Vol 27, No 1 (2025)

In the paper the action of the orthogonal Lie algebra $\mathfrak{o}(V)$ on the exterior powers of a space $V$ is considered for $n$-dimensional vector space $V$ over a perfect field $K$ of characteristic two with a given nondegenerate orthogonal. The exterior algebra is identified with the algebra of truncated polynomials in $n$ variables. The exterior powers of $V$ taken as modules over $\mathfrak{o}(V)$ are identified with homogeneous subspaces of non-alternating Hamiltonian Lie algebra $P(n)$ with respect to the Poisson bracket corresponding to an orthonormal basis of the space $V$ of variables. It is proved that the exterior powers of the standard representation for Lie algebra $\mathfrak{o}(V)$ are irreducible and pairwise nonequivalent. With respect to subalgebra $so(V)$, $n= 2l+1$ or $n= 2l$, there exist $l$ pairwise nonequivalent fundamental representations in the spaces $\Lambda^{r}V$, $r= 1, \ldots, l$. All of them admit a nondegenerate invariant orthogonal form, being irreducible when $n= 2l+1$. When $n= 2l$ the representations of $so(V)$ in $\Lambda^{r}V$, $r= 1, \ldots, l-1$ are irreducible and the space $\Lambda^{l}V$ possesses the only non-trivial proper invariant subspace $M$, which is a maximal isotropic subspace with respect to an invariant form. Two exceptional simple Lie subalgebras $P_{1}(6)$, $P_{2}(6)$ of $P(n)$, of dimension $2^{5}-1$ and $2^{6}-1$, correspondingly, containing the submodule $M$, and exising only in the case of 6 variables, are found.

 

{In this paper, we solve the group classification problem for a nonlinear one-dimensional time-fractional  heat conduction equation with full memory and dual-phase-lag, including thermal relaxation and thermal damping. The characteristic times of relaxation processes are assumed to be small enough and therefore a small parameter for fractional differential relaxation terms is introduced. All thermal properties of a medium are considered as functions of temperature. Group classification is performed with respect to groups of approximate point transformations (groups of approximate symmetries) admitted by the equation up to equivalence transformations. We prove that generally the equation admits five-parameter group of approximate transformations, and the cases of its extension to seven- and nine-parameters groups are found. Also, it is shown that the considered nonlinear equation has an infinite approximate symmetry group if the corresponding unperturbed equation is linear. We find that the equation in question always exactly inherits the symmetries of the unperturbed equation. The obtained results make it possible to construct approximately invariant solutions of equation under consideration. In particular, it follows from the classification found that the equation always has a traveling wave solution. The self-similar solutions can be constructed only if the medium thermal properties have power-law dependences on temperature. Ansatzes of these types of solutions are obtained and symmetry reductions of the equation under consideration to the corresponding ordinary fractional differential equations are performed.

We obtained the sharp estimates of the moduli of the initial Taylor coefficients for functions $f$ of the class $B$ of bounded nonvanishing functions in the unit circle. Two types of estimates are obtained: one for ``large'' values of $|f(0)|$ and another one for ``small'' values of $|f(0)|$. The first type of estimates is asymptotic in the sense that it applies to an increasing number of initial coefficients as $|f(0)|$ increases. Similarly, the second type of estimates is asymptotic in the sense that it applies to an increasing number of initial coefficients as $|f(0)|$ decreases. Both types of estimates are deduced using methods of subordinate function theory and the Caratheodory-Toeplitz theorem for the Caratheodory class. This became possible due to the relation we found between the coefficients of convex univalent functions (class $S^0$) and the coefficients of the majorizing functions in the studied subclasses of the class $B$. The bounds for the applicability of the method are provided depending on $|f(0)|$ and on the coefficient number. The obtained results are applied to the theory of Laguerre polynomials. These results are compared with the previously known ones. The methods outlined here can be applied to arbitrary classes of subordinate functions.