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تسجيل مستخدم جديدAdmissible transformations and Lie symmetries of linear systems of second-order ordinary differential equations
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We comprehensively study admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements parameterizing these systems. For each class from the constructed chain of nested gauged classes of such systems, we single out its singular subclass, which appears to consist of systems being similar to the elementary (free particle) system whereas the regular subclass is the complement of the singular one. This allows us to exhaustively describe the equivalence groupoids of the above classes as well as of their singular and regular subclasses. Applying various algebraic techniques, we establish principal properties of Lie symmetries of the systems under consideration and outline ways for completely classifying these symmetries. In particular, we compute the sharp lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by systems from each of the above classes and subclasses. We also show how equivalence transformations and Lie symmetries can be used for reduction of order of such systems and their integration. As an illustrative example of using the theory developed, we solve the complete group classification problems for all these classes in the case of two dependent variables.
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ty}q(t),dt=infty,$ begin{align*} q_{0}(a)=inf_{xin mathbb R}int_{x-a}^{x+a}q(t),dt=0 quad{rm for ~ any }quad ain (0,infty). end{align*} Under these conditions, the equation ({rm ref{ab}}) is not correctly solvable in $L_{p}(mathbb R)$ for any $p in [1, infty) $. Let $q^{*}(x)$ be the Otelbaev-type average of the function $q(t), tin mathbb{R}$, at the point $t=x$; $theta(x)$ be a continuous positive function for $x in mathbb R$, and begin{align*} L_{p,theta }(mathbb R) = {fin L_{p}^{rm loc}(mathbb R):, int_{-infty}^{infty}|theta(x)f(x)|^{p},dx<infty }, end{align*} begin{align*} |f|_{L_{p,theta}(mathbb R)}=left(int_{-infty}^{infty}|theta(x)f(x)|^{p},dxright)^{1/p} end{align*} We show that if there exists a constant $cin [1, infty)$, such that the inequality $$c^{-1}q^{*}(x)leq theta(x)leq cq^{*}(x)$$ holds for all $x in mathbb{R}$, then under some additional conditions for $q$ the pair of spaces ${L_{p, theta}(mathbb R); L_{p}(mathbb R)}$ is admissible for the equation ({rm ref{ab}}).
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