Abstract:
We consider smooth, double-odd solutions of the two-dimensional Euler equation in $[-1, 1)^2$ with periodic boundary conditions. It is tempting to think that the symmetry in the flow induces possible double-exponential growth in time of the vorticity gradient at the origin, in particular when conditions are such that the flow is "hyperbolic". This is because examples of solutions with $C^{1, \gamma}$-regularity were already constructed with exponential gradient growth by A. Zlatos. We analyze the flow in a small box around the origin in a strongly hyperbolic regime and prove that the compression of the fluid induced by the hyperbolic flow alone is not sufficient to create double-exponential growth of the gradient.

Abstract:
Data describing the growth of the world population in the past 12,000 years are analysed. It is shown that, if unchecked, population does not increase exponentially but hyperbolically. This analysis reveals three approximately-determined episodes of hyperbolic growth: 10,000-500 BC, AD 500-1200 and AD 1400-1950, representing a total of about 89% of the past 12,000 years. It also reveals three demographic transitions: 500 BC to AD 500, AD 1200 to AD 1400 and AD 1950 to present, representing the remaining 11% of the past 12,000 years. The first two transitions were between sustained hyperbolic trajectories. The current transition is to an unknown trajectory. There was never a transition from stagnation because there was no stagnation in the growth of the world population.

Abstract:
Existence-uniqueness theorems are proved for continuous solutions of some classes of non-linear hyperbolic equations in bounded and unbounded regions. In case of unbounded region, certain conditions ensure that the solution cannot grow to infinity faster than exponentially.

Abstract:
We prove the existence of entire solutions with exponential growth for the semilinear elliptic system [\begin{cases} -\Delta u = -u v^2 & \text{in $\R^N$} -\Delta v= -u^2 v & \text{in $\R^N$} u,v>0, \end{cases}] for every $N \ge 2$. Our construction is based on an approximation procedure, whose convergence is ensured by suitable Almgren-type monotonicity formulae. The construction of \emph{some} solutions is extended to systems with $k$ components, for every $k > 2$.

Abstract:
In this article, we study a free boundary problem modeling the growth of tumors with drug application. The model consists of two nonlinear second-order parabolic equations describing the diffusion of nutrient and drug concentration, and three nonlinear first-order hyperbolic equations describing the evolution of proliferative cells, quiescent cells and dead cells. We deal with the radially symmetric case of this free boundary problem, and prove that it has a unique global solution. The proof is based on the L^p theory of parabolic equations, the characteristic theory of hyperbolic equations and the Banach fixed point theorem.

Abstract:
This paper studies asymptotic behavior of solutions of a free boundary problem modeling the growth of tumors with two species of cells: proliferating cells and quiecent cells. In previous literatures it has been proved that this problem has a unique stationary solution which is asymptotically stable in the limit case $\varepsilon=0$. In this paper we consider the more realistic case $0<\varepsilon<<1$. In this case, after suitable reduction the model takes the form of a coupled system of a parabolic equation and a hyperbolic system, so that it is more difficult than the limit case $\varepsilon=0$. By using some unknown variable transform as well as the similarity transform technique developed in our previous work, we prove that the stationary solution is also asymptotically stable in the case $0<\varepsilon<<1$.

Abstract:
In this study an effort has been given attention to establish a mathematical relationship between rate of increase of food and rate of population growth of Malthus`s Theorem. It is seen that Malthus`s Theorem of rate of increase of food and rate of population growth follows positive exponential model. To verify the stability of the model, Cross Validity-Prediction Power (CVPP) is applied in this study. It is found that the coefficient of determination of the fitted model is 100% and the fitted model is also100% stable. And its all parameters are also highly significant.

Abstract:
Various approaches to model the progression of a dynasty in terms of power are discussed. The efficiency-discounted exponential growth (EDEG) approach is presented, and the effects of changing decay type and growth rate are demonstrated. The Russian Romanov dynasty is utilized as an example for several of the approaches.

Abstract:
There is increasing recognition that stochasticity involved in gene regulatory processes may help cells enhance the signal or synchronize expression for a group of genes. Thus the validity of the traditional deterministic approach to modeling the foregoing processes cannot be without exception. In this study, we identify a frequently encountered situation, i.e., the biofilm, which has in the past been persistently investigated with intracellular deterministic models in the literature. We show in this paper circumstances in which use of the intracellular deterministic model appears distinctly inappropriate. In Enterococcus faecalis, the horizontal gene transfer of plasmid spreads drug resistance. The induction of conjugation in planktonic and biofilm circumstances is examined here with stochastic as well as deterministic models. The stochastic model is formulated with the Chemical Master Equation (CME) for planktonic cells and Reaction-Diffusion Master Equation (RDME) for biofilm. The results show that although the deterministic model works well for the perfectly-mixed planktonic circumstance, it fails to predict the averaged behavior in the biofilm, a behavior that has come to be known as stochastic focusing. A notable finding from this work is that the interception of antagonistic feedback loops to signaling, accentuates stochastic focusing. Moreover, interestingly, increasing particle number of a control variable could lead to an even larger deviation. Intracellular stochasticity plays an important role in biofilm and we surmise by implications from the model, that cell populations may use it to minimize the influence from environmental fluctuation.