Vol 518, No 1 (2024)

Let a hyperelliptic curve C of genus g defined over an algebraically closed field K of characteristic 0, given by the equation $y^2=f\left(x\right)$, where the polynomial $f\left(x\right)\in K\left[x\right]$ is square-free and has odd degree $2g+1$. The curve $C$ contains a single “infinite” point $O$, which is the Weierstrass point. There is a classical embedding of $C\left(K\right)$ into the group of $K$-points $J\left(K\right)$ of the Jacobian variety $J$ of the curve $C$, identifying the point $O$with the unit element of the group $J\left(K\right)$. For $2\le g\le 5$, the article explicitly found representatives of birational equivalence classes such hyperelliptic curves $C$ with a marked unique point at infinity $O$ that the set $C\left(K\right)\cap J\left(K\right)$ contains at least 6 torsion points of order $2g+1$. It was previously known that for $g=2$there are exactly 5 such equivalence classes, and for $g\ge 3$ an upper bound was known that depended only on the genus of $g$. We improve the previously known upper bound by almost 36 times.

It is shown that a material point, under the influence of an attractive linear force and a repulsive force inversely proportional to the cube of the distance from the center of attraction, performs a periodic motion, the period of which does not depend on the initial data (tautochronic motion). The problem is reduced to a nonlinear autonomous second-order equation, the general solution of which is expressed in terms of elementary functions. It has also been proven that for other power laws of repulsive force, except for degrees 0, 1 and –3, the movement of a material point is not tautochronous.

This paper is devoted to the localisation of random 3-CNF formulas that are polynomially solvable by the resolution algorithm. It is shown that random formulas with the number of clauses proportional to the square of the number of variables, are polynomially solvable with probability close to unity when the proportionality coefficient exceeds the found threshold.

New cases of integrable dynamical systems of the ninth order homogeneous in terms of variables are presented, in which a system on a tangent bundle to a four-dimensional manifold can be distinguished. In this case, the force field is divided into an internal (conservative) and an external one, which has a dissipation of a different sign. The external field is introduced using some unimodular transformation and generalizes the previously considered fields. Complete sets of both first integrals and invariant differential forms are given.

We perform a comparative analysis of the accuracy of second-order TVD (Total Variation Diminishing), third-order RBM (Rusanov-Burstein-Mirin), and fifth-order in space and third-order in time A-WENO (Alternative Weighted Essentially Non-Oscillatory) difference schemes for solving a special Cauchy problem for shallow water equations with discontinuous initial data. The exact solution of this problem contains a centered rarefaction wave and does not contain a shock wave. It is shown that in the centered rarefaction wave and its influence area, the solutions of these three schemes with different orders converge to different invariants of the exact solution. This leads to a decrease in the accuracy of these schemes when calculating the vector of base variables of the considered Cauchy problem. The P-form of the first differential approximation of the difference schemes is used for the theoretical justification of these numerical results.

This article is devoted to the improvement of signal recognition methods based on the information characteristics of the spectrum. A discrete function of the normalized ordered spectrum is established for a single window function included in the DFT. Lemmas on estimates of entropy, imbalance and statistical complexity in processing a time series of independent Gaussian quantities are proved. New concepts of one-dimensional and two-dimensional spectral complexities are proposed. The theoretical results obtained were verified by numerical experiments, which confirmed the effectiveness of the new information characteristic when detecting a signal mixed with white noise at low signal-to-noise ratios.