One helpful property of functions generating P\'olya frequency sequences

In this work we study the solutions of the equation $z^pR(z^k)=\alpha$ with nonzero complex $\alpha$, integer $p,k$ and $R(z)$ generating a (possibly doubly infinite) totally positive sequence. It is shown that the zeros of $z^pR(z^k)-\alpha$ are simple (or at most double in the case of real $\alpha^k$) and split evenly among the sectors $\{\frac jk \pi\le\operatorname{Arg} z\le\frac {j+1}k \pi\}$, $j=0,\dots, 2k-1$. Our approach rests on the fact that $z(\ln z^{p/k}R(z) )'$ is an $\mathcal R$-function (i.e. maps the upper half of the complex plane into itself). This result guarantees the same localization to zeros of entire functions $f(z^k)+z^p g(z^k)$ and $g(z^k)+z^{p}f(z^k)$ provided that $f(z)$ and $g(-z)$ have genus $0$ and only negative zeros. As an application, we deduce that functions of the form $\sum_{n=0}^\infty (\pm i)^{n(n-1)/2}a_n z^{n}$ have simple zeros distinct in absolute value under a certain condition on the coefficients $a_n\ge 0$. This includes the "disturbed exponential" function corresponding to $a_n= q^{n(n-1)/2}/n!$ when $0<q\le 1$, as well as the partial theta function corresponding to $a_n= q^{n(n-1)/2}$ when $0<q\le q_*\approx 0.7457224107$.Comment: 25 pages, 3 figure

Water waves over a time-dependent bottom: Exact description for 2D potential flows

Two-dimensional potential flows of an ideal fluid with a free surface are considered in situations when shape of the bottom depends on time due to external reasons. Exact nonlinear equations describing surface waves in terms of the so called conformal variables are derived for an arbitrary time-evolving bottom parameterized by an analytical function. An efficient numerical method for the obtained equations is suggested.Comment: revtex4, 7 pages, 19 EPS figures; corrected version with more numerical result

"Breathing" rogue wave observed in numerical experiment

Numerical simulations of the recently derived fully nonlinear equations of motion for weakly three-dimensional water waves [V.P. Ruban, Phys. Rev. E {\bf 71}, 055303(R) (2005)] with quasi-random initial conditions are reported, which show the spontaneous formation of a single extreme wave on the deep water. This rogue wave behaves in an oscillating manner and exists for a relatively long time (many wave periods) without significant change of its maximal amplitude.Comment: 6 pages, 12 figure

Quasi-planar steep water waves

A new description for highly nonlinear potential water waves is suggested, where weak 3D effects are included as small corrections to exact 2D equations written in conformal variables. Contrary to the traditional approach, a small parameter in this theory is not the surface slope, but it is the ratio of a typical wave length to a large transversal scale along the second horizontal coordinate. A first-order correction for the Hamiltonian functional is calculated, and the corresponding equations of motion are derived for steep water waves over an arbitrary inhomogeneous quasi-1D bottom profile.Comment: revtex4, 4 pages, no figure

Epidemiological analysis of herpesviral lesions of the nervous system

Primary herpesvirus infection usually occurs during childhood and may cause several benign self-limited clinical manifestations, followed by a life-long persistence in a latent state with possible reactivation in case of immunodeficiency. Serious health problems, including CNS lesions can occur as a result of HVs reactivation. Herpesvirus encephalitis (HVE) accounts for up to 40% of all viral encephalitis, and are major causes of mortality and long-term neurological sequelae throughout the world even when using antiviral drugs. Since surveillance and recording of herpesvirus infections (HVI) are not common practice, it is difficult to establish exact figures for the prevalence of both HVI and HVE. Despite being an important public health problem, very few population-based studies have been carried out so far in the world and none in Ukraine. We present the clinical and etiological data obtained in prospective single center population study with 107 enrolled adult patients in Ukraine