Showing papers in "Siam Journal on Applied Mathematics in 2003"
TL;DR: A computational scheme based on numerical PDEs is presented, which allows the automatic handling of topologically complex inpainting domains and connects to the earlier works of Bertalmio, Sapiro, Caselles, and Ballester.
Abstract: Image inpainting is a special image restoration problem for which image prior models play a crucial role. Euler's elastica was first introduced to computer vision by Mumford [Algebraic Geometry and its Applications, Springer-Verlag, New York, 1994, pp. 491--506] as a curve prior model. By functionalizing the elastica energy, Masnou and Morel [Proceedings of the 5th IEEE International Conference Image Processing, 3 (1998), pp. 259--263] proposed an elastica-based variational inpainting model. The current paper is intended to contribute to the development of its mathematical foundation and the study of its properties and connections to the earlier works of Bertalmio, Sapiro, Caselles, and Ballester [SIGGRAPH 2000, ACM Press, New York, 2000] and Chan and Shen [J. Visual Comm. Image Rep., 12 (2001), pp. 436--449]. A computational scheme based on numerical PDEs is presented, which allows the automatic handling of topologically complex inpainting domains.
643 citations
TL;DR: Exploiting the fact that standard models of within-host viral infections of target cell populations by HIV, developed by Perelson and Nelson and Nowak and May, work well, this paper proposes a new approach to studying these infections using a probabilistic approach.
Abstract: Exploiting the fact that standard models of within-host viral infections of target cell populations by HIV, developed by Perelson and Nelson [SIAM Rev., 41 (1999), pp. 3--44] and Nowak and May [Vir...
426 citations
TL;DR: It is shown that a backward bifurcation leading to bistability can occur and under mild parameter constraints, compound matrices are used to show that each orbit limits to an equilibrium.
Abstract: Vaccination of both newborns and susceptibles is included in a transmission model for a disease that confers immunity. The interplay of the vaccination strategy together with the vaccine efficacy and waning is studied. In particular, it is shown that a backward bifurcation leading to bistability can occur. Under mild parameter constraints, compound matrices are used to show that each orbit limits to an equilibrium. In the case of bistability, this global result requires a novel approach since there is no compact absorbing set.
315 citations
TL;DR: This work considers the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals and revisits this problem using the notion of a shape derivative and shows that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals.
Abstract: We consider the problem of segmenting an image through the minimization of an energy criterion involving region and boundary functionals. We show that one can go from one class to the otherby solving Poisson's orHelmholtz's equation with well-chosen boundar y conditions. Using this equivalence, we study the case of a large class of region functionals by standard methods of the calculus of variations and derive the corresponding Euler-Lagrange equations. We revisit this problem using the notion of a shape derivative and show that the same equations can be elegantly derived without going through the unnatural step of converting the region integrals into boundary integrals. We also define a larger class of region functionals based on the estimation and comparison to a prototype of the probability density distribution of image features and show how the shape derivative tool allows us to easily compute the corresponding Gateaux derivatives and Euler-Lagrange equations. Finally we apply this new functional to the problem of regions segmentation in sequences of color images. We briefly describe our numerical scheme and show some experimental results.
288 citations
TL;DR: It is shown that, in the presence of time scales between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas.
Abstract: After the celebrated Black--Scholes formula for pricing call options under constant volatility, the need for more general nonconstant volatility models in financial mathematics motivated numerous works during the 1980s and 1990s. In particular, a lot of attention has been paid to stochastic volatility models in which the volatility is randomly fluctuating driven by an additional Brownian motion. We have shown in [Derivatives in Financial Markets with Stochastic Volatility, Cambridge University Press, Cambridge, UK, 2000; Internat. J. Theoret. Appl. Finance, 13 (2000), pp. 101--142] that, in the presence of a separation of time scales between the main observed process and the volatility driving process, asymptotic methods are very efficient in capturing the effects of random volatility in simple robust corrections to constant volatility formulas. From the point of view of PDEs, this method corresponds to a singular perturbation analysis. The aim of this paper is to deal with the nonsmoothness of the payoff...
215 citations
TL;DR: It is shown that by allowing b to be negative ($b > -2\sqrt a$), p(x) is concave up for small values of x > 0 a...
Abstract: We consider a predator-prey system with nonmonotonic functional response: $p(x)=\frac{mx}{ax^2+bx+1}$ By allowing b to be negative ($b > -2\sqrt a$), p(x) is concave up for small values of x > 0 a
200 citations
TL;DR: Qualitative properties of real traffic flow are shown to agree with properties of the solutions of the model, which consists of a scalar conservation law coupled with a 2 × 2 system of conservation laws.
Abstract: This paper provides a mathematical model of the phenomenon of phase transitions in traffic flow. The model consists of a scalar conservation law coupled with a 2 × 2 system of conservation laws. The coupling is achieved via a free boundary, where the phase transition takes place. For this model, the Riemann problem is stated and globally solved. The Cauchy problem is proved to admit a solution defined globally in time without any assumption about the smallness of the initial data or the number of phase boundaries. Qualitative properties of real traffic flow are shown to agree with properties of the solutions of the model.
198 citations
TL;DR: A new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments, of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions is shown.
Abstract: We show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments ("sedimentation with compression"' or "sedimentation-consolidation process"). For N solid particle species, this theory reduces in one space dimension to an $N\times N$ coupled system of quasi-linear degenerate convection-diffusion equations. Analyses of the characteristic polynomials of the Jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic-hyperbolic type for arbitrary N and particle size distributions. Bounds for the eigenvalues of both matrices are derived. The mathematical model for N=3$ is illustrated by a numerical simulation obtained by the Kurganov-Tadmor central difference scheme for convection-diffusion problems. The numerical scheme exploits...
163 citations
TL;DR: It is shown that no shock wave, vacuum, or mass or heat concentration will be developed in a finite time, although the motion of the flow has large oscillations and there is a complex interaction between the hydrodynamic and magnetodynamic effects.
Abstract: An initial-boundary value problem for nonlinear magnetohydrodynamics (MHD) in one space dimension with general large initial data is investigated. The equations of state have nonlinear dependence on temperature as well as on density. For technical reasons the viscosity coefficients and magnetic diffusivity are assumed to depend only on density. The heat conductivity is a function of both density and temperature, with a certain growth rate on temperature. The existence, uniqueness, and regularity of global solutions are established with large initial data in H1 . It is shown that no shock wave, vacuum, or mass or heat concentration will be developed in a finite time, although the motion of the flow has large oscillations and there is a complex interaction between the hydrodynamic and magnetodynamic effects.
148 citations
TL;DR: This work investigates the qualitative differences between three different cell-kill models: log- kill hypo, log-kill hypo and log-killing hypo.
Abstract: Optimal control techniques are used to develop optimal strategies for chemotherapy. In particular, we investigate the qualitative differences between three different cell-kill models: log-kill hypo...
113 citations
TL;DR: A new model for traffic on a multilane freeway (with n lanes), where the car density $\rho$ is taken across all lanes in the freeway, and the average car velocity u is given.
Abstract: We present a new model for traffic on a multilane freeway (with n lanes). Our basic descriptors are the car density $\rho$ (in cars/mile), taken across all lanes in the freeway, and the average car velocity u (in miles/hour). The flux of cars across all lanes is given by $\rho u = {\sum^n_{i=1}}\rho_i u_i$, where $\rho_i$ is the car density in the ith lane, and ui the velocity of cars in the ith lane. We shall track only $\rho$ and u and not what is going on in each individual lane. On such multilane freeways, one often observes distinct stable equilibrium relationships between car velocity and density. Prototypical situations involve two equilibria, $$ v=v_1(\rho) > v = v_2 (\rho), \ \ \ 0 \leq \rho < \rho_{\rm max,} where $v_1(\cdot)$ and $v_2(\cdot)$ are monotone decreasing and satisfy $v_1 (\rho_{\rm max})=v_2(\rho_{\rm max})=0$. The upper curve is typically stable for densities satisfying $0 \leq \rho \leq \rho_1$, whereas the lower curve is stable for densities satisfying $ \rho_2 \leq \rho \leq \rh...
TL;DR: The Holling--Tanner model for predator-prey systems has two Hopf bifurcation points for certain parameters and the dependence of the environmental parameters on the underlying bIfurcation structure is uncovered using two-timing.
Abstract: The Holling--Tanner model for predator-prey systems has two Hopf bifurcation points for certain parameters. The dependence of the environmental parameters on the underlying bifurcation structure is uncovered using two-timing. Emphasis is on how the bifurcation diagram changes as the Hopf bifurcation points separate. Two degenerate cases require a modification of conventional two-timing. When the two Hopf bifurcation points nearly coalesce, the two stable periodic solution branches are shown to be connected. As a ratio of linear growth rates varies, the Hopf bifurcation points separate further and one limit cycle becomes unstable. This situation can correspond to an outbreak in populations. The modified two-timing analysis analytically captures the unstable and stable limit cycles of the new branch.
TL;DR: It is shown that the 2D critical thresholds depend on the relative sizes of three quantities: the initial density, the initial divergence, and the initial spectral gap, that is, the difference between the two eigenvalues of the 2 × 2 initial velocity gradient.
Abstract: We provide a complete description of the critical threshold phenomenon for the two-dimensional localized Euler-Poisson equations, introduced by the authors in [Comm. Math. Phys., 228 (2002), pp. 435-466]. Here, the questions of global regularity vs. finite-time breakdown for the two-dimensional (2D) restricted Euler-Poisson solutions are classified in terms of precise explicit formulae, describing a remarkable variety of critical threshold surfaces of initial configurations. In particular, it is shown that the 2D critical thresholds depend on the relative sizes of three quantities: the initial density, the initial divergence, and the initial spectral gap, that is, the difference between the two eigenvalues of the 2 × 2 initial velocity gradient.
TL;DR: This work analyzes EM resonance in slabs of two-phase dielectric photonic crystal materials using boundary-integral projections for time-harmonic electromagnetic (EM) fields, and presents numerical examples which demonstrate the effects of structural defects on the resonant properties of a crystal slab and surface waves supported by a die...
Abstract: Using boundary-integral projections for time-harmonic electromagnetic (EM) fields, and their numerical implementation, we analyze EM resonance in slabs of two-phase dielectric photonic crystal materials. We characterize resonant frequencies by a complex Floquet--Bloch dispersion relation $\omega = W(\beta)$ defined by the existence of a nontrivial nullspace of a pair of boundary-integral projections parameterized by the wave number $\beta$ and the time-frequency $\omega$. At resonant frequencies, the crystal slab supports a source-free EM field. We link complex resonant frequencies, where the imaginary part is small, to resonant scattering behavior of incident source fields at nearby real frequencies and anomalous transmission of energy through the slab. At a real resonant frequency, the source-free field supported by the slab is a bound state. We present numerical examples which demonstrate the effects of structural defects on the resonant properties of a crystal slab and surface waves supported by a die...
TL;DR: The homogenization of the Maxwell equations at fixed frequency is addressed in this paper and a new a priori estimate is proved as well as a new result on the correctors.
Abstract: The homogenization of the Maxwell equations at fixed frequency is addressed in this paper. The bulk (homogenized) electric and magnetic properties of a material with a periodic microstructure are found from the solution of a local problem on the unit cell by suitable averages. The material can be anisotropic and satisfies a coercivity condition. The exciting field is generated by an incident field from sources outside the material under investigation. A suitable sesquilinear form is defined for the interior problem, and the exterior Calderon operator is used to solve the exterior radiating fields. The concept of two-scale convergence is employed to solve the homogenization problem. A new a priori estimate is proved as well as a new result on the correctors.
TL;DR: In this article, the authors proposed a no response test to locate the support of a scatterer from knowledge of a far field pattern of a scattered acoustic wave using a set of sampling surfaces and a special test response.
Abstract: We describe a novel technique,which we call the no response test,to locate the support of a scatterer from knowledge of a far field pattern of a scattered acoustic wave. The method uses a set of sampling surfaces and a special test response to detect the support of a scatterer without a priori knowledge of the physical properties of the scatterer. Specifically,the method does not depend on information about whether the scatterer is penetrable or impenetrable nor does it depend on any knowledge of the nature of the scatterer (absorbing,reflecting,etc.). In contrast to previous sampling algorithms,the techniques described here enable one to locate obstacles or inhomogeneities from the far field pattern of only one incident field—the no response test is a one- wave method. We investigate the theoretical basis for the no response test and derive a one-wave uniqueness proof for a region containing the scatterer. We show how to find the object within this region. We demonstrate the applicability of the method by reconstructing sound-soft,sound-hard, impedance,and inhomogeneous medium scatterers in two dimensions from one wave with full and limited aperture far-field data.
TL;DR: A delay differential equation model for the interaction between two species, the adult members of which are in competition, and it is proved the existence of a reaction-diffusion extension of the model which involves the maturation delays.
Abstract: This paper is concerned with a delay differential equation model for the interaction between two species, the adult members of which are in competition. The competitive effects are of the Lotka--Volterra kind, and in the absence of competition it is assumed that each species evolves according to the predictions of a simple age-structured model which reduces to a single equation for the total adult population. For each of the two species the model incorporates a time delay which represents the time from birth to maturity of that species. Thus, the time delays appear in the adult recruitment terms.The dynamics of the model are determined, and global stability results are established for each equilibrium. The equilibria of the model involve the maturation delays. The criteria for global convergence to each equilibrium are sharp and involve these delays.A reaction-diffusion extension of the model is also studied for the case when only the adult members of each species can diffuse. We prove the existence of a ...
TL;DR: A quantitative explanation of time reversal and other more general refocusing phenomena for general classical waves in heterogeneous media is presented based on the asymptotic analysis of the Wigner transform of wave fields in the high frequency limit.
Abstract: In time reversal acoustics experiments, a signal is emitted from a localized source, recorded at an array of receivers, time reversed, and finally reemitted into the medium. A celebrated feature of time reversal experiments is that the refocusing of the reemitted signals at the location of the initial source is improved when the medium is heterogeneous. Contrary to intuition, multiple scattering enhances the spatial resolution of the refocused signal and allows one to beat the diffraction limit obtained in homogeneous media. This paper presents a quantitative explanation of time reversal and other more general refocusing phenomena for general classical waves in heterogeneous media. The theory is based on the asymptotic analysis of the Wigner transform of wave fields in the high frequency limit. Numerical experiments complement the theory.
TL;DR: Within the context of Lienard equations, the FitzHugh--Nagumo model with an idealized nonlinearity is presented with an analytical expression for the transient regime corresponding to the emission of a finite number of action potentials (or spikes) and for the asymptotic regime correspondingto the existence of a limit cycle.
Abstract: Within the context of Lienard equations, we present the FitzHugh--Nagumo model with an idealized nonlinearity. We give an analytical expression (i) for the transient regime corresponding to the emission of a finite number of action potentials (or spikes), and (ii) for the asymptotic regime corresponding to the existence of a limit cycle. We carry out a global analysis to study periodic solutions, the existence of which is linked to the solutions of a system of transcendental equations. The periodic solutions are obtained with the help of the harmonic balance method or as limit behavior of the transient regime. We show how the appearance of periodic solutions corresponds either to a fold limit cycle bifurcation or to a Hopf bifurcation at infinity. The resultsobtained are in agreement with local analysis methods, i.e., the Melnikov method and the averaging method. The generalization of the model leads us to formulate two conjectures concerning the number of limit cycles for the piecewise linear Lienard equ...
TL;DR: It is shown how the layered medium can be related to the Toda lattice, which has discrete soliton solutions, and how pseudospectral solutions to these equations agree well with direct solutions of the hyperbolic conservation laws in the layeredmedium.
Abstract: We study longitudinal elastic strain waves in a one-dimensional periodically layered medium, alternating between two materials with different densities and stress-strain relations. If the impedances are different, dispersive effects are seen due to reflection at the interfaces. When the stress-strain relations are nonlinear, the combination of dispersion and nonlinearity leads to the appearance of solitary waves that interact like solitons. We study the scaling properties of these solitary waves and derive a homogenized system of equations that includes dispersive terms. We show that pseudospectral solutions to these equations agree well with direct solutions of the hyperbolic conservation laws in the layered medium using a high-resolution finite volume method. For particular parameters we also show how the layered medium can be related to the Toda lattice, which has discrete soliton solutions.
TL;DR: A long wave model derived asymptotically from the nonlinear potential theory equations leads to a weakly nonlinear, weakly dispersive (variable coefficient) Boussinesq system valid for a wide class of topographies.
Abstract: A long wave model is derived asymptotically from the nonlinear potential theory equations. The flow regime of interest is incompressible, irrotational, and inviscid. Asymptotic analysis leads to a weakly nonlinear, weakly dispersive (variable coefficient) Boussinesq system valid for a wide class of topographies. The mild slope hypothesis is not required and rapidly varying topographies are also considered. In analogy with atmospheric models we use a terrain-following coordinate system. The novelty is that this coordinate system naturally suggests the weighted averaging of terrain-following velocity components, as opposed to the depth-average of horizontal velocity components found in standard shallow water formulations. Furthermore, a Schwarz--Christoffel toolbox is used to provide additional insight on these new results. Regarding applications, the proposed model can be used for studying solitary waves interacting with fine scale inhomogeneities, a theme of great interest. The terrain-following model als...
TL;DR: The rate at which blowup occurs is investigated in settings with certain symmetries, using the method of matched asymptotic expansions to identify a range of blowup scenarios in both finite and infinite time, including degenerate cases.
Abstract: The harmonic map heat flow is a model for nematic liquid crystals and also has origins in geometry. We present an analysis of the asymptotic behavior of singularities arising in this flow for a special class of solutions which generalizes a known (radially symmetric) reduction. Specifically, the rate at which blowup occurs is investigated in settings with certain symmetries, using the method of matched asymptotic expansions. We identify a range of blowup scenarios in both finite and infinite time, including degenerate cases.
TL;DR: The weak framework the authors introduce and its analysis are not restricted to the simple physics of the ice sheet model they consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings.
Abstract: This paper deals with the weak formulation of a free (moving) boundary problem arising in theoretical glaciology. Considering shallow ice sheet flow, we present the mathematical analysis and the numerical solution of the second order nonlinear degenerate parabolic equation modelling, in the isothermal case, the ice sheet non-Newtonian dynamics. An obstacle problem is then deduced and analyzed. The existence of a free boundary generated by the support of the solution is proved and its location and evolution are qualitatively described by using a comparison principle and an energy method. Then the solutions are numerically computed with a method of characteristics and a duality algorithm to deal with the resulting variational inequalities. The weak framework we introduce and its analysis (both qualitative and numerical) are not restricted to the simple physics of the ice sheet model we consider nor to the model dimension; they can be successfully applied to more realistic and sophisticated models related to other geophysical settings.
TL;DR: The solution for the diffraction by a wedge with given face impedances (the Malyuzhinets problem) is obtained in closed form by an explicit factorization of the kernel.
Abstract: Diffraction by impenetrable wedges having arbitrary aperture angle is studied by means of the Wiener--Hopf (W-H) technique. A system of functional equations called generalized Wiener--Hopf equations (GWHE) is obtained. Only for certain values of the aperture angle are these equations recognizable as standard or classical Wiener--Hopf equations (CWHE). However, in all cases a mapping is found that reduces the GWHE to CWHE. It means that the diffraction by an impenetrable wedge always reduces to a standard W-H factorization. The solution for the diffraction by a wedge with given face impedances (the Malyuzhinets problem) is obtained in closed form by an explicit factorization of the kernel.
TL;DR: The Evans function approach extended by the compound matrix method is employed to numerically solve the linear stability problem for the travelling wave solution.
Abstract: In this paper we investigate the linear stability and properties, such as speed, of the planar travelling combustion front. The speed of the front is estimated both analytically, using the matched asymptotic expansion, and numerically, by means of the shooting and relaxation methods. The Evans function approach extended by the compound matrix method is employed to numerically solve the linear stability problem for the travelling wave solution.
TL;DR: A new approach to the design of semiconductor devices is presented, which leads to fast optimization methods whose numerical effort is of the same order as a single forward simulation of the underlying model, the stationary drift-diffusion system.
Abstract: This paper presents a new approach to the design of semiconductor devices, which leads to fast optimization methods whose numerical effort is of the same order as a single forward simulation of the underlying model, the stationary drift-diffusion system. The design goal we investigate is to increase the outflow current on a contact for fixed applied voltage; the natural design variable is the doping profile.By reinterpreting the doping profile as a state variable and the electrostatic potential as the new design variable, we obtain a simpler optimization problem, whose Karush--Kuhn--Tucker conditions partially decouple. This property allows us to construct efficient iterative optimization algorithms, which avoid solving the fully coupled drift-diffusion system, and need only solves of the continuity equations and their adjoints. The efficiency and success of the new approach is demonstrated in several numerical examples.
TL;DR: It is shown that the classical Taylor dispersion phenomena are better described in terms of low dimensional models that are hyperbolic and contain an effective local time or length scale in place of the traditionalTaylor dispersion coefficient.
Abstract: We formulate a general theory, based on a Lyapunov--Schmidt expansion, for averaging thermal and solutal dispersion phenomena in multiphase reactors, with specific attention to the important Taylor mechanism due to transverse intraphase and interphase capacitance-weighted velocity gradients. We show that the classical Taylor dispersion phenomena are better described in terms of low dimensional models that are hyperbolic and contain an effective local time or length scale in place of the traditional Taylor dispersion coefficient. This description eliminates the use of an artificial exit boundary condition associated with parabolic homogenized equations as well as the classical upstream-feedback and infinite propagation speed anomalies. Our approach is also applicable for describing steady dispersion in the presence of reaction and thermal generation or consumption. For two-phase systems, maximum dispersion is found to exist at an optimum fraction $\epsilon _{f}$ of the lower-capacitance phase. For the disp...
TL;DR: It is proved that solving an integro-differential equation for the pressure in this region is equivalent to solving the scattering problem and it is shown that this approach is incomplete when the density is discontinuous.
Abstract: Acoustic scattering problems are considered when the material parameters (density and speed of sound) are functions of position within a bounded region. An integro-differential equation for the pressure in this region is obtained. It is proved that solving this equation is equivalent to solving the scattering problem. Problems of this kind are often solved by regarding the effects of the inhomogeneity as an unknown source term driving a Helmholtz equation, leading to an equation of Lippmann--Schwinger type. It is shown that this approach is incomplete when the density is discontinuous. Analogous scattering problems for elastic waves and for electromagnetic waves are also discussed briefly.
TL;DR: A mechanism of localization in a discrete system of relaxation oscillators globally coupled via inhibition based on the canard phenomenon is studied, showing that the larger the cluster size, the smaller is the corresponding critical value of γ, implying that it is the smaller cluster that oscillates at large amplitude.
Abstract: Localization in a discrete system of oscillators refers to the partition of the population into a subset that oscillates at high amplitudes and anotherthat oscillates at much loweramplitudes. Motivated by experimental results on the Belousov-Zhabotinsky reaction, which oscillates in the relaxation regime, we study a mechanism of localization in a discrete system of relaxation oscillators globally coupled via inhibition. The mechanism is based on the canard phenomenon for a single relaxation oscillator: a rapid explosion in the amplitude of the limit cycle as a parameter governing the relative position of the nullclines is varied. Starting from a parameter regime in which each uncoupled oscillatorhas a lar ge amplitude and no otherper iodic orotherstable solutions, we show that the canard phenomenon can be induced by increasing a global negative feedback parameter γ, with the network then partitioned into low and high amplitude oscillators. For the case in which the oscillators are synchronous within each of the two such populations, we can assign a canard-inducing critical value of γ separately to each of the two clusters; localization occurs when the value for the system is between the critical values of the two clusters. We show that the larger the cluster size, the smaller is the corresponding critical value of γ, implying that it is the smallerclusterthat oscillates at large amplitude. The theory shows that the above results come from a kind of self-inhibition of each cluster induced by the local feedback. In the full system, there are also effects of interactions between the clusters, and we present simulations showing that these nonlocal interactions do not destroy the localization created by the self-inhibition.
TL;DR: This work studies the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere and finds that the stability intervals coincide with those for linear stability determined by Polvani and Dritschel.
Abstract: We study the nonlinear stability of relative equilibria of configurations of identical point-vortices on the surface of a sphere. In particular, we study how the stability changes as a function of the colatitude $\theta$ and of the number of vortices N. By using the integrals of motion, we view the system in asuitable corotating frame where the polygonal vortex configuration is at rest. Then after a sufficient criterion due to Dirichlet, the stability ranges are the $\theta$-intervals for which the Hessian of the Hamiltonian---evaluated at the equilibrium configuration---is positive or negative definite. We find that the stability intervals coincide with those for linear stability determined by Polvani and Dritschel [J. Fluid Mech., 255 (1993), pp. 35--64]. For $N=3$ we recover the result previously established by Pekarsky and Marsden [J. Math. Phys., 39 (1998), pp. 5894--5907].