Finite Singularities Research Articles

Crack Propagation as a Free Boundary Problem

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a sharp interface method and a phase field approach to investigate the model. The simulations confirm analytical predictions for fast crack propagation.

Numerical simulations of gravitational collapse in Einstein-aether theory

We study gravitational collapse of a spherically symmetric scalar field in Einstein-aether theory (general relativity coupled to a dynamical unit timelike vector field). The initial value formulation is developed, and numerical simulations are performed. The collapse produces regular, stationary black holes, as long as the aether coupling constants are not too large. For larger couplings a finite area singularity occurs. These results are shown to be consistent with the stationary solutions found previously.

Non-equilibrium dynamics of spin facilitated glass models

We consider the dynamics of spin facilitated models of glasses in the non-equilibriumageing regime following a sudden quench from high to low temperatures. We briefly reviewknown results obtained for the broad class of kinetically constrained models, and thenpresent new results for the behaviour of the one-spin facilitated Fredrickson–Andersen andEast models in various spatial dimensions. The time evolution of one-time quantities, suchas the energy density, and the detailed properties of two-time correlation and responsefunctions are studied using a combination of theoretical approaches, including exactmappings of master operators and reductions to integrable quantum spin chains,field theory and renormalization group, and independent interval and timescaleseparation methods. The resulting analytical predictions are confirmed by means ofdetailed numerical simulations. The models we consider are characterized by trivialstatic properties, with no finite temperature singularities, but they neverthelessdisplay a surprising variety of dynamic behaviour during ageing, which can bedirectly related to the existence and growth in time of dynamic lengthscales.Well-behaved fluctuation–dissipation ratios can be defined for these models, andwe study their properties in detail. We confirm in particular the existence ofnegative fluctuation–dissipation ratios for a large number of observables. Ourresults suggest that well-defined violations of fluctuation–dissipation relations, of apurely dynamic origin and unrelated to the thermodynamic concept of effectivetemperatures, could in general be present in non-equilibrium glassy materials.

Nonlinear Quantum Shock Waves in Fractional Quantum Hall Edge States

Using the Calogero model as an example, we show that the transport in interacting nondissipative electronic systems is essentially nonlinear and unstable. Nonlinear effects are due to the curvature of the electronic spectrum near the Fermi energy. As is typical for nonlinear systems, a propagating semiclassical wave packet develops a shock wave at a finite time. A wave packet collapses into oscillatory features which further evolve into regularly structured localized pulses carrying a fractionally quantized charge. The Calogero model can be used to describe fractional quantum Hall edge states. We discuss perspectives of observation of quantum shock waves and a direct measurement of the fractional charge in fractional quantum Hall edge states.

Computing Hypergeometric Solutions of Linear Recurrence Equations

We describe a complete algorithm to compute the hypergeometric solutions of linear recurrence relations with rational function coefficients. We use the notion of finite singularities and avoid computations in splitting fields. An implementation is available in Maple 9.

Bi-orthogonal Polynomials on the Unit Circle, Regular Semi-Classical Weights and Integrable Systems

The theory of bi-orthogonal polynomials on the unit circle is developed for a general class of weights leading to systems of recurrence relations and derivatives of the polynomials and their associated functions, and to functional-difference equations of certain coefficient functions appearing in the theory. A natural formulation of the Riemann-Hilbert problem is presented which has as its solution the above system of bi-orthogonal polynomials and associated functions. In particular, for the case of regular semi-classical weights on the unit circle $ w(z) = \prod^m_{j=1}(z-z_j(t))^{\rho_j}, $ consisting of $ m \in {\bf Z}_{> 0} $ finite singularities, difference equations with respect to the bi-orthogonal polynomial degree n (Laguerre-Freud equations or discrete analogs of the Schlesinger equations) and differential equations with respect to the deformation variables $ z_j(t) $ (Schlesinger equations) are derived completely characterising the system.

Semiclassical and quantum Liouville theory

We develop a functional integral approach to quantum Liouville field theory completely independent of the hamiltonian approach. To this end on the sphere topology we solve the Riemann-Hilbert problem for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincaré accessory parameters. This provides the semiclassical four point vertex function with three finite charges and a fourth infinitesimal. Some of the results are extended to the case of n finite charges and m infinitesimal. With the same technique we compute the exact Green function on the sphere on the background of three finite singularities. Turning to the full quantum problem we address the calculation of the quantum determinant on the background of three finite charges and of the further perturbative corrections. The zeta function regularization provides a theory which is not invariant under local conformal transformations. Instead by employing a regularization suggested in the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the correct quantum conformal dimensions from the one loop calculation and we show explicitly that the two loop corrections do not change such dimensions. We then apply the method to the case of the pseudosphere with one finite singularity and compute the exact value for the quantum determinant. Such results are compared to those of the conformal bootstrap approach finding complete agreement.

Dimensions of Julia sets of meromorphic functions with finitely many poles

Let f be a transcendental meromorphic function with finitely many poles such that the finite singularities of f-1 lie in a bounded set. We show that the Julia set of f has Hausdorff dimension strictly greater than one and packing dimension equal to two. The proof for Hausdorff dimension simplifies the earlier argument given for transcendental entire functions.

Phase Field Modeling of Fast Crack Propagation

We present a continuum theory which predicts the steady state propagation of cracks. The theory overcomes the usual problem of a finite time cusp singularity of the Grinfeld instability by the inclusion of elastodynamic effects which restore selection of the steady state tip radius and velocity. We developed a phase-field model for elastically induced phase transitions; in the limit of small or vanishing elastic coefficients in the new phase, fracture can be studied. The simulations confirm analytical predictions for fast crack propagation.

Inhomogenized sudden future singularities

We find that sudden future singularities may also appear in spatially inhomogeneous Stephani models of the universe. They are temporal pressure singularities and may appear independently of the spatial finite density singularities already known to exist in these models. It is shown that the main advantage of the homogeneous sudden future singularities which is the fulfillment of the strong and weak energy conditions may not be the case for inhomogeneous models.

Semiclassical and quantum Liouville theory on the sphere

We solve the Riemann–Hilbert problem on the sphere topology for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincaré accessory parameters. In this way we compute the semiclassical four point vertex function with three finite charges and a fourth infinitesimal. Some of the results are extended to the case of n finite charges and m infinitesimal. With the same technique we compute the exact Green function on the sphere with three finite singularities. Turning to the full quantum problem we address the calculation of the quantum determinant on the background of three finite charges and the further perturbative corrections. The zeta function technique provides a theory which is not invariant under local conformal transformations. Instead by employing a regularization suggested in the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the correct quantum conformal dimensions from the one loop calculation and we show explicitly that the two loop corrections do not change such dimensions. We expect such a result to hold to all order perturbation theory.

Brane solutions of a spherical sigma model in six dimensions

We explore solutions of six-dimensional gravity coupled to a non-linear sigma model, in the presence of codimension-two branes. We investigate the compactifications induced by a spherical scalar manifold and analyze the conditions under which they are of finite volume and singularity free. We discuss the issue of single-valuedness of the scalar fields and provide some special embedding of the scalar manifold to the internal space which solves this problem. These brane solutions furnish some self-tuning features, however they do not provide a satisfactory explanation of the vanishing of the effective four-dimensional cosmological constant. We discuss the properties of this model in relation with the self-tuning example based on a hyperbolic sigma model.

Factorization Theorem for projective varieties with finite quotient singularities

In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show that they are related by a sequence of GIT wall-crossing flips.

Join products and indices of k-fields over differentiable fiber bundles

We use the intrinsic join product to express the index of a k-field with finite singularities over the total space of a smooth fiber bundle F↪ E→ B as the product of the indices of k-fields with finite singularities given on the fiber F and the base B. We present tables for the join product of generators of π n−1 ( V n, k ) for 1< k⩽4 where V n, k denotes the real Stiefel manifold of k-frames in R n . The indices of k-fields with finite singularities on a smooth closed manifold M n are expressed precisely in terms of generators of π n−1 ( V n, k ) for 1< k⩽4 when the indices depend only on the oriented homotopy type of M n . We obtain direct proofs of some results of Becker and Schultz [Quart. J. Math. Oxford (2) 33 (1982) 385] concerning the existence of differentiable fiber bundles as applications.

A remark on the finite time singularity of the heat flow for harmonic maps

In this paper we study the finite time singularities for the solution of the heat flow for harmonic maps. We derive a gradient estimate for the solution across a finite time singularity. In particular, we find that the solution is asymptotically radial around the isolated singular point in space at a finite singular time. It would be more desirable to understand whether the solution is continuous in space at a finite singular time.

Lattices Generated by Orbits of Subspaces Under Finite Singular Classical Groups and Its Characteristic Polynomials

Let be the (2ν + l)-dimensional vector space over the finite field 𝔽 q , and Sp 2ν+l,ν(𝔽 q ) the singular symplectic groups of degree 2ν + l over 𝔽 q . Let ℳ be any orbit of subspaces under Sp 2ν+l,ν(𝔽 q ). Denote by ℒ the set of subspaces which are intersections of subspaces in ℳ and the intersection of the empty set of subspaces of is assumed to be . By ordering ℒ by ordinary or reverse inclusion, two lattices are obtained. This paper studies the inclusion relations between different lattices, a characterization of subspaces contained in a given lattice ℒ, when the lattices form geometric lattice, and the characteristic polynomial of ℒ.

Explicit construction of extremal Hermitian metrics with finite conical singularities on $S^2$

Explicit construction of extremal Hermitian metrics with finite conical singularities on $S^2$

Disjointness preserving and local operators on algebras of differentiable functions

This article is to discuss the automatic continuity properties and the representation of disjointness preserving linear mappings on certain normal Fréchet algebras of complex-valued functions. This class of operators is defined by the condition that any pair of functions with disjoint cozero sets is mapped to functions with disjoint cozero sets, and subsumes the class of local operators. It turns out that such operators are always continuous outside some finite singularity set of the underlying topological space. Our main emphasis is on disjointness preserving operators from Fréchet algebras of differentiable functions. Such operators are shown to admit a canonical representation that involves weighted composition for the derivatives. This result extends the classical characterization of local operators as linear partial differential operators.

HEAT FLOW FOR YANG-MILLS-HIGGS FIELDS, PART I

The Yang-Mills-Higgs field generalizes the Yang-Mills field. The authors establish the local existence and uniqueness of the weak solution to the heat flow for the Yang-Mills-Higgs field in a vector bundle over a compact Riemannian 4-manifold, and show that the weak solution is gauge-equivalent to a smooth solution and there are at most finite singularities at the maximum existing time.

The Hausdorff dimension of Julia sets of hyperbolic meromorphic functions

Let f be a hyperbolic transcendental meromorphic function such that the finite singularities of f−1 are in a bounded set. We show that there exists 0&lt;s(f)[les ]2 such thatformula herefor each point a in the Julia set of f, whereformula hereWe then show that s(f)[les ]dimJ(f), the Hausdorff dimension of the Julia set, and give examples of such functions for which dimJ(f)&gt;s(f). This contrasts with the situation for a hyperbolic rational function f where it is known that dimJ(f) = s(f).