On minimal round functions

We describe the structure of minimal round functions on closed surfaces and three-folds. The minimal possible number of critical loops is determined and typical non-equisingular round function germs are interpreted in the spirit of isolated line singularities. We also discuss a version of Lusternik-Schnirelmann theory suitable for round functions

Critical configurations of planar robot arms

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration

The mathematics of french fries

Although the act of cutting a single potato (Solanum tuberosum) into french fries may appear to be trivial, the questions concerning the efficiency of this process on an industrial scale are quite daunting. Therefore, many producers are looking for a rigorous method to evaluate the market potential of a given potato crop by predicting the number and parameters of the fries that can be cut from it. Applying the methods of geometry and numerical analysis our group was able to propose several algorithms that can be directly incorporated into the existing production process. Keywords: French fries, geometry, cutting, Finite Fry Method, simulations, histogram

On the Heisenberg invariance and the Elliptic Poisson tensors

We study different algebraic and geometric properties of Heisenberg invariant Poisson polynomial quadratic algebras. We show that these algebras are unimodular. The elliptic Sklyanin-Odesskii-Feigin Poisson algebras $q_{n,k}(\mathcal E)$ are the main important example. We classify all quadratic $H-$invariant Poisson tensors on ${\mathbb C}^n$ with $n\leq 6$ and show that for $n\leq 5$ they coincide with the elliptic Sklyanin-Odesskii-Feigin Poisson algebras or with their certain degenerations.Comment: 14 pages, no figures, minor revision, typos correcte

On local invariants of totally real surfaces

We will be basically concerned with germs of totally real two-dimensional surfaces with isolated singularities in C 2 given by polynomial parametrisations. The ultimate goal we have in mind, is to develop some tools whic

Remarks on generalized Sklyanin algebras

We deal with certain polynomial Poisson structures on affine spaces generalizing the well-known Sklyanin algebras [13] which play significant role in the modern theory of exactly solvable models of classical and quantum mechanic

Elliptic cells and Fredholm operators

We deal with collections of differentiable functions on affine domains satisfying certain elliptic system of linear partial differential equations of first order with constant coefficients. Such systems are sometimes called “canonica