Let $f(z)=\sum\limits_{j=0}^{\infty} a_j z^j$ be a transcendental entire function and let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where $\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space $(\Omega, \mathcal{F}, \mu)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher and Steinhaus entire functions. We prove that, for almost all functions in the family and for any constant C > 1, there exist a constant $r_0=r_0(\omega)$ and a set $E\subset [e, \infty)$ of finite logarithmic measure such that, for $r \gt r_0$ and $r\notin E$,

\begin{equation*}|\log M(r, f)- N(r,0, f_\omega)|\le (C/A)^{\frac1{B}}\,\log^{\frac1{B}}\,\log M(r,f) +\log\,\log M(r, f), \qquad a.s.\end{equation*}

where $A, B$ are constants, $M(r, f)$ is the maximum modulus and $N(r, 0, f)$ is the integrated zero-counting function of f. As a by-product of our main results, we prove Nevanlinna’s second main theorem for random entire functions. Thus, the characteristic function of almost all functions in the family is bounded above by an integrated counting function, rather than by two integrated counting functions as in the classical Nevanlinna theory. For instance, we show that, for almost all Gaussian entire functions fω and for any ϵ > 0, there is r0 such that, for $r \gt r_0$,

\begin{equation*}T(r, f) \le N(r,0, f_\omega)+\left(\tfrac12+\epsilon\right) \log T(r, f).\end{equation*}

We study the distribution of the consensus formed by a broadcast-based consensus algorithm for cases in which the initial opinions of agents are random variables. We first derive two fundamental equations for the time evolution of the average opinion of agents. Using the derived equations, we then investigate the distribution of the consensus in the limit in which agents do not have any mutual trust, and show that the consensus without mutual trust among agents is in sharp contrast to the consensus with complete mutual trust in the statistical properties if the initial opinion of each agent is integrable. Next, we provide the formulation necessary to mathematically discuss the consensus in the limit in which the number of agents tends to infinity, and derive several results, including a central limit theorem concerning the consensus in this limit. Finally, we study the distribution of the consensus when the initial opinions of agents follow a stable distribution, and show that the consensus also follows a stable distribution in the limit in which the number of agents tends to infinity.

Self-focusing of Gaussian laser beam has been investigated in quantum plasma under the effect of applied axial magnetic field. The nonlinear differential equation has been derived for studying the variations in the beam-width parameter. The effect of initial plasma electron temperature and the axial magnetic field on self-focusing and normalized intensity are studied. Our investigation reveals that normalized intensity increases to tenfolds where quantum effects are dominant. The normalized intensity further increases to twelvefolds on increasing the magnetic field.

In the present paper, we have studied self-focusing of Gaussian laser beam in weakly relativistic magnetized cold quantum plasma. When interparticle distance is comparable to the de Broglie wavelength of charged particles, we cannot neglect the quantum contribution of plasma constituents. Therefore, propagation characteristics are studied by taking in to account quantum contribution in the presence of static magnetic field applied along the beam propagation. Our results show that the magnetic field plays a key role in achieving additional focusing, it modifies the quiver motion of electrons by adding cyclotron frequency to the natural frequency of oscillating electrons during laser beam propagation. The results are compared with those of weakly relativistic quantum plasma and weakly relativistic magnetized plasma. The self-focusing is found to be more pronounced when axial magnetic field is increased in the present model. We have setup the non-linear differential equation for the evolution of beam-width parameter by well-known paraxial ray approximation and solved it with the help of computational technique.

Annual bluegrass seedhead suppression on golf greens by spring-applied plant growth regulators (PGR) has been erratic between years and locations. To effectively suppress annual bluegrass seedheads on greens, current information suggest ethephon must be applied before floral initiation. Rogue seedheads, however, can sometimes be found in late winter indicating that a variable proportion of annual bluegrass plants may induce floral initiation in winter. Studies were conducted in Blacksburg and Harrisonburg, VA at five separate sites between 2011 and 2012 to determine if winter applications of ethephon or mefluidide would improve annual bluegrass seedhead suppression when applied in advance of a spring, two-treatment program. A spring, two-treatment program of ethephon plus trinexapac-ethyl reduced annual bluegrass seedhead cover 22 to 55% depending on trial and less than similar mefluidide programs. Applying an early application of ethephon in January or February prior to the spring treatment program resulted in 5 to 7 times less seedhead cover at cover maxima than the spring treatment program alone. Ethephon did not injure creeping bentgrass and caused only slight and transient discoloration to annual bluegrass. Mefluidide injured both creeping bentgrass and annual bluegrass and reduced annual bluegrass population density in late spring. Programs that consisted of an early ethephon application, a spring treatment of triademifon, and two spring treatments of ethephon plus trinexapac ethyl improved seedhead suppression and turf quality but slightly increased annual bluegrass injury. Methiozolin suppressed annual bluegrass seedheads primarily through severe injury to annual bluegrass, which led to a decline in turfgrass quality and NDVI but a substantial increase in creeping bentgrass cover. Applying PGRs in winter is a novel concept and may substantially improve ethephon consistency and performance for annual bluegrass seedhead suppression on greens.

In the present paper, we have investigated self-focusing of Gaussian laser beam in relativistic ponderomotive (RP) cold quantum plasma. When de Broglie wavelength of charged particles is greater than or equal to the inter particle distance or equivalently the temperature is less than or equal to the Fermi temperature, quantum nature of the plasma constituents cannot be ignored. In this context, we have reported self-focusing on account of nonlinear dielectric contribution of RP plasma by taking into consideration the impact of quantum effects. We have setup the nonlinear differential equation for the beam-width parameter by paraxial ray and Wentzel Kramers Brillouin approximation and solved it numerically by the Runge Kutta Fourth order method. Our results show that additional self-focusing is achieved in case of RP cold quantum plasma than relativistic cold quantum plasma and classical relativistic case. The pinching effect offered by quantum plasma and the combined effect of relativistic and ponderomotive nonlinearity greatly enhances laser propagation up to 20 Rayleigh lengths.

In the present paper, we have investigated the growth of a Gaussian perturbation superimposed on a Gaussian laser beam. The nonlinearity we have considered is of relativistic type. We have setup the nonlinear differential equations for beam width parameter of the main beam, growth and width of the laser spike by using the WKB and paraxial ray approximation. These are coupled ordinary differential equations and therefore these are simultaneously solved numerically using the Runge Kutta method. It has been observed from the analysis that self-focusing/defocusing of the main beam and the spike determine the growth dynamic of the spike.

The asymptotic properties of the periodogram of a weakly stationary time series for the triangular array of fundamental frequencies is studied. For linear Gaussian processes, results are obtained relating the asymptotic distribution of certain Riemann sums of the periodogram of the process to those of the periodogram of the innovation process.