Quasi-Normal Modes of a Schwarzschild White Hole

We investigate perturbations of the Schwarzschild geometry using a linearization of the Einstein vacuum equations within a Bondi-Sachs, or null cone, formalism. We develop a numerical method to calculate the quasi-normal modes, and present results for the case $\ell=2$. The values obtained are different to those of a Schwarzschild black hole, and we interpret them as quasi-normal modes of a Schwarzschild white hole.Comment: 5 pages, 4 Figure

Stokes phenomenon and matched asymptotic expansions

This paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting Stokes phenomenon. In particular the approach highlights the way in which the local structure and the possibility of finding Stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

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

Spectra of Jacobi operators via connection coefficient matrices

We address the computational spectral theory of Jacobi operators that are compact perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In particular, for finite-rank perturbation we show that the computation of the spectrum can be reduced to a polynomial root finding problem, from a polynomial that is derived explicitly from the entries of a connection coefficient matrix. A formula for the spectral measure of the operator is also derived explicitly from these entries. The analysis is extended to trace-class perturbations. We address issues of computability in the framework of the Solvability Complexity Index, proving that the spectrum of compact perturbations of the free Jacobi operator is computable in finite time with guaranteed error control in the Hausdorff metric on sets

Lie symmetries of Einstein's vacuum equations in N dimensions

We investigate Lie symmetries of Einstein's vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein's equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein's equations), we set to zero the coefficients of derivatives that do not appear in Einstein's equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein's equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations

Continuation treatment of major depressive disorder: is there a case for duloxetine?

Duloxetine is a serotonin–noradrenaline reuptake inhibitor with established efficacy for the short-term treatment of major depressive disorder. Efficacy in continuation treatment (greater than six months of continuous treatment) has been established from both open and placebo-controlled relapse prevention and comparative studies. Seven published studies were available for review and showed that in both younger and older populations (aged more than 65 years) the acute efficacy of duloxetine was maintained for up to one year. Response to treatment was based on accepted criteria for remission of depression and in continuation studies remission rates were greater than 70%. Comparative studies showed that duloxetine was superior to placebo and comparable to paroxetine and escitalopram in relapse prevention. Importantly a study of duloxetine in patients prone to relapse of major depressive disorder showed that the medication was more effective than placebo in this difficult to treat population. Side effects of duloxetine during continuation treatment were predictable on the basis of the known pharmacology of the drug. In particular there were no significant life-threatening events which emerged with continued use of the medication. On the other hand vigilance is required since the data base on which to judge very rare events is limited by the relatively low exposure to the drug. Duloxetine has established both efficacy and safety for continuation treatment but its place as a first-line treatment of relapse prevention requires further experience. In particular further comparative studies against established agents would be useful in deciding the place of duloxetine in therapy

Computation of power law equilibrium measures on balls of arbitrary dimension

We present a numerical approach for computing attractive-repulsive power law equilibrium measures in arbitrary dimension. We prove new recurrence relationships for radial Jacobi polynomials on d-dimensional ball domains, providing a substantial generalization of the work started in Gutleb et al. (Math Comput 9:2247–2281, 2022) for the one-dimensional case based on recurrence relationships of Riesz potentials on arbitrary dimensional balls. Among the attractive features of the numerical method are good efficiency due to recursively generated banded and approximately banded Riesz potential operators and computational complexity independent of the dimension d, in stark constrast to the widely used particle swarm simulation approaches for these problems which scale catastrophically with the dimension. We present several numerical experiments to showcase the accuracy and applicability of the method and discuss how our method compares with alternative numerical approaches and conjectured analytical solutions which exist for certain special cases. Finally, we discuss how our method can be used to explore the analytically poorly understood gap formation boundary to spherical shell support

Orthogonal polynomials on a class of planar algebraic curves

We construct bivariate orthogonal polynomials (OPs) on algebraic curves of the form ym = φ(x) in R2 where m = 1, 2 and φ is a polynomial of arbitrary degree d, in terms of univariate semiclassical OPs. We compute connection coefficients that relate the bivariate OPs to a polynomial basis that is itself orthogonal and whose span contains the OPs as a subspace. The connection matrix is shown to be banded and the connection coefficients and Jacobi matrices for OPs of degree 0, . . . , N are computed via the Lanczos algorithm in O(Nd4) operations

Euler Polynomials and Identities for Non-Commutative Operators

Three kinds of identities involving non-commutating operators and Euler and Bernoulli polynomials are studied. The first identity, as given by Bender and Bettencourt, expresses the nested commutator of the Hamiltonian and momentum operators as the commutator of the momentum and the shifted Euler polynomial of the Hamiltonian. The second one, due to J.-C. Pain, links the commutators and anti-commutators of the monomials of the position and momentum operators. The third appears in a work by Figuieira de Morisson and Fring in the context of non-Hermitian Hamiltonian systems. In each case, we provide several proofs and extensions of these identities that highlight the role of Euler and Bernoulli polynomials.Comment: 20 page