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Vol 211, No 3 (2020)
- Year: 2020
- Articles: 6
- URL: https://journals.rcsi.science/0368-8666/issue/view/7463
On the heritability of the Sylow $\pi$-theorem by subgroups
Abstract
Let $\pi$ be a set of primes. We say that the Sylow $\pi$-theorem holds for a finite group $G$, or $G$ is a $\mathscr D_\pi$-group, if the maximal $\pi$-subgroups of $G$ are conjugate. Obviously, the Sylow $\pi$-theorem implies the existence of $\pi$-Hall subgroups. In this paper, we give an affirmative answer to Problem 17.44, (b), in the Kourovka notebook: namely, we prove that in a $\mathscr D_\pi$-group an overgroup of a $\pi$-Hall subgroup is always a $\mathscr D_\pi$-group. Bibliography: 52 titles.
Matematicheskii Sbornik. 2020;211(3):3-31
3-31
Global extrema of the Delange function, bounds for digital sums and concave functions
Abstract
The sums $S_q(N)$ are defined by the equality $S_q(N)=s_q(1)+…+s_q(N-1)$ for all positive integers $N$ and $q\ge2$, where $s_q(n)$ is the sum of digits of the integer $n$ written in the system with base $q$. In 1975 Delange generalised Trollope's formula and proved that $S_q(N)/N-({q-1})/2\cdot\log_qN=-1/2\cdot f_q(q^{\{\log_q N\}-1})$, where $f_q(x)=(q-1)\log_q x+D_q(x)/x$ and $D_q$ is the continuous nowhere differentiable Delange function. We find global extrema of $f_q$ and, using this, obtain a precise bound for the difference $S_q(N)/N-(q-1)/2\cdot\log_qN$. In the case $q=2$ this becomes the bound for binary sums proved by Krüppel in 2008 and also earlier by other authors. We also evaluate the global extrema of some other continuous nowhere differentiable functions. We introduce the natural concave hull of a function and prove a criterion simplifying the evaluation of this hull. Moreover, we introduce the notion of an extreme subargument of a function on a convex set. We show that all points of global maximum of the difference $f-g$, where the function $g$ is strictly concave and some additional conditions hold, are extreme subarguments for $f$. A similar result is obtained for functions of the form $v+f/w$. We evaluate the global extrema and find extreme subarguments of the Delange function on the interval $[0,1]$. The results in the paper are illustrated by graphs and tables. Bibliography: 16 titles.
Matematicheskii Sbornik. 2020;211(3):32-70
32-70
Multivalued solutions of hyperbolic Monge-Ampère equations: solvability, integrability, approximation
Abstract
Solvability in the class of multivalued solutions is investigated for Cauchy problems for hyperbolic Monge-Ampère equations. A characteristic uniformization is constructed on definite solutions of this problem, using which the existence and uniqueness of a maximal solution is established. It is shown that the characteristics in the different families that lie on a maximal solution and converge to a definite boundary point have infinite lengths. In this way a theory of global solvability is developed for the Cauchy problem for hyperbolic Monge-Ampère equations, which is analogous to the corresponding theory for ordinary differential equations. Using the same methods, a stable explicit difference scheme for approximating multivalued solutions can be constructed and a number of problems which are important for applications can be integrated by quadratures. Bibliography: 23 titles.
Matematicheskii Sbornik. 2020;211(3):71-123
71-123
A generalized theorem on curvilinear three-web boundaries and its applications
Abstract
Suppose that a curvilinear three-web is given by the equation $F(x,y,z)=0$. A specific structure of the derivatives of the function $F$ is established that characterizes regular three-webs. This makes it possible to list all regular three-webs formed by the Cartesian net and a family of circles, and also by the Cartesian net and a family of second-order curves. Bibliography: 4 titles.
Matematicheskii Sbornik. 2020;211(3):124-157
124-157
A connected compact locally Chebyshev set in a finite-dimensional space is a Chebyshev set
Abstract
Let $X$ be a Banach space. A set $M\subset X$ is a Chebyshev set if, for each $x\in X$, there exists a unique best approximation to $x$ in $M$. A set $M$ is locally Chebyshev if, for any point $x\in M$, there exists a Chebyshev set $F_x\subset M$ such that some neighbourhood of $x$ in $M$ lies in $F_x$. It is shown that each connected compact locally Chebyshev set in a finite-dimensional normed space is a Chebyshev set. Bibliography: 11 titles.
Matematicheskii Sbornik. 2020;211(3):158-168
158-168
Birational automorphisms of Severi-Brauer surfaces
Abstract
We prove that a finite group acting by birational automorphisms of a nontrivial Severi-Brauer surface over a field of characteristic zero contains a normal abelian subgroup of index at most $3$. Also, we find an explicit bound for the orders of such finite groups in the case when the base field contains all roots of $1$. Bibliography: 25 titles.
Matematicheskii Sbornik. 2020;211(3):169-184
169-184