Weighted Moment Invariants

A new kind of geometric invariants is proposed in this paper, which is called affine weighted moment invariant (AWMI). By combination of local affine differential invariants and a framework of global integral, they can more effectively extract features of images and help to increase the number of low-order invariants and to decrease the calculating cost. The experimental results show that AWMIs have good stability and distinguishability and achieve better results in image retrieval than traditional moment invariants. An extension to 3D is straightforward

Statistical physics of power fluctuations in mode locked lasers

We present an analysis of the power fluctuations in the statistical steady state of a passively mode locked laser. We use statistical light-mode theory to map this problem to that of fluctuations in a reference equilibrium statistical physics problem, and in this way study the fluctuations non-perturbatively. The power fluctuations, being non-critical, are Gaussian and proportional in amplitude to the inverse square root of the number of degrees of freedom. We calculate explicit analytic expressions for the covariance matrix of the overall, pulse and cw power variables, providing complete information on the single-time power distribution in the laser, and derive a set of fluctuation-dissipation relations between them and the susceptibilities of the steady-state quantities.Comment: 7 pages, 1 figure, RevTe

Conservation laws of scaling-invariant field equations

A simple conservation law formula for field equations with a scaling symmetry is presented. The formula uses adjoint-symmetries of the given field equation and directly generates all local conservation laws for any conserved quantities having non-zero scaling weight. Applications to several soliton equations, fluid flow and nonlinear wave equations, Yang-Mills equations and the Einstein gravitational field equations are considered.Comment: 18 pages, published version in J. Phys. A:Math. and Gen. (2003). Added discussion of vorticity conservation laws for fluid flow; corrected recursion formula and operator for vector mKdV conservation law

On the geometry of lambda-symmetries, and PDEs reduction

We give a geometrical characterization of $\lambda$-prolongations of vector fields, and hence of $\lambda$-symmetries of ODEs. This allows an extension to the case of PDEs and systems of PDEs; in this context the central object is a horizontal one-form $\mu$, and we speak of $\mu$-prolongations of vector fields and $\mu$-symmetries of PDEs. We show that these are as good as standard symmetries in providing symmetry reduction of PDEs and systems, and explicit invariant solutions

Intrinsic Geometry of a Null Hypersurface

We apply Cartan's method of equivalence to construct invariants of a given null hypersurface in a Lorentzian space-time. This enables us to fully classify the internal geometry of such surfaces and hence solve the local equivalence problem for null hypersurface structures in 4-dimensional Lorentzian space-times