Non-linearσ-models on compact Riemann surfaces

The classicalO(3) non-linear -model is generalised to a theory of fields defined on a compact Riemann surfaceM with values in a compact Kähler manifoldV. The dimension of the space of self-dual fields fromM to the complex projective space ^N is calculated and the classifying space for the inequivalent quantisations of the theory is also calculated.work supported by the Science and Engineering Research Council     

A Nyström method for weakly singular integral operators on surfaces

We describe a modified Nyström method for the discretization of the weakly singular boundary integral operators which arise from the formulation of linear elliptic boundary value problems as integral equations. Standard Nyström and collocation schemes proceed by representing functions via their values at a collection of quadrature nodes. Our method uses appropriately scaled function values in lieu of such representations. This results in a scheme which is mathematically equivalent to Galerkin discretization in that the resulting matrices are related to those obtained by Galerkin methods via conjugation with well-conditioned matrices, but which avoids the evaluation of double integrals. Moreover, we incorporate a new mechanism for approximating the singular integrals which arise from the discretization of weakly singular integral operators which is considerably more efficient than standard methods. We illustrate the performance of our method with numerical experiments.     

Photoelectron microscopy — Applications to biological surfaces

Photoelectron microscopy utilizes incident light to eject electrons which are then accelerated and used to form an image of the specimen surface. Photoelectron microscopy is now being developed to the point where it can begin to contribute to the analysis of biological structures including cell surfaces and the organization of cellular organelles. PEM is the electron optics analog of fluorescence microscopy and shares with it many advantages and areas of application. As in fluorescence microscopy, the strategy is not an elemental analysis of components present since proteins and other functional macromolecules usually do not differ significantly in elemental composition, but rather the combination with site-specific antibodies to identify the organization and structure-function relationships in cells. Material contrast is based on photoelectron yield, which depends on the valence electron properties of the surface materials rather than on atomic number differences, as with most surface analytical techniques. Topographic contrast makes very small objects visible on relatively flat surfaces because the low energy electrons emerging from the specimen surface are easily deflected by the accompanying field variations. The surface sensitivity is also high because of the short escape depths of the ejected photoelectrons. These and other characteristics combined with a lateral resolution of about 10 nm makes photoelectron microscopy a promising technique for studies of organic and biological surfaces.     

A global operator formalism on higher genus Riemann surfaces:b−c systems

We explicitly construct bases for meromorphic-differentials over genusg Riemann surfaces. With the help of these bases we introduce a new operator formalism over Riemann surfaces which closely resembles the operator formalism on the sphere. As an application we calculate the propagators forb-c systems with arbitrary integer or half-integer (in the Ramond and Neveu-Schwarz sectors). We also give explicit expressions for the zero modes and for the Teichmüller deformations for a generic Riemann surface.     

Inflation — an alternative to the singular Big Bang

Domains larger than the horizon in which > (a few) ×M _pl are required for the onset of inflation. Two different, equally plausible, arguments lead us to opposite conclusions about the feasibility of the existence of such regions. It seems that inflation does not free us completely from the need for special initial conditions. However, Linde has pointed out that inflation can be eternal. He stresses the fact that inflation will never cease, but this also means that it did not necessarily have a beginning. We argue that this is the simplest solution to the initial value problem and that inflation might not only solve the problems of the Big Bang model, it might also provide us with an alternative that will replace it altogether.This essay received the third award from the Gravity Research Foundation, 1990 —Ed.     

Riemann–Hilbert approach to a generalized sine kernel

Letters in Mathematical Physics - We derive the large-distance asymptotics of the Fredholm determinant of the so-called generalized sine kernel at the critical point. This kernel corresponds to a...     

A Riemann–Hilbert approach to Painlevé IV

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ?¹ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ?¹ are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.     

Riemann?CHilbert Approach to the Elastodynamic Equation: Part I

We develop the Riemann?CHilbert (RH) approach to scattering problems in elastic media. The approach is based on the RH method introduced in the 1990s by Fokas (A unified approach to boundary value problems, CBMS-SIAM, 2008) for studying boundary problems for linear and integrable nonlinear PDEs. A suitable Lax pair formulation of the elastodynamic equation is obtained. The integral representations derived from this Lax pair are applied to Rayleigh wave propagation in an elastic half space and quarter space. The latter problem is reduced to the analysis of a certain underdetermined RH problem. We show that the problem can be re-formulated as a well-determined vector Riemann?CHilbert problem with a shift posed on a torus.     

Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications

We investigate the asymptotic behaviour of a generalised sine kernel acting on a finite size interval [−q ; q]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener–Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models.     

From Euclidean to Minkowski space with the Cauchy–Riemann equations

We present an elementary method to obtain Green’s functions in non-perturbative quantum field theory in Minkowski space from Green’s functions calculated in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes often is unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore, we suggest to use the Cauchy–Riemann equations, which perform the analytical continuation without assuming global information on the function in the entire complex plane, but only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge quantum chromodynamics, which is known from lattice and Dyson–Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy–Riemann equations against high-frequency noise,which makes it difficult to achieve good accuracy. We also point out a few curious details related to the Wick rotation.     

A local index theorem for families of\bar \partial -operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaceson punctured Riemann surfaces and a new Kähler metric on their moduli spaces

We prove a local index theorem for families of \(\bar \partial \) -operators on Riemann surfaces of type (g, n), i.e. of genusg withn>0 punctures. We calculate the first Chern form of the determinant line bundle on the Teichmüller spaceT _g,n endowed with Quillen's metric (where the role of the determinant of the Laplace operators is played by the values of the Selberg zeta function at integer points). The result differs from the case of compact Riemann surfaces by an additional term, which turns out to be the Kähler form of a new Kähler metric on the moduli space of punctured Riemann surfaces. As a corollary of this result we derive, for instance, an analog of Mumford's isomorphism in the case of the universal curve.     

Padé approximant approach to singular properties of quantum gases: the ideal cases

In this paper, we show how to recover the low-temperature and high-density information of ideal quantum gases from the high-temperature and low-density approximation by the Padéapproximant. The virial expansion is a high-temperature and low-density expansion and in practice, often, only the first several virial coefficients can be obtained. For Bose gases, we determine the BEC phase transition from a truncated virial expansion. For Fermi gases, we recover the low-temperature and high-density result from the virial expansion.     

Extending the First—Order Post—Newtonian Scheme in Multiple Systems to the Second—Order Contributions to Light Propagation

We extend the first-order post-Newtonian scheme in multiple systems presented by Damour-Soffel-Xu to the second-order contribution to light propagation without changing the virtue of the scheme on the linear partial differential equations of the potential and vector potential.The spatial components of the metric are extended to the second-order level both in a global coordinates(qij/c^4) and a local coordinates(Qab/c^4).The equations of qij(or Qab)are obtained from the field the field equations.The relationship between qji and Qab are also presented.In the special case of the solar system(isotropic condition is applied(qij=δijq)),we obtain the solution of q.Finally,a further extension of the second-order contributoions in the parametrized post-Newtonian formalism is discussed.     

Riemann–Hilbert approach to multi-time processes: The Airy and the Pearcey cases

We prove that matrix Fredholm determinants related to multi-time processes can be expressed in terms of determinants of integrable kernels à la Its–Izergin–Korepin–Slavnov (IIKS) and hence related to suitable Riemann–Hilbert problems, thus extending the known results for the single-time case. We focus on the Airy and Pearcey processes. As an example of applications we re-deduce a third order PDE, found by Adler and van Moerbeke, for the two-time Airy process.     

Are ghost surfaces quadratic-flux-minimizing?

Two candidates for “almost-invariant” toroidal surfaces passing through magnetic islands, namely quadratic-flux-minimizing (QFMin) surfaces and ghost surfaces, use families of periodic pseudo-orbits (i.e. paths for which the action is not exactly extremal). QFMin pseudo-orbits, which are coordinate-dependent, are field lines obtained from a modified magnetic field, and ghost-surface pseudo-orbits are obtained by displacing closed field lines in the direction of steepest descent of magnetic action, ∮A⋅dl. A generalized Hamiltonian definition of ghost surfaces is given and specialized to the usual Lagrangian definition. A modified Hamilton's Principle is introduced that allows the use of Lagrangian integration for calculation of the QFMin pseudo-orbits. Numerical calculations show QFMin and Lagrangian ghost surfaces give very similar results for a chaotic magnetic field perturbed from an integrable case, and this is explained using a perturbative construction of an auxiliary poloidal angle for which QFMin and Lagrangian ghost surfaces are the same up to second order. While presented in the context of 3-dimensional magnetic field line systems, the concepts are applicable to defining almost-invariant tori in other degree-of-freedom nonintegrable Lagrangian/Hamiltonian systems.