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Register a new userAdmissible 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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Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients are exhaustively described over both the complex and real fields. The exact lower and upper bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems are obtained using an effective algebraic approach.
We shall give bounds on the spacing of zeros of certain functions belonging to the Laguerre-Polya class and satisfying a second order differential equation. As a corollary we establish new sharp inequalities on the extreme zeros of the Hermite, Laguerre and Jacobi polinomials, which are uniform in all the parameters.
This note reports on the recent advancements in the search for explicit representation, in classical special functions, of the solutions of the fourth-order ordinary differential equations named Bessel-type, Jacobi-type, Laguerre-type, Legendre-type.
Aiming at enlarging the class of symmetries of an SDE, we introduce a family of stochastic transformations able to change also the underlying probability measure exploiting Girsanov Theorem and we provide new determining equations for the infinitesimal symmetries of the SDE. The well-defined subset of the previous class of measure transformations given by Doob transformations allows us to recover all the Lie point symmetries of the Kolmogorov equation associated with the SDE. This gives the first stochastic interpretation of all the deterministic symmetries of the Kolmogorov equation. The general theory is applied to some relevant stochastic models.
We consider the differential equation begin{align}label{ab} -y(x)+q(x)y(x)=f(x), quad x in mathbb R, end{align} where $f in L_{p}(mathbb R)$, $pin [1,infty)$, and $0leq q in L_{1}^{rm loc}(mathbb R)$, $intlimits_{-infty}^{0}q(t),dt=intlimits_{0}^{infty}q(t),dt=infty,$ 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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