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Vol 80, No 1 (2025)
- Year: 2025
- Articles: 13
- URL: https://journal-vniispk.ru/0042-1316/issue/view/20352
On exponential algebraic geometry
Abstract
The set of roots of any finite system of exponential sums in the space $\mathbb{C}^n$ is called an exponential variety. We define the intersection index of varieties of complementary dimensions, and the ring of classes of numerical equivalence of exponential varieties with operations ‘addition-union’ and ‘multiplication-intersection’. This ring is analogous to the ring of conditions of the torus $(\mathbb{C}\setminus 0)^n$ and is called the ring of conditions of $\mathbb{C}^n$. We provide its description in terms of convex geometry. Namely, we associate an exponential variety with an element of a certain ring generated by convex polytopes in $\mathbb{C}^n$. We call this element the Newtonization of the exponential variety. For example, the Newtonization of an exponential hypersurface is its Newton polytope. The Newtonization map defines an isomorphism of the ring of conditions to the ring generated by convex polytopes in $\mathbb{C}^n$. It follows, in particular, that the intersection index of $n$ exponential hypersurfaces is equal to the mixed pseudo-volume of their Newton polytopes.Bibliography: 32 titles.
3-58
On stability of equilibria in a pseudo-Riemannian space
Abstract
The stability of equilibria is considered for systems whose kinetic energy is a pseudo-Riemannian metric on the configuration space. Equilibria are critical points of the potential energy. For a linear system with two degrees of freedom the stability diagram is plotted and the bifurcations of eigenvalues are indicated. Points of maximum and minimum of the potential energy are unstable equilibria in the pseudo-Euclidean case. The same conclusion holds for nonlinear analytic systems with two degrees of freedom. Conditions for stability are indicated for multidimensional linear systems in a pseudo-Euclidean space. In particular, an equilibrium is stable if and only if the linear equations of motion can be reduced to a ‘natural’ system with positive definite kinetic energy and, in addition, the potential energy takes a strict minimum at this equilibrium. The influence of dissipative and gyroscopic forces on the stability of equilibria in a pseudo-Riemannian space is investigated. The instability of an isolated equilibrium is proved in the case when dissipative forces with full energy dissipation are added. The instability degree is calculated for linear dissipative systems. Conditions for the stability of linear systems in the case when large gyroscopic forces are applied to them are indicated.Bibliography: 40 titles.
59-84
Scalar approaches to the limit distribution of the zeros of Hermite–Pade polynomials for a Nikishin system
Abstract
The problem of the existence of a limit distribution of the zeros of Hermite–Pade polynomials for a pair of functions forming a Nikishin system is discussed. Two new scalar methods are proposed for the investigation of this problem. The first is based on a potential-theoretic equilibrium problem stated on a two-sheeted Riemann surface and on the use of the Gonchar–Rakhmanov–Stahl ($\operatorname{GRS}$-)method in treating this problem. The second method is based on the existence of a three-sheeted Riemann surface with Nuttall partition into sheets which is associated with a given pair of functions $f$, $f^2$, and it uses only the maximum principle for subharmonic functions. The connection of these methods and the results obtained with Stahl's methods and results of 1987–88 is discussed. Results of numerical experiments are presented.Bibliography: 109 titles.
85-152
153-154
On an inverse problem of approximation theory in the Bloch space
155-156
Sub-Riemannian geodesics on the 3-dimensional Heisenberg nilmanifold
157-158
Dynamics of second-order equations with impulse-type delayed feedback
159-160
Threshold probabilities for colourings of random hypergraphs
161-162
Legendrian Lavrentiev links
163-164
Dmitry Valerievich Treschev (on his sixtieth birthday)
165-170
Albert Nikolaevich Shiryaev (on his 90th birthday)
171-177
Mathematical culture of the society: its value and development
178-183
Preparing the Russian team to the International Mathematical Olympiad for high school students
184-188