The Euler characteristic heuristic has been proposed as a method for approximating the upper tail probability of the maximum of a random field with smooth sample path. When the random field is Gaussian, this method is proved to be valid in the sense that the relative approximation error is exponentially smaller. However, very little is known about the validly of the method when the random field...

We remark on two free-boundary problems for holomorphic curves in nearly-Kähler 6-manifolds. First, we observe that a curve geodesic ball B of the round 6-sphere meets $$\partial B$$ orthogonally must be totally geodesic. Consequently, obtain rigidity results reflection-invariant $$\mathbb {S}^6$$ and associative cones {R}^7$$ . Second, consider with boundary Lagrangian submanifold strict 6-man...

Abstract: We obtain formulae for the expected number and height distribution of critical points of general smooth isotropic Gaussian random fields parameterized on Euclidean space or spheres of arbitrary dimension. The results hold in general in the sense that there are no restrictions on the covariance function of the field except for smoothness and isotropy. The results are based on a charact...

The simplest type of dune is the transverse one, which propagates with invariant profile orthogonally to a fixed wind direction. Here we show, by means of numerical simulations, that transverse dunes are unstable with respect to along-axis perturbations in their profile and decay on the bedrock into barchan dunes. Any forcing modulation amplifies exponentially with growth rate determined by the...

Simple cells in primary visual cortex exhibit contrast-invariant orientation tuning, in seeming contradiction to feed-forward models that rely on lateral geniculate nucleus (LGN) input alone. Contrast invariance has therefore been thought to depend on the presence of intracortical lateral inhibition. In vivo intracellular recordings instead suggest that contrast invariance can be explained by t...

Lattice actions living on the direction of the space of couplings which originates at the fixed point of a renormalization group (RG) transformation and leaves orthogonally the critical surface, are called fixed point (FP) actions. They are classical perfect actions and have many interesting properties [1]. In particular, FP actions possess scale-invariant instanton solutions. Given a RG transf...

This article presents maximum likelihood estimators (MLEs) and log-likelihood ratio (LLR) tests for the eigenvalues and eigenvectors of Gaussian random symmetric matrices of arbitrary dimension, where the observations are independent repeated samples from one or two populations. These inference problems are relevant in the analysis of diffusion tensor imaging data and polarized cosmic backgroun...

Statistical mechanics approach is useful not only in analyzing macroscopic system performance of wireless communication systems, but also in discussing design problems of wireless communication systems. In this paper, we discuss a design problem of spreading sequences in code-division multipleaccess (CDMA) systems, as an example demonstrating the usefulness of statistical mechanics approach. We...