Page 697 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 697
POSTER SESSION
On 2-closures of rank 3 groups
Saveliy Skresanov, skresan@math.nsc.ru
Sobolev Institute of Mathematics, Russian Federation
Let G be a permutation group on Ω. A 2-orbit is an orbit of G in its induced action on Ω × Ω.
Recall that the number of 2-orbits is called the rank of G, and the largest permutation group on
Ω having the same 2-orbits as G is called the 2-closure of G.
If the order of G is even, then a nondiagonal 2-orbit of G induces a strongly regular graph
called a rank 3 graph. The full automorphism group of that graph is precisely the 2-closure of
the corresponding group.
We present the description of 2-closures of rank 3 groups of sufficiently large degree.
Groups are organized into several families, based on the combinatorial structures preserved,
and the full automorphism groups of the corresponding structures are given. The proof heav-
ily relies on the classification of rank 3 groups and on known results about automorphisms of
strongly regular graphs.
Mean value theorems for polynomial solutions of linear elliptic equations
with constant coefficients in the complex plane
Olha Trofymenko, odtrofimenko@gmail.com
Vasyl’ Stus Donetsk National University, Ukraine
The work is devoted to the mean value theorems for solutions of homogeneous linear par-
tial differential equations with constant coefficients in the complex plane whose left hand side
is represented in the form of the product of some non-negative integer powers of the formal
Cauchy derivatives. We consider systems of special type homogeneous convolution equations
defined on smooth functions in a disk, which generalize the classical mean value property over
disks for harmonic functions. A sharp version of the uniqueness theorem for functions satis-
fying such a system in the case of one equation has been established. We also investigate the
case of two equations and prove a theorem consisting the classical Delsarte’s two-radii theo-
rem for dimension two as a special case. The convolution equations generated by distributions
with compact supports and the corresponding mean-value theorems were investigated by many
authors: C. Berenstein and D. Struppa, L. Zalcman, V. Volchkov. In particular, V. Volchkov de-
scribed a wide class of radial distributions with compact supports such that the solutions of the
corresponding convolution equations in open Euclidean balls can be efficiently characterized in
terms of the Bessel functions and proved the general uniqueness and two-radius theorems for
solutions of these equations that go back to the classical results by F. John and J. Delsarte about
spherical means, respectively.
695
On 2-closures of rank 3 groups
Saveliy Skresanov, skresan@math.nsc.ru
Sobolev Institute of Mathematics, Russian Federation
Let G be a permutation group on Ω. A 2-orbit is an orbit of G in its induced action on Ω × Ω.
Recall that the number of 2-orbits is called the rank of G, and the largest permutation group on
Ω having the same 2-orbits as G is called the 2-closure of G.
If the order of G is even, then a nondiagonal 2-orbit of G induces a strongly regular graph
called a rank 3 graph. The full automorphism group of that graph is precisely the 2-closure of
the corresponding group.
We present the description of 2-closures of rank 3 groups of sufficiently large degree.
Groups are organized into several families, based on the combinatorial structures preserved,
and the full automorphism groups of the corresponding structures are given. The proof heav-
ily relies on the classification of rank 3 groups and on known results about automorphisms of
strongly regular graphs.
Mean value theorems for polynomial solutions of linear elliptic equations
with constant coefficients in the complex plane
Olha Trofymenko, odtrofimenko@gmail.com
Vasyl’ Stus Donetsk National University, Ukraine
The work is devoted to the mean value theorems for solutions of homogeneous linear par-
tial differential equations with constant coefficients in the complex plane whose left hand side
is represented in the form of the product of some non-negative integer powers of the formal
Cauchy derivatives. We consider systems of special type homogeneous convolution equations
defined on smooth functions in a disk, which generalize the classical mean value property over
disks for harmonic functions. A sharp version of the uniqueness theorem for functions satis-
fying such a system in the case of one equation has been established. We also investigate the
case of two equations and prove a theorem consisting the classical Delsarte’s two-radii theo-
rem for dimension two as a special case. The convolution equations generated by distributions
with compact supports and the corresponding mean-value theorems were investigated by many
authors: C. Berenstein and D. Struppa, L. Zalcman, V. Volchkov. In particular, V. Volchkov de-
scribed a wide class of radial distributions with compact supports such that the solutions of the
corresponding convolution equations in open Euclidean balls can be efficiently characterized in
terms of the Bessel functions and proved the general uniqueness and two-radius theorems for
solutions of these equations that go back to the classical results by F. John and J. Delsarte about
spherical means, respectively.
695
