Narrow Results

We give an O(n + t) time algorithm to determine whether an NFA with n states and t transitions accepts a language of polynomial or exponential growth. Given an NFA accepting a language of polynomial growth, we can also determine the order of polynomial growth in O(n+t) time. We also give polynomial time algorithms to solve these problems for context-free grammars.

We study FAD-languages, which are regular languages defined by finite sets of forbidden factors, together with their “canonical” recognizing automata. We are mainly interested in the possible asymptotic orders of growth for such languages. We analyze certain simplifications of sets of forbidden factors and show that they “almost” preserve the canonical automata. Using this result and structural properties of canonical automata, we describe an algorithm that effectively lists all canonical automata having a sink strong component isomorphic to a given digraph, or reports that no such automata exist. This algorithm can be used, in particular, to prove the existence of a FAD-language over a given alphabet with a given exponential growth rate. On the other hand, we give an example showing that the algorithm cannot prove non-existence of a FAD-language having a given growth rate. Finally, we provide some examples of canonical automata with a nontrivial condensation graph and of FAD-languages with a “complex” order of growth.

The stability characteristics of nonlinear surface waves propagating at a linear gyrodielectric substrate and a nonlinear dielectric cover have been simulated numerically by using the perturbation method. The growth rate of perturbation is computed by solving the dispersion equation of perturbation. We found that the nonlinear surface waves are unstable when their growth rate of perturbation is real, and stable when their growth rate of perturbation is imaginary. The spatial evolution of the steady-state field amplitude is determined using a computer simulation method.

Molecular dynamics simulations of crystal growth of SiC in the reduced temperature range of 0.51–1.02 have been carried out. In particular, the relationship between the growth rate and the reduced temperature has been investigated by the simulations. The results show that the growth rate increases first with the temperature and then decreases dramatically after passing through a maximum. Calculations of the growth rate according to the Wilson–Frenkel model have been applied to the present system, with the required parameters of the activation energy for atomic diffusion and the free energy changes calculated by molecular dynamics simulations. The temperature dependence of the growth rate, calculated by molecular dynamics, agrees with the prediction of Wilson–Frenkel model, indicating that the crystal growth of SiC is a kind of diffusion limited growth.

Vorticity variation in a supersonic planar mixing layer interacting with an oblique shock wave is investigated analytically and numerically. A model that simplifies the mixing layer to a discontinuous flow is established to solve the post-shock flow parameters, and it is validated through qualitative and quantitative comparisons with the Buttsworth’s model and numerical results. A model to estimate the shock-induced Maximum Vorticity Amplification (MVA) is obtained, which agrees well with the numerical results. The model could estimate the growth rate and maximum vorticity of the shocked mixing layer. The vorticity of the mixing layer is amplified by the shock impingement, even though the vorticity thickness decreases, which can improve the mixing performance for different practical applications.

The Lyapunov exponent gives a measure of the mean decay/growth rates of the flows of nonlinear systems. However, the Lyapunov exponent needs an infinite time interval of flows and the Jacobian matrix of system dynamics. In this paper, we propose an instantaneous decay/growth rate that is a kind of generalized Lyapunov exponent and call the instantaneous Lyapunov exponent (ILE) with respect to a decay function. The instantaneous Lyapunov exponent is one of the measures that estimate the decay and growth rates of flows of nonlinear systems by assigning a comparison function and can apply a stable system whose decay rate is slower than an exponential function. Moreover, we propose a synchronization measure of two signals using the ILE.

In a standard bifurcation of a dynamical system, the stationary points (or more generally attractors) change qualitatively when varying a control parameter. Here we describe a novel unusual effect, when the change of a parameter, e.g. a growth rate, does not influence the stationary states, but nevertheless leads to a qualitative change of dynamics. For instance, such a dynamic transition can be between the convergence to a stationary state and a strong increase without stationary states, or between the convergence to one stationary state and that to a different state. This effect is illustrated for a dynamical system describing two symbiotic populations, one of which exhibits a growth rate larger than the other one. We show that, although the stationary states of the dynamical system do not depend on the growth rates, the latter influence the boundary of the basins of attraction. This change of the basins of attraction explains this unusual effect of the qualitative change of dynamics by growth rate variation.

We investigate the function d_A(n), which gives the size of a least size generating set for Aⁿ, in the case where A has a cube term. We show that if A has a k-cube term and A^k is finitely generated, then d_A(n) ∈ O(log(n)) if A is perfect and d_A(n) ∈ O(n) if A is imperfect. When A is finite, then one may replace "Big O" with "Big Theta" in these estimates.

By applying the Poincaré—Birkhoff—Witt property and the Gröbner—Shirshov bases technique, we find the linear basis of the associative universal enveloping algebra in the sense of Ginzburg and Kapranov of a pair of compatible Lie brackets. We state that the growth rate of this universal enveloping over $n$-dimensional compatible Lie algebra equals $n+1$.

It is shown that the proportion of alternating n-crossing, prime link types amongst all n-crossing prime link types tends to zero exponentially with increasing n. Also, a characterization is established for essential annuli in alternating tangles, and a simple criterion is given for equivalence of alternating tangles.

In the present study, the individual growth process of an organism has been shown with the help of a mathematical model. The surplus energy production rate, i.e. intake rate minus metabolic cost, plays a crucial role in controlling the growth rate. Considering the existence of an optimum mass, which maximizes the surplus energy, it has been found that the scaling exponent for the metabolic cost has to be greater than the exponent for the intake rate. On the basis of the consideration that the system always generates some surplus energy, a relationship among the empirical constants has been established. The growth is found to continue with an ever decreasing rate. When the system attains its optimum mass, the growth rate is found to be the maximum. The mass variation with time has been graphically shown using the expression obtained by solving a differential equation involving surplus energy. Using figures, the dependence of mass variation upon various scaling parameters, has been thoroughly discussed. As mass increases, the surplus energy production rate per unit mass is found to decrease and this may be the probable reason for the smaller number of organisms with larger mass. As the scaling exponent regarding intake increases, the maximum attainable mass increases along with an increase in the time required for its attainment.

Let $\psi :\mathbb{N}\to {ℝ}^{+}$ be a function satisfying $\psi (n)\to \infty$ as $n\to \infty$. Write

The goal of this paper is to identify the factors that precede and may cause sudden changes in the pace of growth of high-growth SMEs or gazelles. A retrospective longitudinal case study of seven high growth SMEs that had undergone a total of 14 sudden shifts in growth reveals that a number of events caused the changes of pace. Some were triggered by the entrepreneur's decisions while others resulted from events beyond his/her control. Management's motivation for growth was an important element and this motivation changed over time, being influenced by both success and problems associated with actual growth. The success of growth strategies also appears to depend on the firm's proximity to its client base and its ability to obtain the information required for sound decision-making. Lastly, the availability of tangible and intangible resources was found essential in allowing the company to seize growth opportunities and proximity to the business milieu may help accessing these resources.

Using the quantum hydrodynamic model (QHDM), quantum corrections (via Bohm potential) on modulational amplification characteristics in semiconductor magneto-plasmas are investigated. Expressions are obtained for the threshold pump amplitude and the growth rate of the modulated beam with including and excluding quantum effects (QEs). Numerical analysis is performed for $n$-InSb/CO₂ laser system. The dependence of the threshold pump amplitude and the growth rate of the modulated beam on wave number, applied magnetic field (via cyclotron frequency), and plasma carrier concentration (via plasma frequency) are explored. The lowering in threshold pump amplitude and enhancement of growth rate of the modulated beam is observed by incorporating the QEs. The analysis provides detailed information of QEs on semiconductor magneto-plasmas and establishes the technological potentiality of semiconductor magneto-plasmas as the hosts for the fabrication of efficient optical modulators.

We study families of time-varying linear systems, where time-variations have to satisfy restrictions on the dwell time, that is, on the minimum distance between discontinuities, as well as on the derivative in between discontinuities. For this class of systems we study continuity properties of the growth rate as a function of the systems' data. It is shown by example, that a straightforward topology on the space of systems does not yield the desired continuity result. A new natural metric is introduced and a continuity result is obtained. Furthermore, local Lipschitz continuity may be shown for the (generic) case of irreducible systems. The methods rely heavily on a recent converse Lyapunov theorem for the class under consideration.

We present results of a detailed numerical investigation of the phase separation kinetic process of the macromolecular microsphere composite (MMC) hydrogel. Based on the Flory-Huggins-de Gennes-like reticular free energy, we use the time-dependent Ginzburg–Landau (TDGL) mesoscopic model (called MMC-TDGL model) to simulate the phase separation process. Domain growth is investigated through the pair correlation function. Then we obtain the time-dependent characteristic domain size, which reflects the growth kinetics of the MMC hydrogel. The results indicate that the growth law based on the MMC-TDGL equation is consistent with the modified Lifshitz–Slyozov theory.

Deligne proved that the Hilbert series of all Artin monoids are rational functions. We give an algorithm to compute the Hilbert series of the braid monoids . We also show that the Hilbert series of the positive words in with a given prefix are rational functions.

In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology.

Using this invariant we show that if $Q$ is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationally elliptic and so it must have certain bounds on its Betti numbers.

A pre-modulated relativistic electron beam (REB) counter propagating to the surface wave in the vacuum region Compton backscatters the surface wave into a high frequency radiation. The surface wave extends into the vacuum region and can be employed as a wiggler for the generation of sub-millimeter waves. The growth rate and gain were evaluated for a typical FEL (Free Electron Laser) parameters and It is found that the growth rate and gain of the surface wave pumped free electron laser increases with the modulation index. Moreover, the growth rate of the FEL (Free electron Laser) instability scales as one-third power of the beam density in the Compton regime.

The numerical calculations of the growth rate in long parallel wavelength are made for a spiraling ion beam propagating through a collision less magnetized dusty plasma cylinder that drives electrostatic ion cyclotron waves to instability via cyclotron interaction. It is found that the growth rate of the instability of the electrostatic ion cyclotron waves increase in the long parallel limit with the density ratio of negatively charged dust grains to electrons. The growth rate of the unstable mode has the maximum value for the modes whose Eigen functions peak at the location of the beam and varies as the one-third power of the beam current in both the limits.