Nonlinear Response of the Semiconductor Laser with Pump Current Modulation

The influence of tuning the lasing monochromatic radiation frequency _g within the amplification band on the nonlinear response of the semiconductor laser with harmonic modulation of pump current is investigated theoretically. It is established that the principal features of the behavior of the nonlinear amplitude-detuning characteristic (ADC) are determined by the relation between the current modulation frequency _m and the main resonance frequency of the laser _r. If _{m r}, then with increase in _g the response decreases monotonically mainly due to the decrease of its dynamic component. The exception is provided by the spectral regions where peaks on the ADC appear because of the explicitly nonlinear lasing regimes (period doubling, chaos, etc.) When _m < _r, the resonance conditions for induced oscillations are satisfied only for definite spectral intervals within the amplification band and a dip appears on the low-frequency side of the ADC. With decreasing _m, the dip boundary shifts to a more high-frequency region of the band corresponding to smaller local resonance frequencies. The peaks on the ADC corresponding to the radiation period doubling shift to the region of smaller values of _g on increase in _m.     

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     

Absolute Continuity of Vitali–Hahn–Saks Measure Convergence Theorems

In this paper, we prove the following improved Vitali–Hahn–Saks measure convergence theorem: Let (L, 0, 1) be a Boolean algebra with the sequential completeness property, (G, ) be an Abelian topological group, be a nonnegative finitely additive measure defined on L, {_n: n N} be a sequence of finitely additive s-bounded G-valued measures defined on L, too. If for each a L, {_n(a)}_{n N} is a -convergent sequence, for each nN, when { (a)} convergent to 0, {_n(a)} is -convergent, then when { (a)} convergent to 0, {_n(a)} are -convergent uniformly with respect to nN     

The tortuosity of occupied crossings of a box in critical percolation

We consider the length of an occupied crossing of a box of size [0,n]×[0, 3n] ^D–1 (in the short direction) in standard (Bernoulli) bond percolation on ^D at criticality. Let ¦s _n¦ be the length of the shortest such crossing. It is believed that ¦s _n¦ ^1+c in some sense for somec>0. Here we show that if the correlation length(p) satisfies (p)p _c}–p) ^– for some <1, then with a probability tending to 1, ¦s _n¦>/C ₁ n ^1/(logn)^–(1–)/. The assumption (p)C ₃(p _c–p)^– with <1 has been rigorously established^(1,2) for largeD, but cannot hold⁽³⁾ forD=2. In the latter case, let ¦l _n¦ be the length of the lowest occupied crossing of the square [0,n]². We outline a proof ofP _pc(¦l_n¦ n ^1+c)n ^– for somec, >0. We also obtain a result about the length of optimal paths in first-passage percolation.     

Some Remarks on a Nongeometrical Interpretation of Gravity and the Flatness Problem

In a nongeometrical interpretation of gravity,the metric g(x) = + (x)is interpreted as an effective metric, whereas(x) is interpreted as afundamental gravitational field, propagated in spacetime which isactually flat. Some advantages and disadvantages of suchan interpretation are discussed. The main advantage isa natural resolution of the flatness problem.     

Transverse waves in a relativistic rigid body

We present relativistic elasticity as a scalar field theory. We apply it to rigid bodies, i.e., relativistic bodies with a nonlinear elastic law and a definite longitudinal wave velocity _l equal to the light velocity,c. We obtain the transverse wave equation with a definite velocity _t , and the relation between _l , _t , and the Poisson coefficient is the classical one. This is an indication that we have the relativistic extension of a classical Hooke elastic law.     

Singular perturbations of discrete spectrum

Even solutions of the Schrödinger equation with retaining potential x² are constructed for singular perturbation potentials |x|^–. It is shown that the perturbation automatically entails an induced point potential, taking account of which the perturbation matrix elements and Rayleigh-Schrödinger series may be constructed when 1 < < 3/2. In the opposite case (3/2 2), although the solutions are analytic with respect to , not even diverging series can be obtained for the energy solutions without solution of the Schrödinger equation. The analogy with quantum field theory is explored.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 58–64, March, 1988.     

True self-avoiding walk on critical percolation clusters and lattice animal

The statistics of true-self-avoiding walk model on two dimensional critical percolation clusters and lattice animals are studied using real-space renormalization group method. The correlation length exponents 's are found to be _TSAW ^pc 0.576 and _TSAW ^LA 0.623 respectively for the critical percolation clusters and lattice animals.     

Geometric localization of the threshold in two-dimensional Ising ±J spin glasses forT=0

Following an approach of Toulouse, ground states in random 2D Ising ±J spin glasses (without external magnetic field), on square lattices, and with concentrations 0p0.5 of antiferromagnetic bonds are studied by means of minimal matchings of frustrated plaquettes. Lete(p) be the ground-state energy per spin in the thermodynamic limit. Then the well-known equatione(p)=–2+(p)f(p) holds, wheref(p) is the concentration of frustrated plaquettes and(p) is the average connection length between paired frustrated plaquettes in minimal matchings. Introducing (p) as the probability that a frustrated plaquette is matched to another frustrated plaquette by a connection of length (in a minimal matching), the average length(p) can be rewritten asgl(p)=(p). The study of(p) and its components (p) leads to an intervalp ^*pp ₂ (p ^*0.121±0.008,p ₂0.161±0.008) where the threshold between ferromagnet and paramagnet forT=0 lies. Analyzing a similar so-called adjoined average lengthl(p) admits further insight.     

The influence of the transformation of waves on the threshold currents of the beam instability

In a hot anisotropic plasma with an inhomogeneity across the magnetic field, eigenmodes are a superposition of long waves of cold plasma and the short-wave Bernstein modes. As a consequence of the fact that the Bernstein modes with non-zero wave numbers are damped more intensively than the waves of cold plasma, the threshold currents of the beam instability increase correspondingly compared with the case of homogeneous plasma. For frequencies between the first and second harmonics of the electron cyclotron frequency the threshold current is from ₀/ _T to ( ₀/ _T )³ times greater, where ₀ is the beam velocity and _T is the thermal velocity of plasma electrons.It is the author's pleasant duty to express his thanks to the theorists of the Institute of Plasma Physics, especially to ing. K. Jungwirth CSc, for valuable discussions he has had with them.     

Correlated majority model

We extend the bichromatic majority model by including (one-dimensional isotropic) correlations and numerically discuss, through Monte Carlo simulations, the simple, 1/3, and 2/3 majority rules. We calculate, as functions of the concentration and correlation degree, the mean finite cluster size, and the order parameterm as well as their respective critical exponents and. We find1 regardless of the correlation degree or the type of majority. Also, a supplementary divergence of is observed at the>0 borderline.     

Real-space renormalization group for directed percolation system

We use the recently proposed real-space renormalization group method to study the critical behavior of directed percolation system in two dimensions. The correlation length exponents and are found to be 1.76 and 1.15. These results are in good agreements with the best known values.     

Singularity theorems and the [General Relativity + additional matter fields] formulation of metric theories of gravitation

By using the [General Relativity + additional matter fields] formulation (which depends on a redefined metrich ) of metric theories of gravitation, the study of singularities characterized by incomplete nonspacelike geodesics is simplified, but may be used only if (at least) the non-spacelike geodesics of the original metricg are conserved under the transformation betweeng and the new metrich . In order that every class of geodesies of a diagonal Bianchi I metric correspond to the same class of geodesies of a diagonal metrich , it is necessary that the transformation between these two metrics be a constant (positive) conformal transformation. We analyse the implications of the previous results for the singularitiesg when the latter is a solution of theories with a quadratic or polynomial Lagrangian.     

Stochastic linear maps and transition probability

Some aspects of the transition probability P(, ) between states , on unital ^*-algebras are discussed. It is shown that P increases under the action of any stochastic linear map T, i.e., P(T, T)P(, ). Some properties of P are derived in starting from a recently-proved characterization of the quantity in question.     

Fluctuations in the flux nucleation process of the current-induced resistive state in type-I superconductors

The fluctuations of the flux-tube nucleation frequency ^* in the current-induced resistive state have been studied using a strictly passive micro-fieldprobe as a flux-tube counter. The measurements were performed with constricted indium films near 2.0K. They included both the bandwidth of the rf signal from the field probe and the temporal variations of ^*. From a comparison of the power spectrum of the noise voltage and of the function ^*(t) the relative importance of the fluctuations in the size and in the nucleation frequency of the flux tubes can be evaluated. In addition to fluctuations ofv ^* around an average value, switching between two frequencies ₁ ^* and ₂ ^* can be observed. As a function of sample voltage the bandwidth shows oscillations which appear to be associated with the change in the time-averaged number of flux tubes traveling simultaneously through the constricted film. Narrow-band flux nucleation with ^*/ ^*<10^–1 is observed only in rather restricted regimes of the sample voltage.Supported by a grant from the Deutsche Forschungsgemeinschaft     

Maximum mass-particle velocities in Kantor's information mechanics

Kantor's information mechanics links phenomena previously regarded as not treatable by a single theory. It is used here to calculate the maximum velocities _m of single particles. For the electron, _m/c1–1.253814×10^–77. The maximum _m corresponds to _m/c1–1.097864×10^–122 for a single mass particle with a rest mass of 3.078496×10^–5g. This is the fastest that matter can move. Either information mechanics or classical mechanics can be used to show that _m is less for heavier particles. That _m is less for lighter particles can be deduced from an information mechanics argument alone.     

Treatment of droplike clusters in nucleation theory using Reiss's method

The expression of the correction factor in nucleation theory is derived by extending the method Reiss used recently. is the factor appearing in the number of critical nuclei (formed as a vapor condenses into liquid drops) as a correction to the conventional theory. It is shown that=p _l / _g /kT, wherep _l is the pressure of the liquid phase inside the drop, _g , is the volume per molecule in the vapor phase,k is the Boltzmann constant, andT is the absolute temperature. The difference between this and Reiss's expression isp _l , which replaces hisp _g (the vapor pressure in equilibrium with the drop). The derived in this paper is compatible with the expression= _g / _l ( _l is the molecular volume in the liquid phase) previously proposed by the present author.     

Weakly singular perturbations of a discrete spectrum

We consider weakly singular perturbations ¦x¦^–(0<<2) of an even restoring potential. We compute the matrix elements of the perturbation together with the additional point potential associated with the perturbation. It is shown that even for unperturbed wave functions, the matrix elements exist when 0 < < 3/2. The series for the Rayleigh-Schrödinger coefficients converge in all orders for the same interval in , regardless of the form of the restoring potential. For odd states, the matrix elements of the perturbation exist when 0 < < 3, while estimates for the Rayleigh-Schrödinger coefficients give the boundary = 2.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 14–18, June, 1989.     

Schrödinger Operators with Empty Singularly Continuous Spectra

Let H be a semibounded perturbation of the Laplacian H ₀ in L ²( ^d ). For an admissible function sufficient conditions are given for the completeness of the scattering system (H), (H ₀). If is the exponential function and if e^{– H} is an integral operator we denote the kernel of the difference D = e^{– H} – e^{– H} ₀ by D (x, y), > 0. The singularly continuous spectrum of H is empty if^d dx ^d dy |D(x,y)| (1 + |y|²)< for some > 1. This result is applied to potential perturbations and to perturbations by imposing Dirichlet boundary conditions.