Three examples of three-dimensional continued fractions in the sense of Klein

The problem of the investigation of the simplest n-dimensional continued fraction in the sense of Klein for $n?2$ was posed by V. Arnold. The answer for the case $n=2$ can be found in the works of E. Korkina (1995) and G. Lachaud (1995). In present Note we study the case $n=3$. To cite this article: O. Karpenkov, C. R. Acad. Sci. Paris, Ser. I 343 (2006).     

The Influence of Milling Modes on the Structure and Magnetic Properties of (Sm,Ho)2Fe17Nx (x = 0, 2.4) Powder Materials

Technical Physics - The composition, structure, and surface topology of as-cast, homogenized, and nitrogenized alloy based on an intermetallic compound Sm2Fe17 with partial substitution of samarium...     

Energies of knot diagrams

We introduce and begin the study of new knot energies defined on knot diagrams. Physically, they model the internal energy of thin metallic solid tori squeezed between two parallel planes. Thus the knots considered can perform the second and third Reidemeister moves, but not the first one. The energy functionals considered are the sum of two terms, the uniformization term (which tends to make the curvature of the knot uniform) and the resistance term (which, in particular, forbids crossing changes). We define an infinite family of uniformization functionals, depending on an arbitrary smooth function f and study the simplest nontrivial case f(x) = x ², obtaining neat normal forms (corresponding to minima of the functional) by making use of the Gauss representation of immersed curves, of the phase space of the pendulum, and of elliptic functions.     

Development of nanostructured magnetic materials based on high-purity rare-earth metals and study of their fundamental characteristics

The effect of the structural state on magnetic and hysteretic properties of compounds with high contents of a 3d transition metal, i.e., R ₂Fe_{14 ? x} Co _x B and RFe_{11 ? x} Co _x Ti (where R = Y, Sm; 0 ≤ x ≤ 8), was studied. Alloys were prepared using high-purity rare-earth metals by two different methods: induction melting and argon-arc melting. Severe plastic deformation and rapid melt-quenching allowed preparation of nanostructured samples. Structural studies of the samples were performed by X-ray powder diffraction and atomic-force microscopy methods. Magnetic hysteretic properties were studied using a PPMS magnetometer in the temperature range of 4.2–300 K in fields to 20 kOe. It was shown that the dependences of fundamental magnetic parameters (Curie temperature, saturation magnetization, and magnetocrystalline anisotropy constant) on the cobalt content exhibit a similarity for both systems. It was found that, depending on sample treatment, the grain size varies from 30 to 70 nm after severe plastic deformation and in wider ranges (from 10 to 100 nm) after rapid quenching, not exceeding the single-domain size. The interrelation between the microstructure and magnetic characteristics was investigated. It was revealed that the concentration dependence of the coercivity for both systems has a maximum at the same cobalt content, i.e., x = 2.     

Continued fractions and the second Kepler law

In this paper we introduce a link between geometry of ordinary continued fractions and trajectories of points that moves according to the second Kepler law. We expand geometric interpretation of ordinary continued fractions to the case of continued fractions with arbitrary elements.     

Euler elasticae in the plane and the Whitney-Graustein theorem

In this paper, we study normal forms of plane curves and knots. We investigate the Euler functional E (the integral of the square of the curvature along the given curve) for closed plane curves, and introduce a closely related functional A, defined for polygonal curves in the plane ?² and its modified version A _R , defined for polygonal knots in Euclidean space ?³. For closed plane curves, we find the critical points of E and, among them, distinguish the minima of E, which give us the normal forms of plane curves. The minimization of the functional A for plane curves, implemented in a computer animation, gives a very visual approximation of the process of gradient descent along the Euler functional E and, thereby, illustrates the homotopy in the proof of the classical Whitney-Graustein theorem. In ?³, the minimization of A _R (implemented in a 3D animation) shows how classical knots (or more precisely thin knotted solid tori, which model resilient closed wire curves in space) are isotoped to normal forms.     

Classification of lattice-regular lattice convex polytopes

We completely describe lattice convex polytopes in ℝ ⁿ (for any dimension n) that are regular with respect to the group of affine transformations preserving the lattice. Supported in part by the RFBR (Grant Nos. SS-1972.2003.1 and 05-01-01012a) and the NWO-RFBR (Grant No. 047.011.2004.026/RFBR No. 05-02-89000-NWO_a).     

On the flexibility of Kokotsakis meshes

In this paper we study geometric, algebraic, and computational aspects of flexibility and infinitesimal flexibility of Kokotsakis meshes. A Kokotsakis mesh is a mesh that consists of a face in the middle and a certain band of faces attached to the middle face by its perimeter. In particular any (3 × 3)-mesh made of quadrangles is a Kokotsakis mesh. We express the infinitesimal flexibility condition in terms of Ceva and Menelaus theorems. Further we study semi-algebraic properties of the set of flexible meshes and give equations describing it. For (3 × 3)-meshes we obtain flexibility conditions in terms of face angles.     

Geometry of Configuration Spaces of Tensegrities

Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in ℝ ^d to admit a nonzero self-stress with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in ℝ ^d . In particular we find surgeries on graphs that give relations between different strata. Further we discuss questions related to geometric conditions defining the strata for plane tensegrities. We conclude the paper with particular examples of strata for tensegrities in the plane with a small number of vertices.     

On an invariant Möbius measure and the Gauss-Kuzmin face distribution

We study Möbius measures of the manifold of n-dimensional continued fractions in the sense of Klein. By definition any Möbius measure is invariant under the natural action of the group of projective transformations PGL(n + 1) and is an integral of some form of the maximal dimension. It turns out that all Möbius measures are proportional, and the corresponding forms are written explicitly in some special coordinates. The formulae obtained allow one to compare approximately the relative frequencies of the n-dimensional faces of given integer-affine types for n-dimensional continued fractions. In this paper we make numerical calculations of some relative frequencies in the case of n = 2.