A Boolean Delay Equation Model of Colliding Cascades. Part I: Multiple Seismic Regimes

We consider a prominent feature of hierarchical nonlinear (complex) systems: persistent recurrence of abrupt overall changes, called here critical transitions. Motivated by the earthquake prediction problem, we formulate a model that uses heuristic constraints taken from the dynamics of seismicity. Our conclusions, though, may apply to hierarchical systems that arise in other areas.We use the Boolean delay equation (BDE) framework to model the dynamics of colliding cascades, in which a direct cascade of loading interacts with an inverse cascade of failures. The elementary interactions of elements in the system are replaced by their integral effect, represented by the delayed switching of an element's state.The present paper is the first of two on the BDE approach to modeling seismicity. Its major results are the following: (i) A model that implements the approach. (ii) Simulating three basic types of seismic regime. (iii) A study of regime switching in a parameter space of the loading and healing rates. The second paper focuses on the earthquake prediction problem.     

The Volume Element of Space-Time and Scale Invariance

Scale invariance is considered in the context of gravitational theories where the action, in the first order formalism, is of the form S= L ₁ d ⁴ x+ L ₂ d ⁴ x where the volume element d ⁴ x is independent of the metric. For global scale invariance, a dilaton has to be introduced, with non-trivial potentials V()=f ₁ e in L ₁ and U()=f ₂ e ² in L ₂ . This leads to non-trivial mass generation and a potential for which is interesting for inflation. Interpolating models for natural transition from inflation to a slowly accelerated universe at late times appear naturally. This is also achieved for Quintessential models, which are scale invariant but formulated with the use of volume element d ⁴ x alone. For closed strings and branes (including the supersymmetric cases), the modified measure formulation is possible and does not require the introduction of a particular scale (the string or brane tension) from the begining but rather these appear as integration constants.     

Leptonic decays of vector mesons and a possible μ+μ− resonance

The possible existence of a vector leptonic resonance (⁰) in the ^+– system is considered. We discuss the effect such a resonance would have on the g factor of the muon and also on the ratio of the partial widths of the muonic and electronic decay modes (R_V = (V ^+–)/G (V e⁺ e^–)) of the neutral vector mesons ⁰,,,, and. From the experimental values of R and R, the following values are obtained for the mass, coupling constant, and partial decay widths of the resonance: M = 872 ± 60 MeV, f²/4 = 4 ± 2) ·10^–4, ( ^{0 + –}) = 0.12±0.06 MeV, and (^{0 0 +–}) = 0.13±0.06 MeV.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 39–43, May, 1977.     

Some twisted self-dual solutions for the Yang-Mills equations on a hypertorus

TheSU(N) Yang-Mills equations are considered in a four-dimensional Euclidean box with periodic boundary conditions (hypertorus). Gauge-invariant twists can be introduced in these boundary conditions, to be labeled with integersn (= –n ), defined moduloN. The Pontryagin number in this space is often fractional. Whenever this number is zero there are solutions to the equationsG =0 HereG is the covariant curl. When this number is not zero we find a set of solutions to the equations , provided that the periodsa of the box satisfy certain relations.Work supported in part by the US Department of Energy under Contract No. DE-AC-03-76ER 00068 and by the Fairchild FoundationOn leave from the Institute for Theoretical Physics, University of Utrecht, P.O. Box 80.006, NL-3508 TA Utrecht, The Netherlands     

Bimetric general relativity and cosmology

A modification of the general relativity theory is proposed (bimetric general relativity) in which, in addition to the usual metric tensorg _v describing the space-time geometry and gravitation, there exists also a background metric tensor _v The latter describes the space-time of the universe if no matter were present and is taken to correspond to a space-time of constant curvature with positive spatial curvature (k=1). Field equations are obtained, and these agree with the Einstein equations for systems that are small compared to the size of the universe, such as the solar system. Energy considerations lead to a generalized form of the De Donder condition. One can set up simple isotropic closed models of the universe which first contract and then expand without going through a singular state. It is suggested that the maximum density of the universe was of the order ofc ^{5 –1} G ^–210⁹³ g/cm³. The expansion from such a high-density state is similar to that from the singular state (big bang) of the general relativity models. In the case of the dust-filled model one can fit the parameters to present cosmological data. Using the radiation-filled model to describe the early history of the universe, one can account for the cosmic abundance of helium and other light elements in the same way as in ordinary general relativity.     

The cohomology of the space of magnetic monopoles

Denote byX _q the reduced space ofSU ₂ monopoles of chargeq in ³. In this paper the cohomology ofX _q , the cohomology with compact supports ofX _q , and the image of the latter in the former are all calculated as representations of /q which acts onX ₂. This provides a non-trivial lower bound for theL ² cohomology ofX _q which is compatible with some conjectures of Sen. It is also shown that, granted some assumptions about the metric onX _q , itsL ² cohomology does not exceed this bound in the situation referred to in the paper as the coprime case.The work described here was carried out partly at the University of Texas at Austin.     

Absence of the partially ordered phase in theq-state models

We investigate theq-state models called (N ,N ) model using an infinitesimal Migdal-Kadanoff renormalization-group method. We distinguish two cases namely the isotropic model and the anisotropic model. The first one presents a critical value ofq,q _c such that forq _c we obtain an Ashkin-Teller phase diagrams while forq>q _c the partially ordered phase disappears then the model exhibits only phase transition between ferromagnetic phase and disordered one. The phase diagrams in the second case are qualitatively similar to one obtained forZ(6) model for all values ofq.     

Periodic nonlinear waves on a half-line

Nontrivial solutions of the equationu _tt=u _xx–g(u) which are 2-periodic int and which decay asx are shown to exist ifg(a)=0 andg(0)>1. Breather-like solutions, which also decay asx –, can be interpreted as homoclinic solutions in thex-dynamics; their existence is still in question for generalg.     

A numerical study of the hierarchical Ising model: High-temperature versus epsilon expansion

We study numerically the magnetic susceptibility of the hierarchical model with Ising spins (=±1) above the critical temperature and for two values of the epsilon parameter. The integrations are performed exactly using recursive methods which exploit the symmetries of the model. Lattices with up to 2¹⁸ sites have been used. Surprisingly, the numerical data can be fitted very well with a simple power law of the form (1-/ ₀)^g for thewhole temperature range considered. This approximate law implies a simple approximate formula for the coefficients of the high-temperature expansion, and, more importantly, approximate relations among the coefficients themselves. We found that some of these approximate relations hold with errors less then 2%. On the other hand,g differs significantly from the critical exponent calculated with the epsilon expansion, even when the fit is restricted to intervals closer to ^c. We discuss this discrepancy in the context the renormalization group analysis of the hierarchical model.     

Bound on the mass gap for finite volume stochastic ising models at low temperature

We consider a sequence of finite volume Z ^d ,d2, reversible stochastic Ising models in the low temperature regime and having invariant measures satisfying free boundary conditions. We show that associated with the models are random hitting times whose expectations, regarded as a function of , grow exponentially in ||( ^d-1)/d ; moreover, the mass gaps for the models shrink exponentially fast in ||( ^d-1)/d . A geometrical lemma is employed in the analysis which states that if a Peierls' contour is sufficiently small relative to the faces of , then the fraction of the contour tangent to the faces is less than a constant smaller than one.     

On the existence of thermodynamics for the generalized random energy model

Derrida's generalized random energy model is considered. Almost sure andL ^p convergence of the free energy at any inverse temperature are proven for an arbitrary numbern of hierarchical levels. The explicit form of the free energy is given in the most general case and the limitn is discussed.     

The use of the isotropic orientation factor in fluorescence resonance energy transfer (FRET) studies of the actin filament

A Dale-Eisinger style analysis (R. E. Daleet al., Biophys. J. 26, 161, 1979) is used to produce three-dimensional plots that display the limits on the average orientation factor k ² that is required to calculate molecular distances in F-actin from fluorescence resonance energy transfer measurements. Maxima and minima plots are generated for the transfer of energy from a donor to a single acceptor and for transfer to multiple acceptors that are related by F-actin helical symmetry. The analysis is performed in terms of dipole cone half-angles rather than depolarization factors, in order to facilitate the modeling of the multiple acceptor problem. Calculations are carried out under the restrictive condition of a single electric dipole moment per fluorophore. In addition, both surface and volume averaging of the donor and acceptor dipoles are considered. Comparisons between the plots show that for the multiple acceptor cases with F-actin symmetry, there is a great reduction in the range for maxima and minima limits on k ². The calculations also suggest guidelines for the choice of fluorescence label that will result in an average orientation factor occurring within acceptable limits, i.e., inside the limits for which k ²=2/3 may be employed. Thus, without having detailed knowledge of the mean donor or acceptor dipole relative orientations, the use of k ²=2/3 in radial coordinate studies of F-actin is more than reasonable and is fairly assured of being correct.     

Weyl's geometry and physics

It is proposed to remove the difficulty of nonitegrability of length in the Weyl geometry by modifying the law of parallel displacement and using standard vectors. The field equations are derived from a variational principle slightly different from that of Dirac and involving a parameter . For =0 one has the electromagnetic field. For <0 there is a vector meson field. This could be the electromagnetic field with finite-mass photons, or it could be a meson field providing the missing mass of the universe. In cosmological models the two natural gauges are the Einstein gauge and the cosmic gauge. With the latter the universe has a fixed size, but the sizes of small systems decrease with time and their masses and energies increase, thus producing the Hubble effect. The field of a particle in this gauge is investigated, and it leads to an interesting solution of the Einstein equations that raises a question about the Schwarzschild solution.     

On the Large-Distance Behavior of Correlations for a Hierarchical N-Component Classical Vector Model in Three Dimensions

We consider the correlation functions for a hierarchical N-component classical vector model in three dimensions. For N = , we find explicitly the eigenvalues and global eigenfunctions of the linearized renormalization group transformation. In a very direct way, this yields the correlation functions for the N = model. In particular, we check that the two-point function has canonical decay.     

Disordered ground states for classical discrete-state problems in one dimension

It is known that one-dimensional lattice problems with a discrete, finite set of states per site generically have periodic ground states (GSs). We consider slightly less generic cases, in which the Hamiltonian is constrained by either spin (S) or spatial (I) inversion symmetry (or both). We show that such constraints give rise to the possibility ofdisordered GSs over a finite fraction of the coupling-parameter space—that is, without invoking any nongeneric fine tuning of coupling constants, beyond that arising from symmetry. We find that such disordered GSs can arise for many values of the number of statesk at each site and the ranger of the interaction. The Ising (k=2) case is the least prone to disorder:I symmetry allows for disordered GSs (without fine tuning) only forr5, whileS symmetry never gives rise to disordered GSs.     

Two repulsive lines on disordered lattices

We investigate the ground-state properties of two lines with on-site repulsion on disordered Cayley tree and (Berker) hierarchical lattices, in connection with the question of multiple pure states for the corresponding one-line problem. Exact recursion relations for the distribution of ground-state energies and of the overlaps are derived. Based on a numerical study of the recursion relations, we establish that the total interaction energy on average is asymptotically proportional to the width of the ground-state energy fluctuation of a single line for both weak and strong (i.e., hard-core) repulsion. When the lengtht of the lines is finite, there is a finite probability of ordert ^–a for (nearly) degenerate, nonoverlapping one-line ground-state configurations, in which case the interaction energy vanishes. We show thata= (t ) on hierarchical lattices. Monte Carlo transfer matrix calculation on a (1+1)-dimensional model yields the same scaling for the interaction energy but ana different from =1/3. Finitelength scalings of the distribution of the interaction energy and of the overlap are also discussed.     

The small momentum transfer k L o C→K S 0 C regeneration at high energies

Results of the first elastic K _S ^o regeneration experiment on carbon, using magnetic spark chamber spectrometer, are presented in the beam momentum interval 10p50 GeV/c. The d ifferentia cross section d/dt is reconstructed in the range 0·0025–t0·02 (GeV/c)² and its slopeB is found to be momentum independent with an average valueB=(65±11) (GeV/c)^–2. The results are in agreement with the calculations using the coherent production model.     

Nonequilibrium Brownian dynamics simulations of shear thinning in concentrated colloidal suspensions

The effect of interparticle forces on shear thinning in concentrated aqueous and nonaqueous colloidal suspensions was studied using nonequilibrium Brownian dynamics. Hydrodynamic interactions among particles were neglected. Systems of 108 particles were studied at volume fractions of 0.2 and 0.4. For the nonaqueous systems, shear thinning could be correlated with the gradual breakup of small flocs present because of the weak, attractive secondary minimum in the interparticle potential. At the highest shear rate for=0.4, the particles were organized into a hexagonally packed array of strings. For the strongly repulsive aqueous systems, the viscosity appeared to be a discontinuous function of the shear rate. For=0.4, this discontinuity coincided with a transition from a disordered state to a lamellar structure for the suspension.     

The Weyl-Cartan Space Problem in Purely Affine Theory

According to Poincaré, only the epistemological sum of geometry and physics is measurable. Of course, there are requirements of measurement to be imposed on geometry because otherwise the theory resting on this geometry cannot be physically interpreted. In particular, the Weyl-Cartan space problem must be solved, i.e., it must be guaranteed that the comparison of distances is compatible with the Levi-Civita transport. In the present paper, we discuss these requirements of measurement and show that in the (purely affine) Einstein-Schrödinger unified field theory the solution of the Weyl-Cartan space problem simultaneously determines the matter via Einstein's equations. Here the affine field ⁱ _kl represents Poincaré's sum, and the solution of the space problem means its splitting in a metrical space and in matter fields, where the latter are given by the torsion tensor ⁱ _[kl].     

Algebraic Decay in Hierarchical Graphs

We study the algebraic decay of the survival probability in open hierarchical graphs. We present a model of a persistent random walk on a hierarchical graph and study the spectral properties of the Frobenius–Perron operator. Using a perturbative scheme, we derive the exponent of the classical algebraic decay in terms of two parameters of the model. One parameter defines the geometrical relation between the length scales on the graph, and the other relates to the probabilities for the random walker to go from one level of the hierarchy to another. The scattering resonances of the corresponding hierarchical quantum graphs are also studied. The width distribution shows the scaling behavior P()1/.