On a class of new and practical performance indexes for approximation of fold bifurcations of nonlinear power flow equations

Efficient measurement of the performance index (the distance of a loading parameter from the voltage collapse point) is one of the key problems in power system operations and planning and such an index indicates the severity of a power system with regard to voltage collapse. There exist many interesting methods and ideas to compute this index. However, some successful methods are not yet mathematically justified while other mathematically sound methods are often proposed directly based on the bifurcation theory and they require the initial stationary state to be too close to the unknown turning point to make the underlying methods practical.This paper first gives a survey of several popular methods for estimating the fold bifurcation point including the continuation methods, bifurcation methods and the test function methods (Seydel's direct solution methods, the tangent vector methods and the reduced Jacobian method) and discuss their relative advantages and problems. Test functions are usually based on scaling of the determinant of the Jacobian matrix and it is generally not clear how to determine the behaviour of such functions. As the underlying nonlinear equations are of a particular type, this allows us to do a new analysis of the determinants of the Jacobian and its submatrices in this paper. Following the analysis, we demonstrate how to construct a class of test functions with a predictable analytical behaviour so that a suitable index can be produced. Finally, examples of two test functions from this class are proposed. For several standard IEEE test systems, promising numerical results have been achieved.     

半线性椭圆方程正解的存在性及熄灭现象

本文在环形域上证明了带有奇异性的半线性椭圆方程边值问题(1)、(2·)在C~2(Ω)∩C~0(Ω)中正解的存在性,且这个解是径向对称的,解关于R_0(R_1)的连续单调性.还给出了解有熄灭现象的充分必要条件.     

Triple Positive Solutions of Boundary Value Problems for High-order Fractional Differential Equation at Resonance with Singularities

In this paper, we investigate the existence of triple positive solutions of boundary value problems for high-order fractional differential equation at resonance with singularities by using the fixed point index theory and the Leggett-Williams theorem. The spectral theory and some new height functions are also employed to establish the existence of triple positive solutions. The nonlinearity involved is arbitrary fractional derivative, and permits singularity.     

Derivation of invariant varieties of periodic points from singularity confinement in the case of Toda map

We have shown in [1] that the invariant varieties of periodic points (IVPP) of all periods of some higher dimensional rational maps can be derived, iteratively, from the singularity confinement (SC). We generalize this algorithm, in this paper, to apply to any birational map, which has more invariants than the half of the dimension.     

Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides

In this paper, we establish the existence of viscosity solutions of Hessian equations with singular right-hand sides and obtain the asymptotic boundary behavior of solutions. The asymptotic results generalize those for Poisson equations and Monge-Ampère equations, and are more precise than obtained from Hopf lemma.     

Peters v. Roseau: Resolution of a conflict on the classical beach problem

A discrepancy is uncovered between two well documented solutions, to the uniquely solvable classical linearised sloping beach problem, by Peters in 1952 and independently by Roseau also in 1952 (though subsequently adjusted by personal correspondence from Roseau and Lehman to authors Wehausen & Laitone in 1960). The source of the main error (and other more minor errors) is identified, quantified and corrected and a consequence is that a study by Morris in 1976 (on a generalisation where the line source is no longer confined to the origin) will also have suffered from this deficiency. That problem therefore requires a re-visit.     

Singularities in Quasi-Banach Spaces with Applications to Semilinear Elliptic Equations I

The paper deals with the extension of the singularity theory which was elaborated for Banach spaces to quasi-Banach spaces. For general quasi-Banach spaces some striking tools fail: not locally convex, the dual space may be trivial,…. Applying the concept of dual rich quasi - Banach spaces and results about Fredholm maps some results of the classical theory can be carried over. As an application we show in the continuation of this part how one can use these results for the investigation of the solution structure of semilinear elliptic boundary value problems in function Rpaces of Besov - Triebel - Lizorkin type.     

Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three-fold Zero Eigenvalue

In this paper we study the singularity at the origin with three-fold zero eigenvalue forsymmetric vector fields with nilpotent linear part and 3-jet C^∞-equivalent to y δ/δx zδ/δy ax^2yδ/δz with a≠0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularityby using the normal form and blow-up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil‘nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel‘nikov methods of a homoclinic orbit to a hyperbolic or non-hyperbolic equilibrium in a highdimensional space.     

Singularity-free cosmological solutions with non-rotating perfect fluids

A conjecture stated by Raychaudhuri which claims that the only physical perfect fluid non-rotating non-singular cosmological models are comprised in the Ruiz–Senovilla and Fernández–Jambrina families is shown to be incorrect. An explicit counterexample is provided and the failure of the argument leading to the result is explicitly pointed out.     

Sign-changing solutions for elliptic problems with critical Sobolev-Hardy exponents

Let be a smooth bounded domain such that 0∈Ω, N?7, 0?s<2, 2∗(s)=2(N−s)/(N−2). We prove the existence of sign-changing solutions for the singular critical problem −Δu−μ(u/|x|2)=(|u|2∗(s)−2/|x|s)u+λu with Dirichlet boundary condition on Ω for suitable positive parameters λ and μ.