Ellipticity Conditions for the Lax Operator of the KP Equations

The Lax pseudo-differential operator plays a key role in studying the general set of KP equations, although it is normally treated in a formal way, without worrying about a complete characterization of its mathematical properties. The aim of the present paper is therefore to investigate the ellipticity condition. For this purpose, after a careful evaluation of the kernel with the associated symbol, the majorization ensuring ellipticity is studied in detail. This leads to non-trivial restrictions on the admissible set of potentials in the Lax operator. When their time evolution is also considered, the ellipticity conditions turn out to involve derivatives of the logarithm of the tau-function.Comment: 21 pages, plain Te

Non-Local Boundary Conditions in Euclidean Quantum Gravity

Non-local boundary conditions for Euclidean quantum gravity are proposed, consisting of an integro-differential boundary operator acting on metric perturbations. In this case, the operator P on metric perturbations is of Laplace type, subject to non-local boundary conditions; by contrast, its adjoint is the sum of a Laplacian and of a singular Green operator, subject to local boundary conditions. Self-adjointness of the boundary-value problem is correctly formulated by looking at Dirichlet-type and Neumann-type realizations of the operator P, following recent results in the literature. The set of non-local boundary conditions for perturbative modes of the gravitational field is written in general form on the Euclidean four-ball. For a particular choice of the non-local boundary operator, explicit formulae for the boundary-value problem are obtained in terms of a finite number of unknown functions, but subject to some consistency conditions. Among the related issues, the problem arises of whether non-local symmetries exist in Euclidean quantum gravity.Comment: 23 pages, plain Tex. The revised version is much longer, and new original calculations are presented in section

The hybrid spectral problem and Robin boundary conditions

The hybrid spectral problem where the field satisfies Dirichlet conditions (D) on part of the boundary of the relevant domain and Neumann (N) on the remainder is discussed in simple terms. A conjecture for the C_1 coefficient is presented and the conformal determinant on a 2-disc, where the D and N regions are semi-circles, is derived. Comments on higher coefficients are made. A hemisphere hybrid problem is introduced that involves Robin boundary conditions and leads to logarithmic terms in the heat--kernel expansion which are evaluated explicitly.Comment: 24 pages. Typos and a few factors corrected. Minor comments added. Substantial Robin additions. Substantial revisio

Wage returns to university disciplines in Greece: are Greek Higher Education degrees Trojan Horses?

This paper examines the wage returns to qualifications and academic disciplines in the Greek labour market. Exploring wage responsiveness across various degree subjects in Greece is interesting, as it is characterised by high levels of graduate unemployment, which vary considerably by field of study, and relatively low levels of wage flexibility. Using micro-data from recently available waves (2002-2003) of the Greek Labour Force Survey (LFS), the returns to academic disciplines are estimated by gender and public/private sector. Quantile regressions and cohort interactions are also used to capture the heterogeneity in wage returns across the various disciplines. The results show considerable variation in wage premiums across the fields of study, with lower returns for those that have a marginal role to play in an economy with a rising services/shrinking public sector. Educational reforms that pay closer attention to the future prospects of university disciplines are advocated

New Kernels in Quantum Gravity

Recent work in the literature has proposed the use of non-local boundary conditions in Euclidean quantum gravity. The present paper studies first a more general form of such a scheme for bosonic gauge theories, by adding to the boundary operator for mixed boundary conditions of local nature a two-by-two matrix of pseudo-differential operators with pseudo-homogeneous kernels. The request of invariance of such boundary conditions under infinitesimal gauge transformations leads to non-local boundary conditions on ghost fields. In Euclidean quantum gravity, an alternative scheme is proposed, where non-local boundary conditions and the request of their complete gauge invariance are sufficient to lead to gauge-field and ghost operators of pseudo-differential nature. The resulting boundary conditions have a Dirichlet and a pseudo-differential sector, and are pure Dirichlet for the ghost. This approach is eventually extended to Euclidean Maxwell theory.Comment: 19 pages, plain Tex. In this revised version, section 5 is new, section 3 is longer, and the presentation has been improve

High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows

We relate high-energy limits of Laplace-type and Dirac-type operators to frame flows on the corresponding manifolds, and show that the ergodicity of frame flows implies quantum ergodicity in an appropriate sense for those operators. Observables for the corresponding quantum systems are matrix-valued pseudodifferential operators and therefore the system remains non-commutative in the high-energy limit. We discuss to what extent the space of stationary high-energy states behaves classically.Comment: 26 pages, latex2

Higher cost of finance exacerbates a climate investment trap in developing economies

Finance is vital for the green energy transition, but access to low cost finance is uneven as the cost of capital differs substantially between regions. This study shows how modelled decarbonisation pathways for developing economies are disproportionately impacted by different weighted average cost of capital (WACC) assumptions. For example, representing regionally-specific WACC values indicates 35% lower green electricity production in Africa for a cost-optimal 2 °C pathway than when regional considerations are ignored. Moreover, policy interventions lowering WACC values for low-carbon and high-carbon technologies by 2050 would allow Africa to reach net-zero emissions approximately 10 years earlier than when the cost of capital reduction is not considered. A climate investment trap arises for developing economies when climate-related investments remain chronically insufficient. Current finance frameworks present barriers to these finance flows and radical changes are needed so that capital is more equitably distributed

Klein-Gordon Solutions on Non-Globally Hyperbolic Standard Static Spacetimes

We construct a class of solutions to the Cauchy problem of the Klein-Gordon equation on any standard static spacetime. Specifically, we have constructed solutions to the Cauchy problem based on any self-adjoint extension (satisfying a technical condition: "acceptability") of (some variant of) the Laplace-Beltrami operator defined on test functions in an $L^2$-space of the static hypersurface. The proof of the existence of this construction completes and extends work originally done by Wald. Further results include the uniqueness of these solutions, their support properties, the construction of the space of solutions and the energy and symplectic form on this space, an analysis of certain symmetries on the space of solutions and of various examples of this method, including the construction of a non-bounded below acceptable self-adjoint extension generating the dynamics

Spectral asymptotics of Euclidean quantum gravity with diff-invariant boundary conditions

A general method is known to exist for studying Abelian and non-Abelian gauge theories, as well as Euclidean quantum gravity, at one-loop level on manifolds with boundary. In the latter case, boundary conditions on metric perturbations h can be chosen to be completely invariant under infinitesimal diffeomorphisms, to preserve the invariance group of the theory and BRST symmetry. In the de Donder gauge, however, the resulting boundary-value problem for the Laplace type operator acting on h is known to be self-adjoint but not strongly elliptic. The latter is a technical condition ensuring that a unique smooth solution of the boundary-value problem exists, which implies, in turn, that the global heat-kernel asymptotics yielding one-loop divergences and one-loop effective action actually exists. The present paper shows that, on the Euclidean four-ball, only the scalar part of perturbative modes for quantum gravity are affected by the lack of strong ellipticity. Further evidence for lack of strong ellipticity, from an analytic point of view, is therefore obtained. Interestingly, three sectors of the scalar-perturbation problem remain elliptic, while lack of strong ellipticity is confined to the remaining fourth sector. The integral representation of the resulting zeta-function asymptotics is also obtained; this remains regular at the origin by virtue of a spectral identity here obtained for the first time.Comment: 25 pages, Revtex-4. Misprints in Eqs. (5.11), (5.14), (5.16) have been correcte