Statistical properties of an iterated arithmetic mapping

We study the (3x+1)/2 problem from a probabilistic viewpoint and show a forgetting mechanism for the lastk binary digits of the seed afterk iterations. The problem is subsequently generalized to a trifurcation process, the (lx+m)/3 problem. Finally the sequence of a set of seeds is empirically shown to be equivalent to a random walk of the variable log₂ x (or log₃ x) though computer simulations.     

Statistical physics analysis of the computational complexity of solving random satisfiability problems using backtrack algorithms

The computational complexity of solving random 3-Satisfiability (3-SAT) problems is investigated using statistical physics concepts and techniques related to phase transitions, growth processes and (real-space) renormalization flows. 3-SAT is a representative example of hard computational tasks; it consists in knowing whether a set of αN randomly drawn logical constraints involving N Boolean variables can be satisfied altogether or not. Widely used solving procedures, as the Davis-Putnam-Loveland-Logemann (DPLL) algorithm, perform a systematic search for a solution, through a sequence of trials and errors represented by a search tree. The size of the search tree accounts for the computational complexity, i.e. the amount of computational efforts, required to achieve resolution. In the present study, we identify, using theory and numerical experiments, easy (size of the search tree scaling polynomially with N) and hard (exponential scaling) regimes as a function of the ratio α of constraints per variable. The typical complexity is explicitly calculated in the different regimes, in very good agreement with numerical simulations. Our theoretical approach is based on the analysis of the growth of the branches in the search tree under the operation of DPLL. On each branch, the initial 3-SAT problem is dynamically turned into a more generic 2+p-SAT problem, where p and 1 - p are the fractions of constraints involving three and two variables respectively. The growth of each branch is monitored by the dynamical evolution of α and p and is represented by a trajectory in the static phase diagram of the random 2+p-SAT problem. Depending on whether or not the trajectories cross the boundary between satisfiable and unsatisfiable phases, single branches or full trees are generated by DPLL, resulting in easy or hard resolutions. Our picture for the origin of complexity can be applied to other computational problems solved by branch and bound algorithms. Received 10 March 2001     

Super-extensions of energy dependent Schrödinger operators

We consider the energy dependent super Schrödinger linear problem which is a direct generalization of the purely even, energy dependent Schrödinger equation discussed in [1]. We show that the isospectral flows of that problem possess (N+1) compatible Hamiltonian structures. We also extend a generalised factorisation approach of [2] to this case and derive a sequence ofN modifications for the 2N component systems. Then ^th such modification possesses (N–n+1) compatible Hamiltonian structures.On leave of absence from Institute of Theoretical Physics, Warsaw University, Hoza 69, PL-00-681 Warsaw, Poland (present address)     

Three-dimensional parton distribution functions g1T and $h_{1L}^{\perp}$ in the polarized proton–antiproton Drell–Yan process

We present predictions of the unweighted and weighted double spin asymmetries related to the transversal helicity distribution g _1T and the longitudinal transversity distribution h_1L^{^}h_{1L}^{\perp}, two of eight leading-twist transverse momentum dependent parton distributions (TMDs) or three-dimensional parton distribution functions (3dPDFs), in the polarized proton–antiproton Drell–Yan process at typical kinematics on the Facility for Antiproton and Ion Research (FAIR). We conclude that FAIR is ideal to access the new 3dPDFs toward a detailed picture of the nucleon structure.     

Riemann-Hilbert method and N-soliton for two-component Gerdjikov-Ivanov equation

We consider the Riemann–Hilbert method for initial problem of the vector Gerdjikov–Ivanov equation, and obtain the formula for its N-soliton solution, which is expressed as a ratio of (N + 1) × (N + 1) determinant and N × N determinant. Furthermore, by applying asymptotic analysis, the simple elastic interactions of N-soliton are confirmed, and the shifts of phase and position are also explicitly displayed.     

A Particle-Picture Approach to Anomalies in Chiral Gauge Theories

The system of a chiral fermion field coupled to a background gauge field is considered. By taking what we call the particle picture and carefully defining the S-matrix in the Heisenberg picture, we investigate anomalous phenomena in this system. It is shown by explicit calculations that the gauge-field configuration with nonvanishing topological-charge causes anomalous production of particles that is directly responsible for the chiral U(1) anomaly. Unlike the chiral U(1) anomaly, the gauge anomaly, that is, gauge non-invariance of the S-matrix is a problem that arises in the phase of the S-matrix. It is shown that this phase is related to the freedom existing in the quantization method, and that a suitably chosen phase which of course is consistent with the equation of motion can remove the gauge anomaly. Finally, a modified form of path-integral quantization for this system is proposed.     

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.     

A geometric generalization of classical mechanics and quantization

A geometrization of classical mechanics is presented which may be considered as a realization of the Hertz picture of mechanics. The trajectories in thef-dimensional configuration spaceV _f of a classical mechanical system are obtained as the projections onV _f of the geodesics in an (f + 1) dimensional Riemannian spaceV _{f + 1}, with an appropriate metric, if the additional (f + 1)th coordinate, taken to be an angle, is assumed to be “cyclic”. When the additional (angular) coordinate is not cyclic we obtain what may be regarded as a generalization of classical mechanics in a geometrized form. This defines new motions in the neighbourhood of the classical motions. It has been shown that, when the angular coordinate is “quasi-cyclic”, these new motions can be used to describe events in the quantum domain with appropriate periodicity conditions on the geodesics inV _{f + 1}.     

Interface Energy in the Edwards-Anderson Model

We numerically investigate the spin glass energy interface problem in three dimensions. We analyze the energy cost of changing the overlap from −1 to +1 at one boundary of two coupled systems (in the other boundary the overlap is kept fixed to +1). We implement a parallel tempering algorithm that simulates finite temperature systems and works with both cubic lattices and parallelepiped with fixed aspect ratio. We find results consistent with a lower critical dimension D _c =2.5. The results show a good agreement with the mean field theory predictions.     

Left-Symmetric Algebras in Hydrodynamics

Left-Symmetric algebras are shown to appear naturally in integrable hydrodynamical systems. First, to a data a Left-Symmetric algebra and an operator of strong deformation on it is attached an infinite commuting hierarchy of integrable systems of hydrodynamical type in 1+1−d. Second, this picture (without deformation) is embedded into an infinite-component integrable hydrodynamic chain.     

Large-scale low-energy excitations in 3-d spin glasses

We numerically extract large-scale excitations above the ground state in the 3-dimensional Edwards-Anderson spin glass with Gaussian couplings. We find that associated energies are O(1), in agreement with the mean field picture. Of further interest are the position-space properties of these excitations. First, our study of their topological properties show that the majority of the large-scale excitations are sponge-like. Second, when probing their geometrical properties, we find that the excitations coarsen when the system size is increased. We conclude that either finite size effects are very large even when the spin overlap q is close to zero, or the mean field picture of homogeneous excitations has to be modified. Received 14 August 2000     

Possible Wormhole Solutions in (4 + 1) Gravity

We extend previous analyses of soliton solutionsin (4 + 1) gravity to new ranges of their definingparameters. The geometry, as studied using invariants,has the topology of wormholes found in (3 + 1) gravity. In the induced-matter picture, thefluid does not satisfy the strong energy conditions, butits gravitational mass is positive. We infer thepossible existence of (4 + 1) which, compared to their (3 + 1) counterparts, are lessexotic.     

On Subalgebras of the Conformal Algebra AC(2,2)

Abstract

Subalgebras of the Lie algebra AC(2, 2) of the group C(2, 2), which is the group of conformal transformations of the pseudo-Euclidean space R _2,2, are studied. All subalgebras of the algebra AC(2, 2) are splitted into three classes, each of those is characterized by the isotropic rank 0, 1, or 3. We present the complete classification of the class 0 subalgebras and also of the class 3 subalgebras which satisfy an additional condition. The results obtained are applied to the reduction problem for the d’Alembert equation □u + λu ³ = 0 in the space R _2,2.     

On global solutions of the Maxwell-Dirac equations

We prove, for the Maxwell-Dirac equations in 1+3 dimensions, that modified wave operators exist on a domain of small entire test functions of exponential type and that the Cauchy problem, inR ⁺×R ³, has a unique solution for each initial condition (att=0) which is in the image of the wave operator. The modification of the wave operator, which eliminates infrared divergences, is given by approximate solutions of the Hamilton-Jacobi equation, for a relativistic electron in an electromagnetic potential. The modified wave operator linearizes the Maxwell-Dirac equations to their linear part.Dedicated to Walter Thirring on his 60^th birthdayThis work is dedicated to Walter Thirring upon the occasion of his sixtieth birthday with appreciation and friendship     

Reconstruction of Random Colourings

Reconstruction problems have been studied in a number of contexts including biology, information theory and statistical physics. We consider the reconstruction problem for random k-colourings on the Δ-ary tree for large k. Bhatnagar et al. [2] showed non-reconstruction when . We tighten this result and show non-reconstruction when , which is very close to the best known bound establishing reconstruction which is . Supported by NSF grants DMS-0528488 and DMS-0548249.     

Long range scattering for non-linear Schrödinger and Hartree equations in space dimensionn≥2

We consider the scattering problem for the non-linear Schrödinger (NLS) equation with a power interaction with critical powerp=1+2/n in space dimensionsn=2 and 3 and for the Hartree equation with potential |x|^–1 in space dimensionn2. We prove the existence of modified wave operators in theL ² sense on a dense set of small and sufficiently regular asymptotic states.Laboratoire associé au Centre National de la Recherche Scientifique     

Rate processes on fractals: Theory,simulations, and experiments

Heterogeneous kinetics are shown to differ drastically from homogeneous kinetics. For the elementary reaction A + A products we show that the diffusion-limited reaction rate is proportional tot ^– h[A]² or to [A]^x, whereh=1- d _s/2, X=1+2/d _s =(h-2)(h-1), andd _s is the effective spectral dimension. We note that ford = d _s =1, h =1/2 andX = 3, for percolating clustersd _s = 4/3,h = 1/3 andX = 5/2, while for dust ds <1, 1 >h > 1/2 and >X > 3. Scaling arguments, supercomputer simulations and experiments give a consistent picture. The interplay of energetic and geometric heterogeneity results in fractal-like kinetics and is relevant to excitation fusion experiments in porous membranes, films, and polymeric glasses. However, in isotopic mixed crystals, the geometric fractal nature (percolation clusters) dominates.     

Ricci-Flat Metrics,Harmonic Forms and Brane Resolutions

 We discuss the geometry and topology of the complete, non-compact, Ricci-flat Stenzel metric, on the tangent bundle of S ⁿ⁺¹ . We obtain explicit results for all the metrics, and show how they can be obtained from first-order equations derivable from a superpotential. We then provide an explicit construction for the harmonic self-dual (p, q)-forms in the middle dimension p+q=(n+1) for the Stenzel metrics in 2(n+1) dimensions. Only the (p, p)-forms are L ² -normalisable, while for (p, q)-forms the degree of divergence grows with . We also construct a set of Ricci-flat metrics whose level surfaces are U(1) bundles over a product of N Einstein-K?hler manifolds, and we construct examples of harmonic forms there. As an application, we construct new examples of deformed supersymmetric non-singular M2-branes with such 8-dimensional transverse Ricci-flat spaces. We show explicitly that the fractional D3-branes on the 6-dimensional Stenzel metric found by Klebanov and Strassler is supported by a pure (2,1)-form, and thus it is supersymmetric, while the example of Pando Zayas-Tseytlin is supported by a mixture of (1,2) and (2,1) forms. We comment on the implications for the corresponding dual field theories of our resolved brane solutions. Received: 22 February 2001 / Accepted: 16 August 2002 Published online: 7 November 2002     

Chiral BRST Cohomology of N= 2 Strings at Arbitrary Ghost and Picture Number

We compute the BRST cohomology of the holomorphic part of the N= 1 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies both display a picture dependence – a phenomenon discovered recently also in the relative Ramond sector of N= 1 strings by Berkovits and Zwiebach [1]. Received: 5 January 1998 / Accepted: 16 November 1998     

Universal exchange algebra for Bloch waves and Liouville theory

We derive a universal formula for the exchange algebra in the Bloch wave basis. The main tool we use is a lattice version of the Coulomb gas picture of conformal field theory, making its quantum group structure explicit from the very beginning. Calulations are then reduced to a factorization problem inU _q (sl ₂).