Within the framework of the well-known curvature models, a fluid lipid bilayer membrane is regarded as a surface embedded in the three-dimensional Euclidean space whose equilibrium shapes are described in terms of its mean and Gaussian curvatures by the so-called membrane shape equation. In the present paper, all solutions to this equation determining cylindrical membrane shapes are found and presented, together with the expressions for the corresponding position vectors, in explicit analytic form. The necessary and sufficient conditions for such a surface to be closed are derived and several sufficient conditions for its directrix to be simple or self-intersecting are given.

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We investigate the dependence of the order parameter profile, local and total susceptibilities on both the temperature and external magnetic field within the mean-filed Ginzburg-Landau Ising type model. We study the case of a film geometry when the boundaries of the film exhibit strong adsorption to one of the phases (components) of the system. We do that using general scaling arguments and deriving exact analytical results for the corresponding scaling functions of these quantities. In addition, we examine their behavior in the capillary condensation regime. Based on the derived exact analytical expressions we obtained an unexpected result -the existence of a region in the phase transitions line where the system jumps below its bulk critical temperature from a less dense gas to a more dense gas before switching on continuously into the usual jump from gas to liquid state in the middle of the system. It is also demonstrated that on the capillary condensation line one of the coexisting local susceptibility profiles is with one maximum, whereas the other one is with two local maxima centered, approximately, around the two gas-liquid interfaces in the system.

The consideration of some non-standard parametric Lagrangian leads to a fictitious dynamical system which turns out to be equivalent to the Euler problem for finding out all possible shapes of the lamina. Integrating the respective differential equations one arrives at novel explicit parameterizations of the Euler's elastica curves. The geometry of the inflexional elastica and especially that of the figure "eight" shape is studied in some detail and the close relationship between the elastica problem and mathematical pendulum is outlined.

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The six-parameter group of three dimensional Euclidean motions is recognized as the largest group of point transformations admitted by the membrane shape equation in Mongé representation. This equation describes the equilibrium shapes of biomembranes being the Euler-Lagrange equation associated with the Helfrich curvature energy functional under the constraints of fixed enclosed volume and membrane area. The conserved currents of six linearly independent conservation laws that correspond to the variational symmetries of the membrane shape equation and hold on its smooth solutions are obtained. All types of non-equivalent group-invariant solutions of the membrane shape equation are identified via an optimal system of onedimensional subalgebras of the symmetry algebra. The reduced equations determining these group-invariant solutions are derived. Special attention is paid to the translationally-invariant solutions of the membrane shape equation.

Abstract. When massless excitations are limited or modified by the presence of material bodies one observes a force acting between them generally called Casimir force. Such excitations are present in any fluid system close to its true bulk critical point. We derive exact analytical results for both the temperature and external ordering field behavior of the thermodynamic Casimir force within the mean-field GinzburgLandau Ising type model of a simple fluid or binary liquid mixture. We investigate the case when under a film geometry the boundaries of the system exhibit strong adsorption onto one of the phases (components) of the system. We present analytical and numerical results for the (temperature-field) relief map of the force in both the critical region of the film close to its finite-size or bulk critical points as well as in the capillary condensation regime below but close to the finite-size critical point.

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