Let $Q$ be the field of rational numbers and $Q^{2}$ be the $2$-dimensional linear space over $Q$. A classification of all non-degenerate symmetric bilinear-metric forms over $Q^{2}$ have obtained. Let $\varphi $ be a non-degenerate symmetric bilinear form on $Q^{2}$. Denote by $O(2,\varphi, Q)$ the group of all $\varphi$-orthogonal (that is the form $\varphi$ preserving) transformations of $Q^{2}$. Put $MO( 2,\varphi, Q)=\left\{F:Q^{2}\rightarrow Q^{2}\mid Fx=gx+b, g\in O(2,\varphi, Q), b\in Q^{2}\right\}$, $SO(2,\varphi, Q)=\left\{ g\in O(2, \varphi, Q)|detg=1\right\}$ and $MSO(2, \varphi, Q)= \left\{F\in M(2, \varphi, Q)|detg=1\right\}$.The present paper is devoted to solutions of problems of $G$-equivalence of $m$-tuples in $Q^{2}$ for groups $G=O(2,\varphi, Q), SO(2,\varphi, Q)$, $MO(2,\varphi, Q)$, $MSO(2,\varphi, Q)$. Complete systems of $G$-invariants of $m$-tuples in $Q^{2}$ for these groups are obtained.
| Mualliflar | Beshimov, Gayrat |
|---|---|
| Jurnal | O‘zbekiston matematika jurnali |
| Nashr sanasi | 2025-11-03 |
| Jild | 69 |
| Son | 4 |
| Betlar | 53-69 |
| Til | Ingliz |
| DOI | 10.29229/uzmj.2025-4-7 |
DOI: 10.29229/uzmj.2025-4-7 · Maqolaning asl sahifasi
Invariant of $m$-tuple; $m$-point invariant.
Let $B$ be a complete Boolean algebra, let $Q(B)$ be the Stone compact of $B$, let $C_\infty (Q(B))$ be the commutative unital algebra of all continuous functions $x:Q(B) \rightarrow [-\infty,+\infty],$…
Let $R$ be a ring, and let $M$ be a right $R$-module. Recently, Baannon and Khalid introduced and studied the concept of $e^*$-essential submodules, extending the notion of essential submodules. In this context…
In this paper we study a ratio-dependent predator-prey model with a free boundary caused by predator-prey interaction over a one dimensional habitat. We study the long time behaviors of the two species and prove a…
In this paper, we classify the symmetric Leibniz algebras whose underlying Lie algebra is almost filiform. We also describe the symmetric Leibniz algebras associated with Heisenberg and triangular Lie algebras…
Hypergeometric functions are divided into complete and confluent functions. For the first time, Srivastava and Karlsson described a method for constructing a set of all possible triple Gaussian hypergeometric series and…
The time-optimal problem is considered for the controllable heat conduction process in a rod. Using the Fourier method, the problem is usually reduced to an infinite system of one-dimensional control equations whose…
In the paper, we consider a class dynamic control system with a discrete parameter and under conditions of uncertainty in the initial data. The optimal control problem of the minimax type is formulated for a non-smooth…
In this paper, we first classify all associative algebra structures on a two-dimensional vector space over an arbitrary base field equipped with a non-degenerate bilinear form. We then determine which of these are…
We analyze an evasion differential game involving one evader and multiple pursuers in $\R^n$. The control functions of the players are subject to exponential integral constraints to ensure bounded energy consumption…
In this work, the problem of stabilizing the equilibrium state for a hyperbolic system with negative nonlocal characteristic velocities and measurement error is investigated. A mixed problem is considered for such…