Sbornik: Mathematics

Our subject is the geometry of planar hinged mechanisms. The article contains a formalization of basic concepts of the theory of hinged-lever constructions, as well as some information from real algebraic geometry needed for their study. We consider mechanisms with variable number of degrees of freedom and mechanisms that have more than one degree of freedom but each hinge of which moves with one degree of freedom. For the last type we find the dimension of the configuration space. We give a number of examples of mechanisms with unusual geometric properties and formulate open questions.

Bibliography: 17 titles.

We say that a zero of a Laurent polynomial that lies on the unit circle with centre $0\in\mathbb C$ is real. We also say that a Laurent polynomial that is real on this circle is real. In contrast with ordinary polynomials, it is known that for random real Laurent polynomials of increasing degree the average proportion of real roots tends to $1/\sqrt 3$ rather than to $0$. We show that this phenomenon of the asymptotically nonvanishing proportion of real roots also holds for systems of Laurent polynomials of several variables. The corresponding asymptotic formula is obtained in terms of the mixed volumes of certain convex compact sets determining the growth of the system of polynomials.

Bibliography: 11 titles.

The topological bifurcations of Liouville foliations on invariant $3$-manifolds that are induced by attaching toric $\Theta$-handles are investigated. It is shown that each marked molecule (Fomenko-Zieschang invariant) can be realized on an invariant submanifold of a closed symplectic manifold with contact singularities which is obtained by attaching toric $\Theta$-handles sequentially to a set of symplectic manifolds, while these latter have the structures of locally trivial fibrations over $S^1$ associated with atoms.

Bibliography: 10 titles.

The time minimization problem with admissible control in a half-disc is considered on the group of motions of a plane. The control system under study provides a model of a car on the plane that can move forwards or rotate in place. Optimal trajectories of such a system are used to detect salient curves in image analysis. In particular, in medical image analysis such trajectories are used for tracking vessels in retinal images. The problem is of independent interest in geometric control theory: it provides a model example when the set of values of the control parameters contains zero at the boundary. The problem of controllability and existence of optimal trajectories is studied. By analysing the Hamiltonian system of the Pontryagin maximum principle the explicit form of extremal controls and trajectories is found. Optimality of the extremals is partially investigated. The structure of the optimal synthesis is described.

Bibliography: 33 titles.

The Golomb-de Boor problem of extremal interpolation of infinite real sequences with smallest $L_p$-norm of the $n$th derivative of the interpolant, $1\leqslant p\leqslant \infty$, on an arbitrary mesh on the real axis is studied under constraints on the norms of the corresponding divided differences. For this smallest norm, lower estimates are obtained for any $n\in \mathbb N$ in terms of $B$-splines. For the second derivative, this quantity is estimated from below and above by constants depending on the parameter $p$.

Bibliography: 13 titles.