48 research outputs found
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, 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
are the main important example. We classify all quadratic
invariant Poisson tensors on with and show that
for 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