Narrow Results

We study concepts of stability associated to a smooth complex curve together with a linear series on it. In particular we investigate the relation between stability of the associated dual span bundle and linear stability. Our results imply that stability of the dual span holds under a hypothesis related to the Clifford index of the curve. Furthermore, in some of the cases, we prove that a stronger stability holds: cohomological stability. Finally, using our results we obtain stable vector bundles of slope 3, and prove that they admit theta-divisors.

In this paper, we study the stability problem for the Einstein metrics on Sasaki Einstein and on complete nearly parallel ${\mathrm{\text{G}}}_2$ manifolds. In the Sasaki case we show linear instability if the second Betti number is positive. Similarly, we prove that nearly parallel ${\mathrm{\text{G}}}_2$ manifolds with positive third Betti number are linearly unstable. Moreover, we prove linear instability for the Berger space $\mathrm{\text{SO}}(5)/\mathrm{\text{SO}}{(3)}_{\mathrm{\text{irr}}}$ which is a $7$-dimensional homology sphere with a proper nearly parallel ${\mathrm{\text{G}}}_2$ structure.

The progress and challenges in thermal lattice-Boltzmann modeling are discussed. In particular, momentum and energy closures schemes are contrasted. Higher order symmetric (but no longer space filling) velocity lattices are constructed for both 2D and 3D flows and shown to have superior stability properties to the standard (but lower) symmetry lattices. While this decouples the velocity lattice from the spatial grid, the interpolation required following free-streaming is just 1D. The connection between fixed lattice vectors and temperature-dependent lattice vectors (obtained in the Gauss–Hermite quadrature approach) is discussed. Some (compressible) Rayleigh–Benard simulations on the 2D octagonal lattice are presented for extended BGK collision operators that allow for arbitrary Prandtl numbers.

In this paper, the linear stability of static Mandal–Sengupta–Wadia (MSW) black holes in (2 + 1)-dimensional gravity against circularly symmetric perturbations is studied. Our analysis only applies to non-extremal configurations, thus leaving out the case of the extremal (2 + 1) MSW solution. The associated fields are assumed to have small perturbations in these static backgrounds. We then consider the dilaton equation and specific components of the linearized Einstein equations. The resulting effective Klein–Gordon equation is reduced to the Schrödinger-like wave equation with the associated effective potential. Finally, it is shown that MSW black holes are stable against the small time-dependent perturbations.

In this paper, we investigate the stability of a recently introduced Bose–Einstein condensate (BEC) which involves logarithmic interaction between atoms. The Gaussian variational approach is employed to derive equations of motion for condensate widths in the presence of a harmonic trap. Then we derive the analytical solutions for these equations and find them to be in good agreement with numerical data. By analyzing deeply the frequencies of collective oscillations, and the mean-square radius, we find that the system is always stable for both negative and positive week logarithmic coupling. However, for strong interaction the situation is quite different: Our condensate collapses for positive coupling and oscillates with fixed frequency for negative one. These special results remain the most characteristic features of the logarithmic BEC compared to that involving two-body and three-body interactions.

In this paper, a new lattice two-lane hydrodynamic model is proposed by considering the lane changing and the optimal current change with memory effect. The linear stability condition of the model is obtained through the linear stability analysis, which depends on both the lane-changing rate and the memory step. A modified Korteweg–de Vries (mKdV) equation is derived through nonlinear analysis to describe the propagating behavior of traffic density wave near the critical point. To verify the analytical findings, numerical simulation was carried out, which confirms that the optimal current change with memory of drivers and the memory step contribute to the stabilization of traffic flow, and that traffic congestion can be suppressed efficiently by taking the lane-changing behavior into account in the lattice model.

Drivers would adjust the speeds in response to not only the external environment, but also the anticipated traffic condition. In this paper, we propose a new continuum model considering the driver’s self-anticipative effect. Such effect is mainly reflected by the difference between the current speed and optimal speed within the anticipation time step. By applying the linear stability theory, the stability condition of the new model is obtained. Through the nonlinear analysis method, the KdV–Burgers equation of the model is provided. The solution describes the evolution of density waves near the neutral stability region. The simulation example verifies that the self-anticipative effect of the driver contributes to suppressing traffic congestion and reducing exhaust emissions effectively. We thus suggest that the traffic flow stability could be improved in an ad hoc manner.

In the real traffic, drivers often perceive traffic information with delay time, while presenting asynchronous reactions that form non-constant delay time. Meanwhile, on-ramps are very common in the real traffic network environment, especially on highway sections. Traffic flow merging from the on-ramps has a significant impact on the traffic stability of the main road. Given these facts, we develop an extended lattice hydrodynamic model with drivers’ continuous delay time and on-ramp. Based on the linear and nonlinear stability analysis, we derive the stability criterion and modified Korteweg–de Vries (mKdV) equation of the proposed lattice model; the kink-antikink soliton wave solution obtained by solving the above mKdV equation can be used to describe the propagation and evolution characteristics of density waves near the neutral stable curve. Results show that the ramp inflow ratio and delay time have a significant impact on the stability of traffic flow on the freeway main lane.

This work is motivated by the importance of vertical vibration in nanofluids suspension, because of its wide applications in heat exchangers from one fluid to another, controlling the mixing in microvolumes, pharmaceutical industry, and engineering. The current study analyzes the effects of high-frequency vertical vibration on a horizontal Casson nanofluid layer heated from below saturated between rigid-rigid, rigid-free, and free-free boundary conditions. Field equations are derived by using time-average method, which describes the vibrational thermo-convection. These field equations are solved by utilizing linear stability analysis and Galerkin technique to obtain the thermal stability curve. By using this curve, the impacts of vertical vibration and physical governing factors on temperature profile are discussed. High-frequency vertical vibration is observed to have a stabilizing effect on Casson nanofluid suspension, thereby reducing thermo-convective heat transfer from one fluid to another. Stability threshold of vibrated suspension is increased approximately as $5.748\%$ and $81.937\%$ when $R_v\to 50$ and $R_v\to 100$, respectively. The significance of Brownian motion and thermophoretic force on convective heat transfer with vertical vibration is discussed. The resisting nature of Casson nanofluid on vibrational convective heat transfer is also tabulated.

In this paper we investigate the onset of instabilities in a model describing the propagation of the steady planar premixed combustion wave. In particular, we are interested in determining the Bogdanov–Takens bifurcation condition, which is investigated semi-analytically. We derive an analytic condition for the existence of this type of bifurcation and based on this criterion we numerically determine the parameter values for which the Bogdanov–Takens bifurcation occurs. This numerical method is found to be more efficient than the previous methods.

This paper shows the effects of optimal control techniques on pattern formation in a three-dimensional Rayleigh–Bénard problem with nonhomogeneous heating from below. In particular, we consider a cylindrical domain with a heating profile at the bottom localized around the origin. An axisymmetric basic state appears as soon a nonzero horizontal temperature gradient is imposed. The basic state may bifurcate to different solutions as spiral waves, stationary patterns, etc., depending on the vertical and lateral temperature gradients and on the shape of the heating function. An optimal control problem that determines the thermal boundary condition minimizing the enstrophy is proposed. The control boundary condition obtained changes the leading mode in the bifurcation and allows to eliminate the instabilities for some values of the parameters.

The reduced basis method is a suitable technique for finding numerical solutions to partial differential equations that must be obtained for many values of parameters. This method is suitable when researching bifurcations and instabilities of stationary solutions for partial differential equations. It is necessary to solve numerically the partial differential equations along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions for a large number of bifurcation parameter values. In this paper, the reduced basis method has been used to solve eigenvalue problems derived from the linear stability analysis of stationary solutions in a two-dimensional Rayleigh–Bénard convection problem. The bifurcation parameter is the Rayleigh number, which measures buoyancy. The reduced basis considered belongs to the eigenfunction spaces derived from the eigenvalue problems for different types of solutions in the bifurcation diagram depending on the Rayleigh number. The eigenvalue with the largest real part and its corresponding eigenfunction are easily calculated and the bifurcation points are correctly captured. The resulting matrices are small, which enables a drastic reduction in the computational cost of solving the eigenvalue problems.

A systematic study of families of planar symmetric periodic orbits of the elliptic restricted three-body problem is presented, in exoplanetary systems. We find families of periodic orbits that surround only one of the primaries (Satellite-Type), that are moving around both primaries (Planet-Type), and also moving about the collinear Lagrange points. The linear stability of every periodic orbit is calculated, and the families are interpreted through stability diagrams. We focus on quasi-satellite motions of test particles that are associated with the known family $f$ that consists of 1:1 resonant retrograde Satellite-Type orbits. Over the last years, quasi-satellite orbits are of special interest due to the many applications in the design of spacecraft missions around moons and asteroids. We find the critical simple (1:1 resonant) periodic orbits of the basic families of the circular problem from which we calculate new families of the elliptic problem. Additionally, families of the elliptic problem which bifurcate from the main family $f$, for various resonances, are also presented and discussed. Hundreds of critical orbits (bifurcation points), from which families of the elliptic problem of higher multiplicity emerge, are found and the corresponding resonances are identified.

The application of a proper orthogonal decomposition (POD) reduced order method to a two-dimensional instability problem of Rayleigh–Bénard convection is described in this work. The partial differential equations that model the problem along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions are numerically solved for different values of the bifurcation parameter, i.e. the Rayleigh number $R$. The POD reduced order method considers spaces derived from different stationary solutions depending on $R$ as snapshots. The eigenvalue with the largest real part and its corresponding eigenfunction are calculated for the POD solutions. The resulting matrices of the eigenvalue problems are low-dimensional. There is a drastic reduction in the computational cost.

This work is devoted to the study of a singular reaction–diffusion system arising in modelling the introduction of a lethal pathogen within an invading host population. In the absence of the pathogen, the host population exhibits a bistable dynamics (or Allee effect). Earlier numerical simulations of the singular SI model under consideration have exhibited stable travelling waves and also, under some circumstances, a reversal of the wave front speed due to the introduction of the pathogen. Here we prove the existence of such travelling wave solutions, study their linear stability and give analytical conditions yielding a reversal of the wave front speed, i.e. the invading host population may eventually retreat following the introduction of the lethal pathogen.

Motivated by empirical observations of spatio-temporal clusters of crime across a wide variety of urban settings, we present a model to study the emergence, dynamics, and steady-state properties of crime hotspots. We focus on a two-dimensional lattice model for residential burglary, where each site is characterized by a dynamic attractiveness variable, and where each criminal is represented as a random walker. The dynamics of criminals and of the attractiveness field are coupled to each other via specific biasing and feedback mechanisms. Depending on parameter choices, we observe and describe several regimes of aggregation, including hotspots of high criminal activity. On the basis of the discrete system, we also derive a continuum model; the two are in good quantitative agreement for large system sizes. By means of a linear stability analysis we are able to determine the parameter values that will lead to the creation of stable hotspots. We discuss our model and results in the context of established criminological and sociological findings of criminal behavior.

Urban crime such as residential burglary is a social problem in every major urban area. As such, many mathematical models have been proposed to study the collective behavior of these crimes. In [V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, M. B. Short, M. R. D’Orsogna and L. B. Chayes, A statistical model of crime behavior, Math. Methods Appl. Sci107 (2008) 1249–1267; M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst.9 (2010) 462–483], Short et al. proposed an agent-based statistical model of residential burglary to model the crime hotspot phenomena. From the point of view of reaction–diffusion systems, the model is a chemotactic system with cross diffusion that exhibit hotspot phenomena. In this paper, we first construct a radial hotspot solution of this system, then study the linear stability of this hotspot solution by studying a nonlocal eigenvalue problem. It turns out that the stability of the hotspot is completely different depending on which spatial dimension the system is on. The main mathematical difficulty of the system involves treating the steep change of diffusion near the core of the hotspot, because of the quasilinearity induced by the cross diffusion. We believe that the techniques used in this paper can be developed to treat many other chemotactic systems.

We exactly calculate the quasinormal frequencies of the electromagnetic and Klein–Gordon perturbations propagating in a five-dimensional dilatonic black hole. Furthermore, we exactly find the quasinormal frequencies of the massive Dirac field. Using these results we study the linear stability of this black hole. We compare our results for the quasinormal frequencies and for the linear stability of the five-dimensional black hole with those already published.

This work proposes a hybrid block numerical method of tenth order for the direct solution of fifth-order initial value problems. The formulas that constitute the block method are derived from a continuous approximation obtained through interpolation and collocation techniques. In order to obtain better accuracy, sixth-order derivatives are incorporated to develop the formulas. The main characteristics of the method are analyzed, namely, the order, local truncation errors, zero-stability, consistency and convergence. The proposed strategy performs well, as shown by some numerical examples and the corresponding efficiency curves. Compared to existing numerical methods in the literature, the proposed method is competitive and the numerical approximations it provides are significantly close to the precise solutions.

Within $R^2$ gravity, we study the linear stability of strongly gravitating spherically symmetric configurations supported by a polytropic fluid. All calculations are carried out in the Jordan frame. It is demonstrated that, as in general relativity, the transition from stable to unstable systems occurs at the maximum of the curve mass-central density of the fluid.