Showing papers on "Singularity published in 1978"

On the Pomeranchuk singularity in the quantum chromodynamics

Abstract: The high-energy asymptotic form of the scattering amplitude of colorless particles in quantum chromodynamics is obtained in the leading logarithmic approximation. It is argued that such a calculation is justified for the amplitudes of scattering in which charmed quarks participate. The cross section for formation of two pairs of charmed quarks in ..gamma gamma.. collisions is found in explicit form.

Singularity structure of the two-point function in quantum field theory in curved spacetime

Abstract: In the point-splitting prescription for renormalizing the stress-energy tensor of a scalar field in curved spacetime, it is assumed that the anticommutator expectation valueG(x, x′)=〈o(x)o(x′)+o(x′)o(x)〉 has a singularity of the Hadamard form asx→x′. We prove here that ifG(x,x′) has the Hadamard singularity structure in an open neighborhood of a Cauchy surface, then it does so everywhere, i.e., Cauchy evolution preserves the Hadamard singularity structure. In particular, in a spacetime which is flat below a Cauchy surface, for the “in” vacuum stateG(x,x′) is of the Hadamard form everywhere, and thus the point-splitting prescription in this case has been rigorously shown to give meaningful, finite answers.

Two-dimensional Ising Field Theory in a Magnetic Field: Breakup of the Cut in the Two Point Function

Abstract: We demonstrate that the cut which is present as the leading singularity in the two-point function of the Ising field theory for $Tl{T}_{c}$ and $H=0$ breaks up into a sequence of poles for $H\ensuremath{ e}0$. Both the positions and the residues of the low-lying poles are calculated.

Transition elements to be used with quarter-point crack-tip elements

Abstract: A generalization of the (r−1/2) strain or stress singularity capability of the quadratic isoparametric element is presented. It is shown that, by variable placement of the side-nobe between quarter- and mid-point, the point of singularity sensed by the element can be controlled. By using eight-noded quadrilateral isoparametric elements with appropriately placed side-nodes as transition elements between the quarter-point crack-tip trinagular and remaining non-singular elements, stress intensity factors may be computed with higher accuracy.

The signature of smoothings of complex surface singularities

Abstract: Let f : (1~3, 0)-"~(t~, 0) be the germ of a complex analytic function with an isolated critical point at the origin. For e > 0 suitably small and 6 yet smaller, the space V ' = f l ( 6 ) ~ D , (where D~ denotes the closed disk of radius e about 0) is a real oriented four-manifold with boundary whose diffeomorphism type depends only on f It has been proved that V' has the homotopy type of a wedge of two-spheres; the number p o f two-spheres is readily computable. Recently an interesting formula for g was given in terms of analytic invariants of a resolution of the singularity at 0 of the complex surface f-1(0) [13]. This formula is proved by applying the Riemann-Roch theorem to the projective completions o f f l ( 0 ) and fi (6) , then canceling terms coming from the parts away from the origin. The purpose of this paper is to find a similar formula for the signature of the intersection pairing on the two-dimensional homology of the manifold V', using the Hirzebruch signature theorem instead of the Riemann-Roch theorem. Various other signature formulas are known, in higher dimensions as well as in dimension two. For f (x , y, z) of the form g(x, y) + z 2, the intersection pairing of V' is the same as a symmetrized Seifert matrix of the compound torus link {g(x, y )=0}~S 3. There is a simple formula for the signature of the symmetrized Seifert matrix of a compound link of one component [20]; hence if g-1(0) is irreducible, there is a simple formula for the signature of V' in terms of the Puiseux pairs ofg. If g1(0) has several branches at the origin, it is possible to find a Seifert matrix for the link defined by g and compute the signature [17], but this process is tedious for all but the simplest links. Formulas also exist for the signature when f(x, y, z) is of the type x a + yb + z c [10], or when f is weighted homogeneous [22]. There is in addition a formula for the signature in terms of mixed Hodge structure [233. The genus (or geometric genus) of the singularity f 1(0) (assumed Stein) is the dimension of HI(V,, C~), where ~" is a resolution of f-1(0). By combining the formulas for the signature and the number # it is possible to show that twice the genus of the singularity f 1(0) is equal to the number of positive plus the number of

Essential Singularity in Percolation Problems and Asymptotic Behavior of Cluster Size Distribution

Abstract: It is rigorously proved that the analog of the free energy for the bond and site percolation problem on\(\mathbb{Z}^v \) in arbitrary dimensionΝ (Ν> 1) has a singularity at zero external field as soon as percolation appears, whereas it is analytic for small concentrations. For large concentrations at least, it remains, however, infinitely differentiable and Borel-summable. Results on the asymptotic behavior of the cluster size distribution and its moments, and on the average surface-to-size ratio, are also obtained. Analogous results hold for the cluster generating function of any equilibrium state of a lattice model, including, for example, the Ising model, but infinite-range andn-body interactions are also allowed.

Nonadiabaticity in mirror machines

Abstract: An analytic technique for calculating magnetic‐moment jumps Δμ of particles in magnetic traps, previously derived for particular two‐dimensional vacuum fields, is generalized to nonvacuum fields of arbitrary complexity and applied to high‐β mirror machines. The size of a jump depends on the behavior of the magnetic‐field strength B (s) near the singularities of B in the complex s plane, where real s measures position along a field line. It is demonstrated that an intrinsic complication of mirror‐machine magnetic configurations is the presence of multiple singularities of B, which become closely spaced for field lines near the axis. An expansion in r2 is used to determine Δμ in the closely spaced regime. The analytic theory is compared with results from a particle‐orbit code for several axisymmetric nonvacuum fields, and is found to be in excellent agreement in both the well separated and closely spaced singularity regimes. Finite‐β effects are examined using axisymmetric model fields derived from the long...

The topology of holomorphic flows with singularity

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Friedmann-like cosmological models without singularity

Abstract: The Einstein-Cartan theory of gravitation (“general relativity with spin”) provides a specific spin-spin contact interaction of matter, in addition to the usual long-range gravity. This new interaction enables us to prevent singularities in Cosmological models. It is shown how this mechanism works in the case when the standard Einstein-Cartan equations are valid at a microphysical level, and some spin-spin terms remain from the averaging procedure for randomly distributed spins. In contrast with the case of aligned spin distributions, it is possible to take over the isotropic and spatially homogeneous (i.e., Friedmannian) models into the Einstein-Cartan theory. These models can be made free from singularity, thanks to the self-interaction of spinning fluid.

Thermal particle production in a radiation dominated Robertson-Walker universe

Abstract: Creation of massive spin-1/2 particles in a 3-flat Robertson-Walker universe with expansion law R approximately t1/2 (radiation dominated universe) is studied with Fock space methods. The universe is thereby completed in 'passing through the singularity' by a time-symmetric (mirror-like) contracting universe preceding the singularity. The respective procedure to do so for Dirac test fields (conformal method) is discussed in detail. For the asymptotic in- and out-regions a WKB particle interpretation is applied. It is found that particles are created with a non-relativistic thermal spectrum. The expressions for the number density, energy density and pressure of the created particles confirm this result (equation of state). The study of the development in time of the creation process shows that 99% of the particles created all together are created at about the Compton time.

Creation of particles by singularities in asymptotically flat spacetimes

Abstract: The creation of massless scalar particles by naked singularities in asymptotically flat spacetimes is investigated within the geometrical-optics approximation. To avoid the need to impose boundary conditions on the singularity, we consider models in which a curvature singularity arises at a finite time in the past. We consider two particular types of models. One is a shell-crossing singularity formed in the gravitational collapse of a dust cloud. The energy flux of the created particles remains finite up to the time of formation of the singularity. In the particular case when the singularity forms on the event horizon, geometrical optics yields the exact flux, in spite of the high curvature of spacetime. The radiation obtained is identical to the thermal Hawking radiation emitted by black holes. The other models considered are those of charged shells for which the charge exceeds the mass. If these shells collapse to form naked singularities (which is possible if the proper mass is negative or if Einstein's equations are not imposed), an infinite flux of created particles results. In the cases examined here, the flux is negative for two-dimensional models and for the minimally coupled scalar field in four-dimensional models, whereas it is positive for the conformally coupled scalar field in four-dimensional models. In either case, the back reaction from particle creation will be large and may prevent formation of a naked singularity.

Application of the singularity method for the formulation of plane elastostatical boundary value problems as integral equations

Abstract: A method for the formulation of elastostatical boundary value problems as integral equations is presented, the basic idea of which consists of superimposing in a suitable fashion singular solutions for the infinite medium. Since mechanical aspects play an important role in the concept of the method, all quantities in the equations can be interpreted physically. The applicability of the method is illustrated by examples of the geometrical and statical boundary value problem ofplane elastostatics for which 32 different formulations as integral equations are established. The second aim of the paper consists of revealing an analogy between the most important notions of the singularity method, viz. between state variables and singularities. The analogy is manifested by certain symmetries of influence functions, and enables the systematical representation of the basic relations and their interpretation within a larger context.

Essential Singularity in the Percolation Model

Abstract: It is rigorously proved that the analog of free energy for the percolation models has a singularity at zero external field as soon as percolation appears. The singularity is an essential one at least for large concentrations. Results on the asymptotic behavior of the cluster-size distribution are also obtained for percolation models and for the Ising model at low temperatures.

Action of a force near the planar surface between two semi-infinite immiscible liquids at very low Reynolds numbers

Abstract: Explicit expressions for the Green's functions due to a point force in one of two half space fluids are presented for the case when inertial effects of the fluid are negligible (Stokes flow) and the interface between the two fluids is considered to be flat due to the action of surface tension. The analytic expressions are discussed in terms of singularity diagrams. For the case of a force parallel to the interface a first approximation to the interface displacement is made.

Equivalence and Singularity of Gaussian Measures and Applications

Abstract: Keywords: reproducing kernel Hilbert spaces;;; equivalence and singularity;;; Gaussian measures;;; expository paper;;; Feldman-Hajek dichotomy for Gaussian measures;;; stationary Gaussian processes;;; absolute continuity and singularity of probability measures Reference GPRO-CHAPTER-1978-003 Record created on 2010-05-25, modified on 2016-08-08

Regge behavior of spontaneously broken non-Abelian gauge theory

Abstract: The high-energy, fixed-momentum-transfer behavior of spontaneously broken non-Abelain gauge theory, with SU(2) gauge group, has previously been calculated through tenth order in perturbation theory in the leading-logarithm approximation. We interpret the complicated results as due to the exchange of the gauge-meson Regge trajectory plus associated cuts generated by Reggeon field theory. There are only two Reggeon couplings consistent with the leading-logarithm perturbation theory calculation. The coupling strengths are highly overdetermined by the perturbation calculation, demonstrating the consistency of the assumption of the applicability of Reggeon field theory. For large values of the momentum transfer our results are in agreement with the calculations of Carruthers, Fishbane, and Zachariasen and Cornwall and Tiktopoulos to all orders in perturbation theory. When all of the contributions of the moving cuts are summed in the weak-coupling approximation, the leading $J$-plane singularity in the $I=0$ channel is a fixed cut. We show that this singularity remains fixed, at least in the weak-coupling approximation, even when asymptotic freedom is taken into account.

The numerical evaluation of a class of generalized stress intensity factors by use of the Lobatto-Jacobi numerical integration rule

Abstract: The Lobatto-Jacobi method of numerical solution of Cauchy type singular integral equations and determination of stress intensity factors, based on the use of the corresponding numerical integration rule, was modified so as to become applicable to the evaluation of a special class of generalized stress intensity factors associated with a real singularity. The same technique can also be used for the evaluation of generalized stress intensity factors associated with a pair of complex conjugate singularities. An application of the method to a plane elasticity problem is also made. Finally, the modification of the method valid for a pair of complex conjugate singularities is illustrated in detail in a numerical example.

Classical instability of a naked singularity

Abstract: The effective potential of the equatorial circular geodesics in Kerr metric show a sharp discontinuity from positive to negative values, when a→m. It is shown that when a is in the small range 1.088 m > a > m, stable circular orbits exist with negative energy (with respect to infinity); this implies that, if accretion takes place, a Kerr naked singularity is slowed down. The efficiency of this process increases as a→m. Therefore, in the above range of a, Kerr naked singularities should decay to a black-hole state.

Transformation methods in the analysis of series for critical properties

Abstract: During the past twenty years, considerable use has been made of conformal transformations as an aid to series analysis in the study of critical phenomena; however, there has been no evident systematic approach to the task of choosing the most suitable transformation for a given series. In this review we discuss the theory of transformations and provide such a rational and systematic method of approach. The purpose of transformation is to map the singularity of interest (usually the critical point) significantly closer to the origin than any other singularity, so that it dominates the later coefficients of the series; extrapolation methods based on Darboux's theorems may then be employed. We show that, for series likely to arise in thermodynamics, it is always possible to find transformations which achieve this purpose. We provide a set of conditions, which should be satisfied by any transformation function, to ensure straightforward and valid analysis, and we discuss the basic types of transforma...

The spectra of some integral operators for plane elastostatical boundary value problems

Abstract: In this paper 16 integral equations for the geometrical and statical boundary value problem of plane elastostatics are systematically compiled. As the equations are formulated with the aid of the singularity method it is possible to interpret mechanically all quantities occurring in them.

Unsteady hydrodynamics of spherical gravitational collapse.

Abstract: Similarity solutions are obtained which describe the time evolution of spherical, self-gravitating gas clouds with initial density distribution rhoapprox.r/sup ..cap alpha../, where ..cap alpha..>-3 but otherwise arbitrary and with pressure P=kapparho/sup ..gamma../, where ..gamma.. is constant and kappa is constant along the trajectory of any fluid element. The case ..gamma..=1, ..cap alpha..=-2 is the isothermal case studied by Shu. The qualitative features of the isothermal collapse solutions found by Shu remain in the more general case, 1 -3. For all 1 5/3.

On finite amplitude Bénard convection in a cylindrical container

Abstract: A study analogous to that of Daniels (1977) is undertaken for finite amplitude Benard convection in a shallow cylindrical container with an imperfectly insulated sidewall. A novel feature of the investigation is the singularity which develops in the amplitude function as the centre of the cylinder is approached. This singularity results in an unexpectedly large cell amplitude at the origin, and a slight increase in the value of the Rayleigh number at which the convection cells spread throughout the fluid from that predicted by linear theory.

Remark on “Algorithm 535: The QZ Algorithm to Solve the Generalized Eigenvalue Problem for Complex Matrices [F2]”

Abstract: Three FORTRAN subroutines are provided that implement a complex form of the QZ algorithm for finding lambda and z such that Az = lambda Bz, where A and B are complex N by N matrices. The complex QZ algorithm is unaffected by singularity or near singularity of B. Subroutie CQZHES implements the first step of the algorithm wherein A and B are simultaneously reduced by unitary transformations to upper Hessenberg and upper triangular form, respectively. Subroutine CQZVAL implements an iterative process that reduces A to upper triangular form while maintaining the trianglar form of B. The eigenvalues are derivable from the corresponding diagonal elements of the reduced A and B. Subroutine CQZVEC applies the accumulated transformations from the two earlier steps onto the eigenvectors of the triangular problem. No facility is provided for obtaining just a few eigenvectors or, for balancing A and B. A long-precision IBM version of the subroutines was tested on a 370/195. There are no machine-dependent constants in the subroutines, so the standard version should run directly on different machines. (RWR)

Action of a force near the planar surface between semi-infinite immiscible liquids at very low Reynolds numbers: Addendum

Abstract: In the paper by Aderogba and Blake [1], the first order approximation to the shape of the interface between two immiscible liquids, at which surface tension acted, was obtained in the case of a point force directed parallel to the interface. In the case of the normal force no solution was obtained due to the logarithmic singularity associated with problems of this type. However, the problem is not physically well-posed in the case of the normal force. Surface tension is not suitable because it cannot alone balance the induced stress on the interface due to a point force. We need an additional force to balance the action of the point force on the interface. The obvious solution is to include a density difference Δρ* between the two fluidswhere and are the densities of the lower and upper fluid respectively.