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Starting with identifications of the very fundamental geometric characteristics of a Mylar balloon such as the profile curve, height, volume, arclength, surface area, crimping factor, etc., using the geometrical moments In(x) and In, we present explicit formulas for them and those of the mechanical moments of both solid and hollow balloons of arbitrary order. This is achieved by relying on the recursive relationships among elliptic integrals and the final results are expressed via the fundamental mathematical constants such as π, lemniscate constant ω˜, and Gauss’s constant G. An interesting periodicity modulo 4 was detected and accounted for in the final formulas for the moments. The principal results are illustrated by two tables, a few graphics, and some direct relationships with other fundamental areas in mathematics, physics and geometry are pointed out.

As the title itself suggests here we are presenting extremely reach two/three parametric families of non-bending rotational surfaces in the three dimensional Euclidean space and provide the necessary details about their natural classifications and explicit parameterizations. Following the changes of the relevant parameters it is possible to trace out the ``evolution'' of these surfaces and even visualize them through their topological transformations. Many, and more deeper questions about their metrical properties, mechanical applications, etc. are left for future explorations.

We explore an interesting three parametric family of essentially non-circular torus-like thin shells of revolution deforming under symmetrical loading without bending. It was found that these non-bending torus-like shells split naturally into two classes of shells lying either outside or inside of a given right circular cylinder. For each of these classes we obtain uniform parameterizations in terms of elliptic integrals. Explicit formulas for the aspect ratio of the non-bending toroids measuring their non-circularity are presented as well.

Here we consider the problem of finding the equilibrium configurations of a rotating liquid drop, paying special attention to the cases when the droplet takes the shape of a Delaunay surface. By making use of the canonical forms of the elliptic integrals and the Jacobian elliptic functions we have derived several explicit parameterizations of the Delaunay surfaces. They are expressed relying on two independent real parameters accounting respectively the size and the shape so that all possible Delaunay surfaces are represented in a unified way.

The problem of a fluid body rotating with a constant angular velocity and subjected to uniform external pressure is of real interest in both fluid dynamics and nuclear theory. Besides, from the geometrical viewpoint the sought equilibrium configuration of such system turns out to be equivalent to the problem of determining the surface of revolution with a prescribed mean curvature. In the simply connected case, the equilibrium surface can be parameterized explicitly via elliptic integrals of the first and second kind.

Here we consider a family of axially symmetric surfaces modeling the shape of thin mechanical shells that are deformable without bending under uniform loading. With the exception of very few surfaces, like the well known right circular cylinder and the sphere, the surfaces of this family have no closed form description in elementary functions. Our main goal is to present their explicit parameterizations including both classes of open and closed families. We distinguish four classes of non-bending surfaces differing by their canonical representations using the normal elliptic integrals and the Jacobian elliptic functions.

We consider a special class of symmetrically loaded thin shells of revolution, which in the presence of certain disturbances of the equilibrium deform without bending. The whole family of such surfaces can be regarded as complementary set to the four classes of equilibrium states of non-bending surfaces that we have studied in some depth recently. The new surfaces (with a few exceptions) have not a closed form description in elementary functions. Here we present their explicit parameterizations in terms of elliptic integrals.

We consider a class of linear Weingarten surfaces of revolution whose principal curvatures, meridional $k_{\mu}$ and parallel $k_{\pi}$, satisfy the relation $k_{\mu}=(n+1)k_{\pi}$, $n=0,\,1,\,2,\ldots\, .$ The first two members of this class of surfaces are the sphere $(n=0)$ and the Mylar balloon $(n=1)$. Elsewhere the Mylar balloon has been parameterized via the Jacobian and Weierstrassian elliptic functions and elliptic integrals. Here we derive six alternative parameterizations describing the third type of surfaces when $n=2$. The so obtained explicit formulas are applied for the derivation of the basic geometrical characteristics of this surface.

The plate-ball problem concerns the shortest trajectories traced by a rolling sphere on a horizontal plane between the prescribed initial and final states meaning the positions and orientations of the sphere. Here we present an explicit parametric representation of these trajectories in terms of the Jacobian elliptic functions and elliptic integrals.

Here we consider an interesting class of free of bending deformations of thin axial symmetric shells subjected to uniform normal pressure. The meridional $k_{\mu}$ and the parallel $k_{\pi}$ principal curvatures of the middle surfaces of such shells obey the non-linear relationship $k_{\mu}=2ak_{\pi}^2+3k_{\pi}$, $a=const$. These non-bending shells depend on two arbitrary parameters, which are the principal radii $r_{\mu}$ and $r_{\pi}$ of some fixed parallel of the shell. Besides, these surfaces have no closed form description in elementary functions. Our principle aim here is to present such a parameterization of the whole class of non-bending closed surfaces by making use of the canonical forms of the elliptic integrals. The obtained explicit formulas are then applied for the derivation of the basic geometrical characteristics of these surfaces.

Here we present a new characterization of Euler elastica via the Weierstrassian functions.

With the aid of modern electronics the speed of light was directly measured by timing the delay of a light pulse from a laser in reflecting from a mirror in experiment performed in educational Physics Laboratory.

Two coupled nonlinear Schrodinger equations describing the transmission of subpicosecond pulses in strongly birefringent fibers with Raman scattering are studied by a symmetry reduction method and a perturbative technique. Two families of approximate vector solitary waves are obtained. Reduction of the Raman frequency shift in birefringent fibers is established.

Here we present a new parametrization of the Mylar balloon via the Weierstrassian functions which is used for the derivation of the basic geometrical characteristics of the balloon.

A laser speckle pattern (LSP) technique for designing an optical computer mouse is presented. The LSP is used to determine the velocity of the device in a way that allows working on a wide class of surfaces. The operational principles of the laser speckle pattern technique are explained. The results of computer simulation tests, carried out for proving its ability to work and the efficiency of the device, are given.

The goal of this chapter is to present the results of the Lie group analysis in application to the Helfrich spontaneous curvature model. Special attention is paid to the translationally invariant solutions and the corresponding cylindrical equilibrium shapes. Graphs of closed diretrices of the obtained cylindrical surfaces in fixed and moving reference frame are presented.

The governing equation of the Helfrich spontaneous-curvature model is the Helfrich equation. It is a coordinate free equation that describes the equilibrium shapes of biological (fluid) membranes. We make use of the conformal metric representation of the Helfrich equation and by applying the symmetry group reduction method we obtain a translationally invariant solution. Based on that solution, we derive analytic expressions for the position vector of special cylindrical equilibrium shapes. Plots of the graphs of some closed directrices of these shapes are presented.

A set of two coupled nonlinear Schr\"odinger equations is systematically analyzed by means of Lie group technique. The physical situations under consideration include nonlinear propagation in strongly birefringent and multimode optical fibers. The most general Lie group of point symmetries, its Lie algebra, and a group of adjoint representations that correspond to the Lie algebra are identified. As a result, a complete list of group-invariant exact solutions is obtained and compared with earlier results. The corresponding laws of conservation are derived employing Noether's theorem.

Here we consider the Helfrich's membrane shape model from a group-theoretical viewpoint. By making use of the conformal metric on the associated surface the model is represented by a system of four second order nonlinear partial differential equations. In order to construct the determining system for the symmetries of the metric we rely on the previously developed package LieSymm-PDE within ${\tt Mathematica}^{\!®}$. In this way we have obtained the determining system consisting of 206 equations. Using the above mentioned programs we have solved the equations in a semi-automatic way. As a result we end up with an infinite dimensional symmetry Lie algebra of the Helfrich`s model in conformal metric representation which we present here in explicit form.

This paper is devoted to finding an optimal system of one‐dimensional subalgebras of an eight‐dimensional Lie algebra of point symmetry transformations, admitted by a system of two coupled nonlinear Schrodinger equations.

A MATHEMATICA$^{®}$ package for finding Lie symmetries of partial differential equations is presented. The package has been applied to perform a full Lie group analysis of basic models of nonlinear fiber optics. As a result of this group invariant solutions have been obtained. Comparisons with earlier published computer algebra implementations of the Lie group method are discussed.

By applying the Lie group reduction method a full symmetry classification of one parameter group invariant solutions of two coupled nonlinear Schrodinger equations is presented. The physical situations under consideration include propagation of two polarization modes in weak and strong birefringent fibers, propagation of two waves at different carrier wavelengths, and nonlinear directional couplers.

A set of couled nonlinear equations describing propagation of two polarization light modes inside optical fibers with stimulated Raman scattering are considered. Using a Lie group technique, the most general Lie algebra is derived for the two cases: with and without orthogonal Raman gain. The optimal set of reduced systems of ordinary differential equations for the case without orthogonal Raman gain is obtained.

Two coupled nonlinear Schrodinger equations describing propagation of light pulses through nonlinear media with negative cross-phase modulation and different group-velocity dispersion regions for the two polarization modes are considered. Applying a Lie group technique, the most general Lie algebra and the corresponding Lie group of point symmetries permitted by the considered equations are derived. As a result, the optimal set of one-dimensional Lie algebras is obtained.© (2005) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

A set of two coupled nonlinear Schroedinger equations, describing nonlinear propagation in strongly birefringent and multimode optical fibers is analyzed by means of Lie group technique. As a result complete list of group- invariant exact solutions is obtained. A new exact solution is presented. The so-called symbiotic periodic and soliton solutions are included in this classification in natural ways. Laws of conservation are also derived.© (1996) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.