Existence and globally asymptotical stability of periodic solutions for two-species non-autonomous diffusion competition n-patch system with time delay and impulses

By using Mawhin continuation theorem and constructing suitable Lyapunov functional, a set of easily verifiable sufficient conditions ensuring the existence and global asymptotical stability of positive periodic solutions for two-species non-autonomous diffusion competition n-patch system with time delay and impulses are established. Finally, applications and an illustrative example are given. The main results generalize and improve the previously known results in Zhang and Wang (J. Math. Anal. Appl. 265:38–48, 2002), Dong et al. (Chaos Solitons Fractals 32:1916–1926, 2007).     

A minimum entropy principle of high order schemes for gas dynamics equations

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy (Tadmor in Appl Numer Math 2:211–219, 1986). First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes (Khobalatte and Perthame in Math Comput 62:119–131, 1994) also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations in Zhang and Shu (J Comput Phys 229:8918–8934, 2010) and Zhang et?al. (J Scientific Comput, in press), to enforce the minimum entropy principle for high order finite volume and discontinuous Galerkin (DG) schemes.     

Foulkes characters, Eulerian idempotents, and an amazing matrix

John Holte (Am. Math. Mon. 104:138?C149, 1997) introduced a family of ??amazing matrices?? which give the transition probabilities of ??carries?? when adding a list of numbers. It was subsequently shown that these same matrices arise in the combinatorics of the Veronese embedding of commutative algebra (Brenti and Welker, Adv. Appl. Math. 42:545?C556, 2009; Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009) and in the analysis of riffle shuffling (Diaconis and Fulman, Am. Math. Mon. 116:788?C803, 2009; Adv. Appl. Math. 43:176?C196, 2009). We find that the left eigenvectors of these matrices form the Foulkes character table of the symmetric group and the right eigenvectors are the Eulerian idempotents introduced by Loday (Cyclic Homology, 1992) in work on Hochschild homology. The connections give new closed formulae for Foulkes characters and allow explicit computation of natural correlation functions in the original carries problem.     

On Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime

We apply the well know Omori?CYau generalized maximum principle (Omori in J Math Soc Jpn 19:205?C214, 1967; Yau in Commun Pure Appl Math 28:201?C228, 1975), as well as a suitable extension of it that was established in a joint work with Caminha (Caminha and de Lima in Gen Relat Grav 41:173?C189, 2009), in order to investigate Bernstein-type properties of complete spacelike hypersurfaces immersed in a generalized Robertson?CWalker spacetime, which is supposed to obey a standard convergence condition.     

Blow up for the critical generalized Korteweg–de Vries equation. I: Dynamics near the soliton

We consider the quintic generalized Korteweg–de Vries equation (gKdV) $$u_t + (u_{xx} + u^5)_x =0,$$ which is a canonical mass critical problem, for initial data in H ¹ close to the soliton. In earlier works on this problem, finite- or infinite-time blow up was proved for non-positive energy solutions, and the solitary wave was shown to be the universal blow-up profile, see [16], [26] and [20]. For well-localized initial data, finite-time blow up with an upper bound on blow-up rate was obtained in [18]. In this paper, we fully revisit the analysis close to the soliton for gKdV in light of the recent progress on the study of critical dispersive blow-up problems (see [31], [39], [32] and [33], for example). For a class of initial data close to the soliton, we prove that three scenarios only can occur: (i) the solution leaves any small neighborhood of the modulated family of solitons in the scale invariant L ² norm; (ii) the solution is global and converges to a soliton as t → ∞; (iii) the solution blows up in finite time T with speed $$\|u_x(t)\|_{L^2} \sim \frac{C(u_0)}{T-t} \quad {\rm as}\, t\to T.$$ Moreover, the regimes (i) and (iii) are stable. We also show that non-positive energy yields blow up in finite time, and obtain the characterization of the solitary wave at the zero-energy level as was done for the mass critical non-linear Schrödinger equation in [31].     

A computational study of the weak Galerkin method for second-order elliptic equations

The weak Galerkin finite element method is a novel numerical method that was first proposed and analyzed by Wang and Ye (2011) for general second order elliptic problems on triangular meshes. The goal of this paper is to conduct a computational investigation for the weak Galerkin method for various model problems with more general finite element partitions. The numerical results confirm the theory established in Wang and Ye (2011). The results also indicate that the weak Galerkin method is efficient, robust, and reliable in scientific computing.     

A discontinuous Galerkin scheme for front propagation with obstacles

We are interested in front propagation problems in the presence of obstacles. We extend a previous work (Bokanowski et al. SIAM J Sci Comput 33(2):923–938, 2011), to propose a simple and direct discontinuous Galerkin (DG) method adapted to such front propagation problems. We follow the formulation of Bokanowski et al. (SIAM J Control Optim 48(7):4292–4316, (2010)), leading to a level set formulation driven by $\min (u_t + H(x,\nabla u), u-g(x))=0$ , where $g(x)$ is an obstacle function. The DG scheme is motivated by the variational formulation when the Hamiltonian $H$ is a linear function of $\nabla u$ , corresponding to linear convection problems in the presence of obstacles. The scheme is then generalized to nonlinear equations, written in an explicit form. Stability analysis is performed for the linear case with Euler forward, a Heun scheme and a Runge-Kutta third order time discretization using the technique proposed in Zhang and Shu (SIAM J Numer Anal 48:1038–1063, 2010). Several numerical examples are provided to demonstrate the robustness of the method. Finally, a narrow band approach is considered in order to reduce the computational cost.     

Existence of solutions for models of shallow water in a basin with a degenerate varying bottom

We prove the existence of solutions for the great lake equations. These equations are obtained from the three-dimensional Euler equations in a basin with a free upper surface and a spatially varying bottom topography by taking a low aspect ratio, i.e., low wave speed and small wave amplitude expansion. These equations are rewritten in an abstract form by considering generalized Euler equations as in Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996). This paper is an extension of Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996), where the varying bottom was assumed to be nondegenerate. Here, we discuss the degenerate case and obtain similar results as in Levermore et?al. (Indiana Univ Math J 45:479?C510, 1996).     

A series of new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation

In this paper, we extend the algebraic method proposed by Fan (Chaos, Solitons & Fractals 20 (2004) 609) and the improved extended tanh method by Yomba (Chaos, Solitons and Fractals 20 (2004) 1135) to uniformly construct a series of soliton-like solutions and double-like periodic solutions for nonlinear partial differential equations (NPDE). Some new soliton-like solutions and double-like periodic solutions of a (2 + 1)-dimensional dispersive long wave equation are obtained.     

Linearly Implicit Approximations of Diffusive Relaxation Systems

Diffusive relaxation systems provide a general framework to approximate nonlinear diffusion problems, also in the degenerate case (Aregba-Driollet et al. in Math. Comput. 73(245):63–94, 2004; Boscarino et al. in Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, 2011; Cavalli et al. in SIAM J. Sci. Comput. 34:A137–A160, 2012; SIAM J. Numer. Anal. 45(5):2098–2119, 2007; Naldi and Pareschi in SIAM J. Numer. Anal. 37:1246–1270, 2000; Naldi et al. in Surveys Math. Indust. 10(4):315–343, 2002). Their discretization is usually obtained by explicit schemes in time coupled with a suitable method in space, which inherits the standard stability parabolic constraint. In this paper we combine the effectiveness of the relaxation systems with the computational efficiency and robustness of the implicit approximations, avoiding the need to resolve nonlinear problems and avoiding stability constraints on time step. In particular we consider an implicit scheme for the whole relaxation system except for the nonlinear source term, which is treated though a suitable linearization technique. We give some theoretical stability results in a particular case of linearization and we provide insight on the general case. Several numerical simulations confirm the theoretical results and give evidence of the stability and convergence also in the case of nonlinear degenerate diffusion.     

Non-Decaying Solutions to the Navier Stokes Equations in Exterior Domains

We establish a new theorem of existence (and uniqueness) of solutions to the Navier-Stokes initial boundary value problem in exterior domains. No requirement is made on the convergence at infinity of the kinetic field and of the pressure field. These solutions are called non-decaying solutions. The first results on this topic dates back about 40 years ago see the references (Galdi and Rionero in Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980; Knightly in SIAM J. Math. Anal. 3:506–511, 1972). In the articles Galdi and Rionero (Ann. Mat. Pures Appl. 108:361–366, 1976, Arch. Ration. Mech. Anal. 62:295–301, 1976, Arch. Ration. Mech. Anal. 69:37–52, 1979, Pac. J. Math. 104:77–83, 1980) it was introduced the so called weight function method to study the uniqueness of solutions. More recently, the problem has been considered again by several authors (see Galdi et al. in J. Math. Fluid Mech. 14:633–652, 2012, Quad. Mat. 4:27–68, 1999, Nonlinear Anal. 47:4151–4156, 2001; Kato in Arch. Ration. Mech. Anal. 169:159–175, 2003; Kukavica and Vicol in J. Dyn. Differ. Equ. 20:719–732, 2008; Maremonti in Mat. Ves. 61:81–91, 2009, Appl. Anal. 90:125–139, 2011).     

Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

We apply the discrete version of Calderón??s reproducing formula and Littlewood?CPaley theory with weights to establish the $H^{p}_{w} \to H^{p}_{w}$ (0<p<??) and $H^{p}_{w}\to L^{p}_{w}$ (0<p??1) boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume w??A _??. The bounds will be expressed in terms of the A _q constant of w if q>q _w =inf?{s:w??A _s }. Our results can be regarded as a natural extension of the results about the growth of the A _p constant of singular integral operators on classical weighted Lebesgue spaces $L^{p}_{w}$ in Hytonen et al. (arXiv:1006.2530, 2010; arXiv:0911.0713, 2009), Lerner (Ill.?J.?Math. 52:653?C666, 2008; Proc. Am. Math. Soc. 136(8):2829?C2833, 2008), Lerner et?al. (Int.?Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149?C156, 2009), Lacey et?al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355?C1375, 2007; Proc. Am. Math. Soc. 136(4):1237?C1249, 2008), and Petermichl and Volberg (Duke Math. J. 112(2):281?C305, 2002). Our main result is stated in Theorem?1.1. Our method avoids the atomic decomposition which was usually used in proving boundedness of singular integral operators on Hardy spaces.     

Efficient model for interval goal programming with arbitrary penalty function

Penalty function is a key factor in interval goal programming (IGP), especially for decision makers weighing resources vis-à-vis goals. Many approaches (Chang et al. J Oper Res Soc 57:469–473, 2006; Chang and Lin Eur J Oper Res 199, 9–20, 2009; Jones et al. Omega 23, 41–48, 1995; Romero Eur J Oper Res 153, 675–686, 2004; Vitoriano and Romero J Oper Res Soc 50, 1280–1283, 1999)have been proposed for treating several types of penalty functions in the past several decades. The recent approach of Chang and Lin (Eur J Oper Res 199, 9–20, 2009) considers the S-shaped penalty function. Although there are many approaches cited in literature, all are complicated and inefficient. The current paper proposes a novel and concise uniform model to treat any arbitrary penalty function in IGP. The efficiency and usefulness of the proposed model are demonstrated in several numeric examples.     

A remark on doubly critical elliptic systems

We obtain positive solutions for some doubly critical elliptic systems via variational methods. This result improves some existence results of Abdellaoui, Felli and Peral (Calc Var 34:97–137 2009). Our arguments are completely different from those in (Calc Var 34:97–137 2009).     

Shorter identity-based encryption via asymmetric pairings

We present efficient identity-based encryption (IBE) under the symmetric external Diffie–Hellman (SXDH) assumption in bilinear groups; our scheme also achieves anonymity. In our IBE scheme, all parameters have constant numbers of group elements, and are shorter than those of previous constructions based on decisional linear (DLIN) assumption. Our construction uses both dual system encryption (Waters, CRYPTO 2009) and dual pairing vector spaces (Okamoto and Takashima, Pairing 2008; ASIACRYPT 2009). Specifically, we show how to adapt the recent DLIN-based instantiation of Lewko (EUROCRYPT 2012) to the SXDH assumption. To our knowledge, this is the first work to instantiate either dual system encryption or dual pairing vector spaces under the SXDH assumption. Furthermore, our work could be extended to many other functional encryption. In Particular, we show how to instantiate our framework to inner product encryption and key-policy functional encryption. All parameters of our constructions are shorter than those of DLIN-based constructions.     

Nonparametric density estimation for randomly perturbed elliptic problems III: convergence, computational cost, and generalizations

This is the third in a series of three papers on nonparametric density estimation for randomly perturbed elliptic problems. In the previous papers by Estep, M?lqvist, and Tavener (SIAM J. Sci. Comput. 31:2935?C2959,?2009; Int. J. Numer. Methods Eng. 80:846?C867,?2009), we derive an a posteriori error estimate for a computed probability distribution and an efficient adaptive algorithm for propagation of uncertainty into a quantity of interest computed from numerical solutions of an elliptic partial differential equation. We also test the algorithm on various problems including an example relevant to oil reservoir simulation. In this paper, we derive a convergence result for the method based on the assumption that the underlying domain decomposition algorithm converges geometrically. The main ideas of the proof can be applied to a large class of domain decomposition algorithms. We also present several generalizations of the method and an analysis of the computational cost.     

Irreversible investments with delayed reaction: an application to generation re-dispatch in power system operation

In this article we consider how the operator of an electric power system should activate bids on the regulating power market in order to minimize the expected operation cost. Important characteristics of the problem are reaction times of actors on the regulating market and ramp-rates for production changes in power plants. Neglecting these will in general lead to major underestimation of the operation cost. Including reaction times and ramp-rates leads to an impulse control problem with delayed reaction. Two numerical schemes to solve this problem are proposed. The first scheme is based on the least-squares Monte Carlo method developed by Longstaff and Schwartz (Rev Financ Stud 14:113–148, 2001). The second scheme which turns out to be more efficient when solving problems with delays, is based on the regression Monte Carlo method developed by Tsitsiklis and van Roy (IEEE Trans Autom Control 44(10):1840–1851, 1999) and (IEEE Trans Neural Netw 12(4):694–703, 2001). The main contribution of the article is the idea of using stochastic control to find an optimal strategy for power system operation and the numerical solution schemes proposed to solve impulse control problems with delayed reaction.     

Dynamical system solutions for homogeneous linear functional equations of higher order

We present the solution of a large class of homogeneous linear functional equations of higher order by using ideas from dynamical systems. A particularly simple example from this class is the functional equation $$f(x) = \frac{1}{2}f \left(\frac{x}{2}\right) + \frac{1}{2}f \left(\frac{x+1}{2}\right), \quad 0 < x < 1.$$ Equations such as these have found important applications in wavelet theory by Hilberdink (Aequa Math 61(1–2):179–189, 2001) where they are called dilation equations and are usually solved by Fourier methods by Daubechies (Comm Pure Appl Math 41(7):909–996, 1988) or iteration methods of Daubechies (SIAM J Math Anal 22(5):1388–1410, 1991). A recent result of Góra (Ergod Theory Dyn Syst 29(5):1549–1583, 2009) allows us to represent the solution as an infinite series that is determined by the dynamics of a map that is defined by the functional equation. In this problem the interplay between dynamical systems and solutions of functional equations is brought into sharp focus.     

On the Hald–Gesztesy–Simon Theorem

The method (Martynyuk and Pivovarchik, Inverse Probl. 26(3):035011, 2010) of recovering the potential of the Sturm–Liouville equation on a half of the interval by the spectrum of a boundary value problem and by the restriction of the potential onto the other half of the interval is used for treating the missing eigenvalue problem (Trans. Am. Math. Soc. 352:2765–3789, 2000, J. R. Astr. Soc. 62:41–48, 1980, J. Math. Pures Appl. 91:468–475, 2009, J. Math. Soc. Japan 38:39–65, 1986). The latter arises in the case of the half-inverse (Hochstadt–Lieberman) problem with Robin boundary conditions and lies in the fact that in many cases all the eigenvalues but one are needed to recover the potential and the Robin condition at one of the ends.     

Slow Modulations of Periodic Waves in Hamiltonian PDEs,with Application to Capillary Fluids

Since its elaboration by Whitham almost 50 years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction–diffusion systems were first considered by Doelman et al. (Mem Am Math Soc 199(934):viii+105, 2009), and viscous systems of conservation laws have been addressed by Johnson et al. (Invent Math, 2013). Here, only nondissipative models are considered, and a most basic question is investigated, namely, the expected link between the hyperbolicity of modulated equations and the spectral stability of periodic traveling waves to sideband perturbations. This is done first in an abstract Hamiltonian framework, which encompasses a number of dispersive models, in particular the well-known (generalized) Korteweg–de Vries equation and the less known Euler–Korteweg system, in both Eulerian coordinates and Lagrangian coordinates. The latter is itself an abstract framework for several models arising in water wave theory, superfluidity, and quantum hydrodynamics. As regards its application to compressible capillary fluids, attention is paid here to untangle the interplay between traveling waves/modulation equations in Eulerian coordinates and those in Lagrangian coordinates. In the most general setting, it is proved that the hyperbolicity of modulated equations is indeed necessary for the spectral stability of periodic traveling waves. This extends earlier results by Serre (Commun Partial Differ Equ 30(1–3):259–282, 2005), Oh and Zumbrun (Arch Ration Mech Anal 166(2):99–166, 2003), and Johnson et al. (Phys D 239(23–24):2057–2065, 2010). In addition, reduced necessary conditions are obtained in the small-amplitude limit. Then numerical investigations are carried out for the modulated equations of the Euler–Korteweg system with two types of “pressure” laws, namely, the quadratic law of shallow-water equations and the nonmonotone van der Waals pressure law. Both the evolutionarity and the hyperbolicity of the modulated equations are tested, and regions of modulational instability are thus exhibited.