The problem of normal oscillations of a viscous stratified fluid with an elastic membrane

Normal oscillations of a viscous stratified fluid partially filling an arbitrary vessel and bounded above by an elastic horizontal membrane are studied. In this case, we consider a scalar model problem that reflects the main features of the vector spatial problem. The characteristic equation for the eigenvalues of the model problem is obtained, the structure of the spectrum and the asymptotics of the branches of the eigenvalues are studied. Assumptions are made about the structure of the oscillation spectrum of a viscous stratified fluid bounded by an elastic membrane for an arbitrary vessel. It is proved that the spectrum of the problem is discrete, located in the right complex half-plane symmetrically with respect to the real axis, and has a single limit point $+\infty$. Moreover, the spectrum is localized in a certain way in the right half-plane, the location zone depends on the dynamic viscosity of the fluid.

Polynomials and Reciprocals of Eisenstein series

Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane. These led to asymptotic results for the partition numbers and proven and unproven formulas for the coefficients of the reciprocals of Eisenstein series [Formula: see text], especially of weight 4. Berndt et al. finally proved them all. Recently, Bringmann and Kane generalized Petersson’s approach via Poincaré series, to handle the general case. We introduce a third approach. We attach recursively defined polynomials to reciprocals of Eisenstein series. This provides easy access to the signs of the Fourier coefficients of reciprocals of Eisenstein series, sheds some light on reciprocals of [Formula: see text] of general weight, and provides some upper and lower bounds for their growth.

Dynamics of poles in two-dimensional hydrodynamics with free surface: new constants of motion

We address the problem of the potential motion of an ideal incompressible fluid with a free surface and infinite depth in a two-dimensional geometry. We admit the presence of gravity forces and surface tension. A time-dependent conformal mapping $z(w,t)$ of the lower complex half-plane of the variable $w$ into the area filled with fluid is performed with the real line of $w$ mapped into the free fluid’s surface. We study the dynamics of singularities of both $z(w,t)$ and the complex fluid potential $\unicode[STIX]{x1D6F1}(w,t)$ in the upper complex half-plane of $w$. We show the existence of solutions with an arbitrary finite number $N$ of complex poles in $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ which are the derivatives of $z(w,t)$ and $\unicode[STIX]{x1D6F1}(w,t)$ over $w$. We stress that these solutions are not purely rational because they generally have branch points at other positions of the upper complex half-plane. The orders of poles can be arbitrary for zero surface tension while all orders are even for non-zero surface tension. We find that the residues of $z_{w}(w,t)$ at these $N$ points are new, previously unknown, constants of motion, see also Zakharov & Dyachenko (2012, authors’ unpublished observations, arXiv:1206.2046) for the preliminary results. All these constants of motion commute with each other in the sense of the underlying Hamiltonian dynamics. In the absence of both gravity and surface tension, the residues of $\unicode[STIX]{x1D6F1}_{w}(w,t)$ are also the constants of motion while non-zero gravity $g$ ensures a trivial linear dependence of these residues on time. A Laurent series expansion of both $z_{w}(w,t)$ and $\unicode[STIX]{x1D6F1}_{w}(w,t)$ at each poles position reveals the existence of additional integrals of motion for poles of the second order. If all poles are simple then the number of independent real integrals of motion is $4N$ for zero gravity and $4N-1$ for non-zero gravity. For the second-order poles we found $6N$ motion integrals for zero gravity and $6N-1$ for non-zero gravity. We suggest that the existence of these non-trivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics in deep water. Analytical results are solidly supported by high precision numerics.

NEW INTEGRALS OF MOTION AND NON-CANONICAL HAMILTONIAN STRUCTURE FOR 2D HYDRODYNAMICS WITH FREE SURFACE

We consider a potential motion of ideal incompressible fluid with a free surface and infinite depth in two dimensions with gravity forces and surface tension. A time-dependent conformal mapping z(w, t) of the lower complex half-plane of the variable w into the area filled with fluid is performed with the real line of w mapped into the free fluid’s surface. We study the dynamics of singularities of both z(w, t) and the complex fluid potential Π(w, t) in the upper complex half-plane of w. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian structure for a pair of the Hamiltonian variables (Dyachenko et al., submitted), the imaginary part of z(w, t) and the real part at Π(w, t) (both evaluated of fluid’s free surface). The corresponding Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. New Hamiltonian structure is a generalization of the canonical Hamiltonian structure of (Zakharov, 1968) (valid only for solutions for which the natural surface parametrization is single valued, i.e. each value of the horizontal coordinate corresponds only to a single point on the free surface). In contrast, new non-canonical Hamiltonian equations are valid for arbitrary nonlinear solutions (including multiple-valued natural surface parametrization) and are equivalent to Euler equations. We also consider a generalized hydrodynamics with the additional physical terms in the Hamiltonian beyond the Euler equations as in (Lushnikov and Zubarev, 2018) with the powerful reductions which allowed to find general classes of particular solutions. In Eulerian case we show the existence of solutions with an arbitrary finite number N of complex poles in zw(w, t) and Πw(w, t) which are the derivatives of z(w, t) and Π(w, t) over w (Dyachenko et al., submitted). These solutions are not purely rational because they generally have branch points at other positions of the upper complex halfplane with generally the infinite number of sheets of the Riemann surface for z(w, t) and Π(w, t) (Lushnikov, 2016). The order of poles is arbitrary for zero surface tension while all orders are even for nonzero surface tension. We find that the residues of zw(w, t) at these N points are new, previously unknown constants of motion. These constants of motion commute with each other with respect to the Poisson bracket. There are more integrals of motion beyond these residues. If all poles are simple then the number of independent real integrals of motion is 4N for zero gravity and 4N-1 for nonzero gravity. For higher order poles the number of the integrals is increasing. These nontrivial constants of motion provides an argument in support of the conjecture of complete integrability of free surface hydrodynamics. Work of A. Dyachenko, P. Lushnikov and V. Zakharov was supported by state assignment «Dynamics of the complex materials».

On an Exact Relation between ζ″(2) and the Meijer G -Functions

In this paper we consider some integral representations for the evaluation of the coefficients of the Taylor series for the Riemann zeta function about a point in the complex half-plane ℜ ( s ) > 1 . Using the standard approach based upon the Euler-MacLaurin summation, we can write these coefficients as Γ ( n + 1 ) plus a relatively smaller contribution, ξ n . The dominant part yields the well-known Riemann’s zeta pole at s = 1 . We discuss some recurrence relations that can be proved from this standard approach in order to evaluate ζ ″ ( 2 ) in terms of the Euler and Glaisher-Kinkelin constants and the Meijer G -functions.

Stability analysis of Beck's column over a fractional-order hereditary foundation

This paper considers the case of Beck's column resting on a hereditary bed of independent springpots. The springpot possesses an intermediate rheological behaviour among linear spring and linear dashpot. It is defined by means of couple ( C β , β ) that characterize the material of the element and is ruled by a Caputo's fractional derivative. In this paper, we investigate the critical load of the column under the action of a follower load by means of a novel complex transform that allows to use the Routh–Hurwitz theorem in the complex half-plane for the stability analysis.

Continued Fractions and Conformal Mappings for Domains with Angel Points

Here we construct the conformal mappings with the help of the continued fraction approximations. We first show that the method of [19] works for conformal mappings of the unit disk onto domains with acute external angles at the boundary. We give certain illustrative examples of these constructions. Next we outline the problem with domains which boudary possesses acute internal angles. Then we construct the method of rational root approximation in the right complex half-plane. First we construct the square root approximation and consider approximative properties of the mapping sequence in Theorem 1. Then we turn to the general case, namely, the continued fraction approximation of the rational root function in the complex right half-plane. These approximations converge to the algebraic root functions , , , . This is proved in Theorem 2 of the aricle. Thus we prove convergence of this method and construct conformal approximate mappings of the unit disk onto domains with angles and thin domains. We estimate the convergence rate of the approximation sequences. Note that the closer the point is to zero or infinity and the lower is the ratio k/N the worse is the approximation. Also we give the examples that illustrate the conformal mapping construction.