CLT with explicit variance for products of random singular matrices related to Hill’s equation

We prove a central limit theorem (CLT) for the product of a class of random singular matrices related to a random Hill’s equation studied by Adams–Bloch–Lagarias. The CLT features an explicit formula for the variance in terms of the distribution of the matrix entries and this allows for exact calculation in some examples. Our proof relies on a novel connection to the theory of [Formula: see text]-dependent sequences which also leads to an interesting and precise nondegeneracy condition.

Optimality conditions for nonconvex problems over nearly convex feasible sets

In this paper, we study the optimization problem (P) of minimizing a convex function over a constraint set with nonconvex constraint functions. We do this by given new characterizations of Robinson’s constraint qualification, which reduces to the combination of generalized Slater’s condition and generalized sharpened nondegeneracy condition for nonconvex programming problems with nearly convex feasible sets at a reference point. Next, using a version of the strong CHIP, we present a constraint qualification which is necessary for optimality of the problem (P). Finally, using new characterizations of Robinson’s constraint qualification, we give necessary and sufficient conditions for optimality of the problem (P).

Codimension-3 Flip Bifurcation of a Class of Difference Equations

In this paper, we consider a one-dimensional difference equation with three parameters, the derivative of which at a fixed point has an eigenvalue [Formula: see text] as the parameters are all zero. In the case that both nondegeneracy conditions of the flip bifurcation and the generalized flip bifurcation are not satisfied, by computing normal form, we give the nondegeneracy condition and transversality condition of the codimension-3 flip bifurcation. Moreover, by discussing the number of positive zeros of a cubic function in a neighborhood of the origin, we show the bifurcation scenario and give the parameter conditions, respectively, that the normal form of the equation possesses three 2-cycles, two 2-cycles, only one 2-cycle or none.

Improved Convergence Properties of the Relaxation Schemes of Kadrani et al. and Kanzow and Schwartz for MPEC

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

-PURITY VERSUS LOG CANONICITY FOR POLYNOMIALS

In this article, we consider the conjectured relationship between $F$-purity and log canonicity for polynomials over $\mathbb{C}$. In particular, we show that log canonicity corresponds to dense $F$-pure type for all polynomials whose supporting monomials satisfy a certain nondegeneracy condition. We also show that log canonicity corresponds to dense $F$-pure type for very general polynomials over $\mathbb{C}$. Our methods rely on showing that the $F$-pure and log canonical thresholds agree for infinitely many primes, and we accomplish this by comparing these thresholds with the thresholds associated to their monomial term ideals.

On the Dynamics Near a Homoclinic Network to a Bifocus: Switching and Horseshoes

We study a homoclinic network associated to a nonresonant hyperbolic bifocus. It is proved that on combining rotation with a nondegeneracy condition concerning the intersection of the two-dimensional invariant manifolds of the equilibrium, switching behavior is created: close to the network, there are trajectories that visit the neighborhood of the bifocus following connections in any prescribed order. We discuss the existence of suspended horseshoes which accumulate on the network and the relation between these horseshoes and the switching behavior.

NONPARAMETRIC IDENTIFICATION OF POSITIVE EIGENFUNCTIONS

Important features of certain economic models may be revealed by studying positive eigenfunctions of appropriately chosen linear operators. Examples include long-run risk–return relationships in dynamic asset pricing models and components of marginal utility in external habit formation models. This paper provides identification conditions for positive eigenfunctions in nonparametric models. Identification is achieved if the operator satisfies two mild positivity conditions and a power compactness condition. Both existence and identification are achieved under a further nondegeneracy condition. The general results are applied to obtain new identification conditions for external habit formation models and for positive eigenfunctions of pricing operators in dynamic asset pricing models.

NONAUTONOMOUS BIFURCATION OF BOUNDED SOLUTIONS: CROSSING CURVE SITUATIONS

Carathéodory differential equations naturally occur as path-wise realization of random differential equations and are amenable for deterministic calculus. In the setup of such nonautonomous differential equations with only measurable time-dependence, we present an approach to a bifurcation theory based on a topological change in the set of bounded entire solutions. In such a setting of at least planar equations, we provide sufficient criteria for a nonhyperbolic entire solution to bifurcate into two branches of bounded or homoclinic solutions. As opposed to transcritical or pitchfork bifurcations, no trivial solution branch is supposed to exist in advance. In particular, we discuss a degenerate fold bifurcation pattern, where the transversality assumption is replaced by a nondegeneracy condition on the second-order derivative. Both bifurcation patterns are intrinsically nonautonomous and do not occur for time-invariant equations. Our notion of a nonhyperbolic solution is based on the fact that the associate variational equation possesses exponential dichotomies on both semiaxes with compatible projectors. The resulting Fredholm theory allows one to apply recent abstract bifurcation results due to Liu, Shi and Wang (2007).