Counting spectrum via the Maslov index for one dimensional θ−\theta-periodic Schr\"odinger operators

We study the spectrum of the Schr\"odinger operators with $n\times n$ matrix valued potentials on a finite interval subject to $\theta-$periodic boundary conditions. For two such operators, corresponding to different values of $\theta$, we compute the difference of their eigenvalue counting functions via the Maslov index of a path of Lagrangian planes. In addition we derive a formula for the derivatives of the eigenvalues with respect to $\theta$ in terms of the Maslov crossing form. Finally, we give a new shorter proof of a recent result relating the Morse and Maslov indices of the Schr\"odinger operator for a fixed $\theta$

Approximation of Random Slow Manifolds and Settling of Inertial Particles under Uncertainty

A method is provided for approximating random slow manifolds of a class of slow-fast stochastic dynamical systems. Thus approximate, low dimensional, reduced slow systems are obtained analytically in the case of sufficiently large time scale separation. To illustrate this dimension reduction procedure, the impact of random environmental fluctuations on the settling motion of inertial particles in a cellular flow field is examined. It is found that noise delays settling for some particles but enhances settling for others. A deterministic stable manifold is an agent to facilitate this phenomenon. Overall, noise appears to delay the settling in an averaged sense.Comment: 27 pages, 9 figure

Results and conjectures on simultaneous core partitions

An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that (2n)- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type C_n, generalizing a result of Fishel--Vazirani for type A. We also introduce a major statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate simultaneous (2n)- and (2n+1)-core partitions that yield q-analogues of the Coxeter-Catalan numbers of type A and type C. We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q,t-Catalan numbers.Comment: 17 pages; to appear in the European Journal of Combinatoric

A Morse index theorem for elliptic operators on bounded domains

Given a selfadjoint, elliptic operator $L$, one would like to know how the spectrum changes as the spatial domain $\Omega \subset \mathbb{R}^d$ is deformed. For a family of domains $\{\Omega_t\}_{t\in[a,b]}$ we prove that the Morse index of $L$ on $\Omega_a$ differs from the Morse index of $L$ on $\Omega_b$ by the Maslov index of a path of Lagrangian subspaces on the boundary of $\Omega$. This is particularly useful when $\Omega_a$ is a domain for which the Morse index is known, e.g. a region with very small volume. Then the Maslov index computes the difference of Morse indices for the "original" problem (on $\Omega_b$) and the "simplified" problem (on $\Omega_a$). This generalizes previous multi-dimensional Morse index theorems that were only available on star-shaped domains or for Dirichlet boundary conditions. We also discuss how one can compute the Maslov index using crossing forms, and present some applications to the spectral theory of Dirichlet and Neumann boundary value problems.Comment: 21 pages; weaker regularity assumptions than in the first versio

On the Existence and Stability of Fast Traveling Waves in a Doubly-Diffusive FitzHugh-Nagumo System

The FitzHugh-Nagumo equation, which was derived as a simplification of the Hodgkin-Huxley model for nerve impulse propagation, has been extensively studied as a paradigmatic activator-inhibitor system. We consider the version of this system in which two agents diffuse at an equal rate. Using geometric singular perturbation theory, we prove the existence and stability of fast traveling pulses. The stability proof makes use of the Maslov index--an invariant of symplectic geometry--to count unstable eigenvalues for the linearization about the wave. The calculation of the Maslov index is carried out by tracking the evolution of the unstable manifold of the rest state using the timescale separation. This entails a careful consideration of how the transition from fast to slow dynamics occurs in the tangent bundle over the wave. Finally, we observe in the calculation that the Maslov index lacks monotonicity in the spatial parameter, which distinguishes this application of the Maslov index from similar analyses of Hamiltonian systems.Comment: 3 figure

A sex difference in the context-sensitivity of dominance perceptions

Although dominance perceptions are thought to be important for effective social interaction, their primary function is unclear. One possibility is that they simply function to identify individuals who are capable of inflicting substantial physical harm, so that the perceiver can respond to them in ways that maximize their own physical safety. Another possibility is that they are more specialized, functioning primarily to facilitate effective direct (i.e., violent) intrasexual competition for mates, particularly among men. Here we used a priming paradigm to investigate these two possibilities. Facial cues of dominance were more salient to women after they had been primed with images of angry men, a manipulation known to activate particularly strong self-protection motivations, than after they had been primed with images of angry women or smiling individuals of either sex. By contrast, dominance cues were more salient to men after they had been primed with images of women than when they had been primed with images of men (regardless of the emotional expressions displayed), a manipulation previously shown to alter men's impressions of the sex ratio of the local population. Thus, men's dominance perceptions appear to be specialized for effective direct competition for mates, while women's dominance perceptions may function to maximize their physical safety more generally. Together, our results suggest that men's and women's dominance perceptions show different patterns of context-sensitivity and, potentially, shed new light on the routes through which violence and intrasexual competition have shaped dominance perceptions