Global properties of Gowdy spacetimes with T3 × R topology

We study the global properties of the Gowdy metrics generated by Cauchy data on the 3-torus. We show that the boundaries of the maximal Cauchy developments of Gowdy initial data sets are always “crushing singularities” in the sense of Eardley and Smarr. This means that each solution admits a slicing in which tr K(t) (the trace of the second fundamental form induced on the surface Σ_t of the slicing) uniformly blows up as t approaches its limiting value. A theorem of Hawking shows that the maximal Cauchy development cannot extend beyond the boundary at which tr K blows up and our result shows that no singularities arise to halt the evolution until this boundary is reached. Thus each maximal Cauchy development is always as large as it can be, consistent with Hawking's theorem. We discuss the relevance of this result to the strong cosmic censorship conjecture and the question of when the crushing singularities are in fact curvature singularities.     

Evaluation of the spin wave Green's functions for rutile structure Heisenberg antiferromagnets with exchange between ions both on the same and opposite sublattices

A straightforward method is presented for the evaluation of the spin wave Green's function appropriate to the Raman scattering in the xy and xz geometry in rutile structure Heisenberg antiferromagnets with exchange between ions both on the same and on opposite sublattices and arbitrary local anisotropy. The analytical asymptotic behaviour of the Green's functions near the singularities is explicitly given and the problem of the numerical evaluation of their real and imaginary parts is discussed. Tables of the imaginary parts, calculated at the points corresponding to a Gaussian quadrature procedure in the appropriate interval, are supplied on request.     

Impact parameter representation from the Watson-Sommerfeld transform

Using the Watson-Sommerfeld transform the elastic scattering amplitude of two spinless particles is shown to have an exact and unique impact parameter, or Fourier-Bessel (FB) representation. The representation is valid for all physical energies and scattering angles. Wallace's recent work is found to be an asymptotic expansion of the FB amplitude obtained from the partial-wave expansion. The way singularities of the partial-wave amplitude in the l-plane enter in the FB amplitude is also explicitly shown.     

How to regularize the infrared and mass singularities in QCD

The parton cross sections in perturbative QCD are mass (M) and infrared (IR) divergent. The singularities are prevented by on/off-shell parton masses or by dimensional regularization. All mass assignments must be chosen in agreement with Kinoshita's double-cut rule. The “off-shell” assignment, often used in the past, is incorrect. We demonstrate that Drell-Yan mass factorization in order α_S is independent of all regularizatiohns; the coffcient function is unique.     

The universe as a stochastic process

A stochastic radiation-filled world model is constructed in which the equation of state is perturbed with a “white noise”. The corresponding Fokker-Planck equation is solved. It turns out that the maximum probability path of the world's evolution is that of the Friedman model. However, singularities are no obstacles for the stochastic evolution, and σ² → ∞ as t → ∞.     

Analysis of the generic singularity studies by Belinskii,Khalatnikov, and Lifschitz

We apply some modern mathematical methods of global analysis to a series of studies undertaken by Belinskii, Khalatnikov and Lifschitz (BKL) to elucidate the structure of space-time near a general cosmological singularity. A brief summary of BKL's large body of work on inhomogeneous cosmological models is given (their work on homogeneous models is not under discussion here). Various theorems are proven and analyses of a mathematical and physical nature are made to show that the constructions of BKL cannot be general and in some cases do not give Lorentz manifolds. We conclude that although the work of BKL has led to very significant advances in our understanding of the dynamics of homogeneous cosmological models, the local techniques they employ do not extend to give us reliable information about the global structure of generic space-times. A detailed discussion of stability, generality, function counting, linearization stability, physical singularities and fictitious singularities is given together with an outline of various physical considerations which might be useful in future studies of the structure of generic space-times.     

Proprietes optiques et photoelectriques du GaSb

Reflectivity of single crystals of GaSb were measured between 1,3 and 70 eV; optical and dielectric constants were deduced by Kramers-Kronig's analysis in the interband transition region. An interpretation of observed singularities is given.Optical studies were completed by determination of quantum yield and energy distribution curves between 7,7 and 25 eV. The analysis of these curves gives initial and final energies of excitations and allows transitions assignments.By reflectivity measurements we also observed core transitions from d levels of gallium and antimony.     

Thermodynamic critical amplitudes in the dipolar critical region of a uniaxial ferromagnet

In zero magnetic field susceptibility, magnetization and specific heat of GdCl₃ follow below t = |T/T_c - 1| ≈0.015 mean field laws with logarithmic corrections. Their amplitudes characterizing the strengths of the singularities are related by an equation predicted by the theory of Larkin and Khmel'nitzkii.     

Chiral solitons and current algebra

We investigate the possibility of a realistic hadrodynamics based solely on observable currents. The basic idea is to exploit the soliton generation in bosonic chiral theories as a mechanism for finding the fermionic representations of current algebra. A prototype realization is Skyrme's O(4) invariant theory of pions and nucleons. A comprehensive reexamination of this model in the context of chiral dynamics suffices to reveal a strikingly self-consistent dynamical picture. First a differential geometric formulation gives the proper framework for a chiral invariant quantum theory of solitons and allows a compact derivation of Skyrme's main results. While no exact analytic solution is found, the solitons are sufficiently localized so that their singularities can be properly isolated out for analysis. Using Witten's ansatz, a determination of the form of the 1-soliton singularity is obtained from the field equations. It is given by Cayley's steoreographic projection from S³ to $\text{R}{}^3\text{? {∞} ≈ S}{}^3$; a most suitable form for the proof of spinor structure. Williams' proof that the quantized 1-soliton sector gives rise to fermionic spin states is recalled. It is argued that the topological dynamics of this sector induce an invariance group K = SU(2)_I ? SU(2)_J and its associated strong coupling isobaric spectrum for the nucleons. The associated current algebra is derived and resolves the main difficulties of the Sugawara-Sommerfield program. The signature of a field theoretical bootstrap is clear: massive nucleons as soliton bound states of Nambu-Goldstone bosons illustrate a dynamical mechanism dual to that of Nambu and Jona-Lasinio.     

A thermodynamic discussion of the possibility of singular yield conditions in plasticity theory

A generalization is given of the author's theory for plasticity phenomena. The generalization leads to the possibility that the yield surface has singularities. From the theory a formula may be derived which is analogous to a formula proposed by Koiter for plastic flow in media with singular yield surfaces. The possibility of elastic relaxation phenomena in the preplastic range is included in the developed formalism.     

Singularities of the second derivatives of tunnel current

It is shown, that though paramagnetic impurities wash out the density of electronic states at the gap edge, the singularities of the second derivatives in the model of Abricosov-Gor'kov are not eliminated.     

Applications of catastrophe theory to the physical sciences

Catastrophe Theory was introduced by René Thom in the late 1960's, as an attempt to model morphogentic changes in nature using ideas from topological dynamics and the theory of singularities of mappings. Thom envisaged a very general approach to topological changes in the solutions to parametrized systems of equations (such as differential and difference equations), and in particular discussed the special case of ‘elementary’ catastrophe theory: singularities of smooth real-valued functions. Popular expositions have tended to overemphasize this special case, but it remains the major source of ideas and methods.Here we survey the applications of Catastrophe Theory to the physical sciences (physics, chemistry, engineering, fluid mechanics, etc.). For brevity we confine attention to an area lying between ‘elementary’ and ‘general’ Catastrophe Theory, usually known as Singularity Theory. This is the theory of singularities of smooth vector-valued functions, which mathematically is a straightforward (though non-trivial) generalization of the real-valued case. In the last few years it has developed into a powerful and useful technique in several areas of theoretical physics, notably optics and bifurcation theory. Equivariant Catastrophe Theory, taking account of symmetry, is likely to prove especially interesting.     

Effects of impurities on heat capacities at paramagnetic to ferromagnetic transitions

The influence of controlled impurities upon the detailed shape of the heat capacity of Gd near the Curie temperature is studied experimentally. The experimental results can be interpreted in terms of Watson's theoretical analyses which show that small scale, random inhomogeneities do not alter the shape of the heat capacity curve, whereas large scale inhomogeneities are found to reduce the sharpness of the critical singularities.     

Non-linear σ-model and CP(n−1) at 2 + ε dimensions

Two systems, O(n) non-linear σ-model and CP^(n?1), are studied in the light of Elitzur's theorem, on the disappearance of infrared singularities at two dimensions. The consequences of the theorem are expressed in dimensional regularization, and issues like the proper analytic continuation to d = 2 + ε, the peculiarities of momentum-space Green functions near d = 2 and their renormalization, and the exponentiation of Green functions are clarified.The analysis is applied to compute the renormalization constants, and the gauge-invariant critical exponent η associated with the wave function of CP^(n?1) at one order higher than previously done. Finally, we conjecture on a possible connection between infrared finiteness and renormalizability.     

Singularities in Bianchi cosmologies

We discuss how infinite density singularities may be shown to occur in Friedmann-Robertson-Walker universes and orthogonal spatially homogeneous universes, but how very different behaviours are possible in tilted homogeneous cosmologies. After considering various possibilities that arise in this case, we illustrate them by examining the behaviour of exact solutions of Einstein's equations for a homogeneous cosmology which is a locally rotationally symmetric tilted Bianchi type V universe. These universes - which can be arbitrarily similar to a Robertson-Walker universe at late times - show a variety of singular behaviours quite different from those in the ‘orthogonal’ case. In particular, there exist such universes in which two singularities occur at the early stages of the universe, but in which the density of matter is finite at all times.     

XI. The Borel transform and the renormalization group

We study the consequences of the renormalization group on the large momenta behaviour of the Borel transform of the Green's functions. We discuss the implications on the structure of the ultraviolet singularities of the Borel transform.     

The Numerical Methods for Oscillating Singularities in Elliptic Boundary Value Problems

The singularities near the crack tips of homogeneous materials are monotone of type r^α and r^α log^δr (depending on the boundary conditions along nonsmooth domains). However, the singularities around the interfacial cracks of the heterogeneous bimaterials are oscillatory of type r^α sin( log r). The method of auxiliary mapping (MAM), introduced by Babu ka and Oh, was proven to be successful in dealing with r^α type singularities. However, the effectiveness of MAM is reduced in handling oscillating singularities. This paper deals with oscillating singularities as well as the monotone singularities by extending MAM through introducing the power auxiliary mapping and the exponential auxiliary mapping.     

De Rham currents,extended singularities of fields and magnetic monopoles

We apply De Rham's concept of “currents” to the treatment of extended singularities or discontinuities of physical fields in a coordinate independent way. In particular, we obtain a generalization of the theory of magnetic monopoles when applied to the electromagnetic field.     

Phase and polarization singularities of nonparaxial Gaussian vortex beams

The analytical expression for nonparaxial Gaussian vortex beams propagating in free space is derived, which enables us to study phase and polarization singularities in nonparaxial vector wavefields. Differing from the polarization singularities formed by two transverse electric-field components in the paraxial regime, the polarization singularities can be formed by the transverse and longitudinal electric-field components of nonparaxial beams, and there exist C-points and L-lines. The variation of the beam parameters and propagation distance will result in a shift of phase and polarization singularities, but their position relation remains unchanged and the topological relationship holds true.