On Huisman’s conjectures about unramified real curves

Let X ⊂ ℙ n be an unramified real curve with X(ℝ) ≠ 0. If n ≥ 3 is odd, Huisman [9] conjectured that X is an M-curve and that every branch of X(ℝ) is a pseudo-line. If n ≥ 4 is even, he conjectures that X is a rational normal curve or a twisted form of such a curve. Recently, a family of unramified M-curves in ℙ3 providing counterexamples to the first conjecture was constructed in [11]. In this note we construct another family of counterexamples that are not even M-curves. We remark that the second conjecture follows for generic curves of odd degree from the de Jonquières formula.

Application of the Riccati hereditary mathematical model to the study of the dynamics of radon accumulation in the storage chamber

The article proposes a mathematical model of radon accumulation in a chamber, which takes into account the hereditary properties of the environment in which radon migrates. The model equation is the fractional Riccati equation with a derivative of a fractional variable order of the Gerasimov-Caputo type, taking into account heredity, as well as taking into account nonlinearity, which is responsible for the mechanisms of radon entry into the chamber. The obtained model curves of the accumulation process are compared with real data. It is shown that the model described in the work gives a better agreement between the model and real curves of radon accumulation and can be used for a more accurate description of the processes occurring in the chamber.

A New Objective Function for the Recovery of Gielis Curves

The superformula generates curves called Gielis curves, which depend on a small number of input parameters. Recovering parameters generating a curve that adapts to a set of points is a non-trivial task, thus methods to accomplish it are still being developed. These curves can represent a great variety of forms, such as living organisms, objects and geometric shapes. In this work we propose a method that uses a genetic algorithm to minimize a combination of three objectives functions: Euclidean distances from the sample points to the curve, from the curve to the sample points and the curve length. Curves generated with the parameters obtained by this method adjust better to real curves in relation to the state of art, according to observational and numeric comparisons.

Research of the process of radon accumulation in the accumulating chamber taking into account the nonlinearity of its entrance

The paper presents a mathematical model of radon accumulation in a chamber, which takes into account the hereditary properties of the environment in which radon migrates, and also uses a nonlinear function that is responsible for the mechanisms of radon entering the chamber. The simulation of accumulation is performed in comparison with real data. It is shown that the model presented in this work gives a better agreement between the model and real curves of radon accumulation and can be used for a more accurate description of the processes occurring in the chamber.

A recursion for counts of real curves in ℂℙ2n−1: Another proof

In a recent paper, we obtained a WDVV-type relation for real genus 0 Gromov–Witten invariants with conjugate pairs of insertions; it specializes to a complete recursion in the case of odd-dimensional projective spaces. This paper provides another, more complex-geometric, proof of the latter. The main part of this approach readily extends to real symplectic manifolds with empty real locus, but not to the general case.

Bifurcation values of families of real curves

The detection of the bifurcation set of polynomial mapping ℝn → ℝp, n ⩾ p, in more than two variables remains an unsolved problem. In this note we provide a solution for n = p + 1 ⩾ 3.