The space-time cut locus

Let (M, g) be a space-time with Lorentzian distance functiond. If (M, g) is distinguishing andd is continuous, then (M, g) is shown to be causally continuous. Furthermore, a strongly causal space-time (M, g) is globally hyperbolic iff the Lorentzian distance is always finite valued for all metricsg conformal tog. Lorentzian distance may be used to define cut points for space-times and the analogs of a number of results holding for Riemannian cut loci may be established for space-time cut loci. For instance in a globally hyperbolic space-time, any timelike (or respectively, null) cut pointq of p along the geodesicc must be either the first conjugate point ofp or else there must be at least two maximal timelike (respectively, null) geodesics fromp toq. Ifq is a closest cut point ofp in a globally hyperbolic space-time, then eitherq is conjugate top or elseq is a null cut point. In globally hyperbolic space-times, no point has a farthest nonspacelike cut point.     

Invariants and chaotic maps

A one-dimensional mapf(x) is called an invariant of a two-dimensional mapg(x, y) ifg(x, f(x))=f(f(x)). The logistic map is an invariant of a class of two-dimensional maps. We construct a class of two-dimensional maps which admit the logistic maps as their invariant. Moreover, we calculate their Lyapunov exponents. We show that the two-dimensional map can show hyperchaotic behavior.     

Vacuum type N space-times admitting homothetic vector fields with isolated fixed points

We consider vacuum space-times (M, g) which are of Petrov type N on an open dense subset ofM, and which admit (proper) homothetic vector fields with isolated fixed points. We prove that if such is the case then, at the fixed point, (M,g) is flat and the homothetic bivector,X _[a;b] , is necessarily simple-timelike. Furthermore, we prove that if the homothetic bivector remains simple-timelike in some neighbourhood of the fixed point then, around the fixed point, the space-time in question is a pp-wave. The paper ends with a local characterization and some examples of space-tunes satisfying these conditions.     

Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity

An example of a finite dimensional factorizable ribbon Hopf -algebra is given by a quotientH=u _q (g) of the quantized universal enveloping algebraU _q (g) at a root of unityq of odd degree. The mapping class groupM _g,1 of a surface of genusg with one hole projectively acts by automorphisms in theH-moduleH ^*g , ifH ^* is endowed with the coadjointH-module structure. There exists a projective representation of the mapping class groupM _g,n of a surface of genusg withn holes labeled by finite dimensionalH-modulesX ₁, ...,X _n in the vector space Hom _H (X ₁ ... X _n ,H ^*g ). An invariant of closed oriented 3-manifolds is constructed. Modifications of these constructions for a class of ribbon Hopf algebras satisfying weaker conditions than factorizability (including most ofu _q (g) at roots of unityq of even degree) are described.This work was supported in part by the EPSRC research grant GR/G 42976.     

Mirror supersymmetry in the space of (super) extended Poincaré algebras p(p,q)

We classify extended Poincaré Lie superalgebras and Lie algebras of any signature (p, q), i.e. Lie superalgebras and ₂-graded Lie algebras g = g₀ + g₁, where g₀ = s₀(V) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = ^{p, q} of signature (p, q) and g₁ is a spin 1/2 s₀(V)-module extended to a s₀-module with kernel V.As a result of the classification, we obtain, if g₁ = S is the spinor module, the numbers L ⁺(n, s) (resp. L ^–(n, s)) of independent such Lie super algebras (resp. Lie algebras), which are periodic functions of the dimension n=p+q (mod 8) and the signature s=p–q (mod 8) and satisfy: L ⁺(–n, s)=L ^– (n, s).Supported by Max-Planck-Institut für Mathematik (Bonn).Supported by the Alexander von Humboldt Foundation, MSRI (Berkeley) and SFB 256 (Bonn University).     

How to Find Separation Coordinates for the Hamilton–Jacobi Equation: A Criterion of Separability for Natural Hamiltonian Systems

The method of separation of variables applied to the natural Hamilton–Jacobi equation (u/q _i )²+V(q)=E consists of finding new curvilinear coordinates x _i (q) in which the transformed equation admits a complete separated solution u(x)=u _(i)(x _i ;). For a potential V(q) given in Cartesian coordinates, the main difficulty is to decide if such a transformation x(q) exists and to determine it explicitly. Surprisingly, this nonlinear problem has a complete algorithmic solution, which we present here. It is based on recursive use of the Bertrand–Darboux equations, which are linear second order partial differential equations with undetermined coefficients. The result applies to the Helmholtz (stationary Schrödinger) equation as well.     

Unitary Correlations and the Fejér Kernel

Let M be a unitary matrix with eigenvalues t _j , and let f be a function on the unit circle. Define X _f (M)=f(t _j ). We derive exact and asymptotic formulae for the covariance of X _f and X _g with respect to the measures |(M)|²dM where dM is Haar measure and an irreducible character. The asymptotic results include an analysis of the Fejér kernel which may be of independent interest.     

The mechanism of the increase of the generalized dimension of a filtered chaotic time series

The determination of the attractor dimension from an experimental time series may be affected by the influence of filters which are incorporated into many measuring processes. While this is expected from the Kaplan-Yorke conjecture, we show that for one-dimensional maps a weak filter can induce a self-similarity which is responsible for the increase of the Hausdorff dimension. We are able to calculate the increase of the generalized dimensionD _q for the filtered time series of the logistic mapx _i +1=rx _i (1–x _i ) atr=4 analytically.     

Disentangling q-Exponentials: A General Approach

We revisit the q-deformed counterpart of the Zassenhaus formula, expressing the Jackson q-exponential of the sum of two non-q-commuting operators as an (in general) infinite product of q-exponential operators involving repeated q-commutators of increasing order, E_q(A+B) = E_q0(A)E_q1 (B) _i=2 E_qi. By systematically transforming the q-exponentials into exponentials of series and using the conventional Baker–Campbell–Hausdorff formula, we prove that one can make any choice for the bases ^qi, i=0, 1, 2, ..., of the q-exponentials in the infinite product. An explicit calculation of the operators C _i in the successive factors, carried out up to sixth order, also shows that the simplest q-Zassenhaus formula is obtained for ₀ = ₁ =1, and ₂ = 2, and ₃ = 3. This confirms and reinforces a result of Sridhar and Jagannathan, on the basis of fourth-order calculations.     

Dynamics of forest fire under rotational constraint

A rotationally constrained forest fire model is studied on square and triangular lattices of size 400×400. The critical probabilityp _c for onset of fire propagation is determined. The scaling relationsMt ^d _r, R_gt^v andMR _g ^d _f are analysed at fire propagation probabilityp=p _c whereM is the number of burnt trees,R _g the radius of gyration andd _f the fractal dimension of the cluster of burnt trees at timet. Numerical estimates ofd _t, v andd _f have been obtained.     

Supersymmetric solitons

The generalization of the two-dimensional soliton to four space-time dimensions is a magnetic monopole. If the Higgs field lies in the adjoint representation electric and magnetic chargesq andg can be defined. If further the Higgs self-interaction vanishes all the particle states obey the universal, dual symmetric mass formulaM=a (q ²+g ²).There is evidence that this is exact in quantum mechanics if the theory is madeSO(2) supersymmetric.Invited talk at the Symposium on Mathematical Methods in the Theory of Elementary Particles, Liblice castle, Czechoslovakia, June 18–23, 1978.     

Quantum Hamiltonian for the Rigid Rotator

An excess term exists when using hermitian form of Cartesian momentum p _i (i = 1,2,3) in usual kinetic energy 1/(2) p ² _i for the rigid rotator, and the correct kinetic energy turns to be 1/(2) (1/f _i ) p _i f _i p _i where f _i are dummy factors in classical mechanics and nontrivial in quantum mechanics.     

Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line

It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrödinger equation on the half-line can be expressed through the solution of a 2×2 matrix Riemann–Hilbert problem. This problem is specified by the spectral functions {a(k),b(k)} which are defined in terms of the initial condition q(x,0)=q ₀(x), and by the spectral functions {A(k),B(k)} which are defined in terms of the specified boundary condition q(0,t)=g ₀(t) and the unknown boundary value q _x (0,t)=g ₁(t). Furthermore, it has been shown that given q ₀ and g ₀, the function g ₁ can be characterized through the solution of a certain 'global relation' coupling q ₀, g ₀, g ₁, and (t,k), where satisfies the t-part ofthe associated Lax pair evaluated at x=0. We show here that, by using a Gelfand–Levitan–Marchenko triangular representation of , the global relation can be explicitly solved for g ₁.     

Map dependence of the fractal dimension deduced from iterations of circle maps

Every orientation preserving circle mapg with inflection points, including the maps proposed to describe the transition to chaos in phase-locking systems, gives occasion for a canonical fractal dimensionD, namely that of the associated set of for whichf =+g has irrational rotation number. We discuss how this dimension depends on the orderr of the inflection points. In particular, in the smooth case we find numerically thatD(r)=D(r ^–1)=r ^–1/8.     

q-Gaussians in the porous-medium equation: stability and time evolution

The stability of q-Gaussian distributions as particular solutions of the linear diffusion equation and its generalized nonlinear form, , the porous-medium equation, is investigated through both numerical and analytical approaches. An analysis of the kurtosis of the distributions strongly suggests that an initial q-Gaussian, characterized by an index q_i, approaches asymptotically the final, analytic solution of the porous-medium equation, characterized by an index q, in such a way that the relaxation rule for the kurtosis evolves in time according to a q-exponential, with a relaxation index q_rel ≡q_rel(q). In some cases, particularly when one attempts to transform an infinite-variance distribution (q_i ≥ 5/3) into a finite-variance one (q < 5/3), the relaxation towards the asymptotic solution may occur very slowly in time. This fact might shed some light on the slow relaxation, for some long-range-interacting many-body Hamiltonian systems, from long-standing quasi-stationary states to the ultimate thermal equilibrium state.     

Reduced heat kernels on nilpotent Lie groups

LetU be a basis representation of an irreducible unitary representation of a nilpotent Lie groupG inL ₂(R ^k) and letdU denote the representation of the Lie algebrag obtained by differentiation. Ifb ₁,...,b _d is a basis ofg andB _i =dU(b _i ) we consider the operators

Statistical mechanics with homogeneous first-degree Lagrangians

The equilibrium statistical mechanics is investigated of any system whose LagrangianL ₀(v, q) is a convex homogeneous function of generalized velocitiesv, with coordinatesq in a bounded setD. A member of a canonical ensemble, the system has a conjugate HamiltonianH ₀(p, q) that vanishes identically in some subsetC×D of its phase space. The subsetC may also be specified, in some systems with a finite functionf(p, q), convex inpL ₀/, and thenL ₀ is also convex and homogeneous inv. In either case, ifC is bounded and convex, thenC or the convex functionf constitutes the fundamental constraint on the system. Under this fundamental constraint, it is shown that the so-called partition function becomes a phase-space volumeG (classical) or a numberW of microstates (quantum) from which follows the thermodynamic fundamental relation, entropySk InG (ork lnW).     

Proper homothetic maps and fixed points

Let f be a proper homothetic map of the pseudo-Riemannian manifold M and assume f has a fixed point p. If all of the eigenvalues of either f* _p or f ^-1*_p have absolute values less than unity, then M is topologically R _n and M has a flat metric. This yields three characterizations of Minkowski spacetime. In general, a homothetic map of a complete pseudo-Riemannian manifold need not have fixed points. Furthermore, an example shows the existence of a proper homothetic map with a fixed point does not imply M is flat. The scalar curvature vanishes at a fixed point, but some of the sectional curvatures may be nonzero.     

Causality criteria

By causality of matter one means its property not to admitsuperluminal excitations, i.e. excitations that propagate faster than the vacuum speed of lightc. In discussing the propagation of small excitations, one has to distinguish betweenphase velocities _j /k, (1jg=number of dispersion branches),group velocities d _j /dk, a front velocityv _f : = and the propagation speedv _q :=(dp/d)^1/2 of isotropic quasistatic (small) perturbations. We discuss some of their properties. In particular, the (maximal) speedv _s of small signals is not smaller thanv _f , and equalsv _f whenever the dispersion branches _j (k) behave reasonably at infinity of the complexk-plane. In essence stronger conditions guaranteev _q <v _f (in which casev _q c would imply superluminal behaviour).     

Instabilities of certain empty Robertson-Walker space-times

Results are established concerning perturbations of each empty Robertson-Walker space-time (M, g) with a nonvanishing cosmological constant. The perturbed space-times have the general form ( ) with an extension ofM, and lying in an open neighborhood of g in a type ofW ^m topology. These results indicate that large classes of such perturbations give rise to space-times which suffer from one of two types of incompleteness.