Let Qn(x) = ∑n i=0 Aix i be a random algebraic polynomial where the coefficients A0, A1, · · · form a sequence of centered Gaussian random variables. Moreover, assume that the increments ∆j = Aj−Aj−1, j = 0, 1, 2, · · · are independent, assuming A−1 = 0. The coefficients can be considered as n consecutive observations of a Brownian motion. We obtain the asymptotic behaviour of the expected numb...

We establish an upper bound for the Thurston–Bennequin number of a Legendrian link using the Khovanov homology of the underlying topological link. This bound is sharp in particular for all alternating links, and knots with nine or fewer crossings.

We compute the maximal Thurston-Bennequin number for a Legendrian two-bridge knot or oriented two-bridge link in standard contact R, by showing that the upper bound given by the Kauffman polynomial is sharp. As an application, we present a table of maximal Thurston-Bennequin numbers for prime knots with nine or fewer crossings.

In this paper, we present an approach for image reconstruction from local phase vectors in the monogenic scale space. The local phase vector contains not only the local phase but also the local orientation of the original signal, which enables the simultaneous estimation of the structural and geometric information. Consequently, the local phase vector preserves a lot of important information of...

We give sharp two-sided linear bounds of the crosscap number (nonorientable genus) of alternating links in terms of their Jones polynomial. Our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings. We also discuss generalizations of our results for classes of n...

The influence theorem for product measures on the discrete space {0, 1}N may be extended to probability measures with the property of monotonicity (which is equivalent to ‘strong positive-association’). Corresponding results are valid for probability measures on the cube [0, 1]N that are absolutely continuous with respect to Lebesgue measure. These results lead to a sharp-threshold theorem for ...

We present a diabatization technique in combination with the recently developed hyperspherical close coupling (HSCC) method. In contrast to the strict diabatization, our simple diabatization procedure transforms only sharp avoided crossings in the adiabatic hyperspherical potential curves into real crossings. With this approach, the weak collision channels can be removed from the close-coupling...

Brain fiber pathways are presumed to follow smooth curves but recent high angular resolution diffusion MRI (dMRI) suggests that instead they follow 3 primary axes often nearly orthogonal. To investigate this, we analyzed axon pathways under monkey primary motor cortex with (1) dMRI tractography, (2) axon tract tracing, and (3) axon immunohistochemistry. dMRI tractography shows the predicted cro...