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This paper provides an overview of the latest research on the two-sided estimates of classical characteristics of growth of entire functions such as the type and the lower type in terms of the ordinary or average densities of the distribution of zeros. We give also accurate estimates of the type of an entire function, taking into account additionally the step and the lacunarity index of the sequence of zeros. The results under consideration are based on the solution of extremal problems in classes of entire functions with restrictions on the behavior of the zero set. Particular attention is paid to the following important cases of the location of zeros: on a ray, on a straight line, on a number of rays, in the angle, or arbitrarily in the complex plane. [ABSTRACT FROM AUTHOR]

We consider an open extremal problem in geometric function theory of complex variables on the maximum of the functional [ABSTRACT FROM AUTHOR]

Some extremal problems of the geometric theory of functions of a complex variable related to the estimates of functionals defined on systems of non-overlapping domains are considered. Till now, many such problems have not been solved, though some partial solutions are available. In the paper, the improved method is proposed for solving the problems on extremal decomposition of the complex plane. The main results generalize and strengthen some known results in the theory of non-overlapping domains with free poles to the case of an arbitrary arrangement of systems of points on the complex plane. [ABSTRACT FROM AUTHOR]

The problems of extremal decomposition with free poles on a circle are well known in the geometric theory of functions of a complex variable. One of such problems is the problem of maximum of the functionalInγ=rγB00∏k=1nrBk,ak,where γ ∈ (0, n], B0, B1, B2,...,Bn, n ≥ 2, are pairwise disjoint domains in C¯,a0=0,ak=1,k=1,n¯ are different points of the circle, r(B, a) is the inner radius of the domain B ⊂ C¯ relative to the point a ∈ B. We consider a more general problem, in which the restriction ak=1,k=1,n¯, is replaced by a more general condition. [ABSTRACT FROM AUTHOR]

Sharp estimates of a product of the inner radii of pairwise disjoint domains are obtained. In particular, the extremal problem in the case of any finite number of free poles on a radial system of points is solved. [ABSTRACT FROM AUTHOR]

Applications of the method of modules of curve families to extremal problems of geometric function theory are presented. For these problems, the associated quadratic differentials have a large number of free poles. Bibliography: 13 titles. [ABSTRACT FROM AUTHOR]

Results and applications of the method of modules in geometric function theory are presented. The method was originally created by J. A. Jenkins, and further developed in works of the Leningrad-St. Petersburg mathematical school. A retrospective description of the origin of the method is given, and the determining role of Jenkins in the development of the method of the extremal metric is pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the notions of separating value and tantiles of continuous random variables and examine their extremal properties and extremal properties of quantiles. We obtain estimates for the differences between expected values, medians, and separating values of continuous random variables. [ABSTRACT FROM AUTHOR]

The symmetry/asymmetry of functions providing sharp constants in the embedding theorems[InlineMediaObject not available: see fulltext.] for various r and k is studied. The sharp constants for all r > k in the cases k = 4 and k = 6 are calculated explicitly as well. Bibliography: 16 titles. [ABSTRACT FROM AUTHOR]

For every real number γ ≥ 1 we denote by K( γ) the least possible value of the constant term of an even nonnegative trigonometric polynomial with monotone coefficients such that all its coefficients, save for the constant term, are not lesser than 1 and the sum of these coefficients equals γ. In this paper, the asymptotic estimate of K( γ) is found and some extremal problems on the minimum of the constant term of an even nonnegative trigonometric polynomial are studied. [ABSTRACT FROM AUTHOR]

We study properties of extremals of the variational problem about the shortest paths in the Carnot–Carathéodory metric on Heisenberg groups. We prove that the Carnot–Carathéodory balls in Heisenberg groups satisfy the Carnot–Carathéodory homogeneous cone condition. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

We consider the variational problem of equilibrium of an elastic plate (the Timoshenko model) containing a crack. On the curve describing the crack, we impose boundary conditions in the form of inequalities. We study the dependence of the solutions on the curve shape. We prove the existence of an extremal curve shape and show that the solutions weakly converge in the Sobolev space as the parameter describing the crack shape tends to zero. The convergence is strong if the external forces satisfy the Lipschitz condition. Bibliography: 19 titles. Illustrations: 1 figure. [ABSTRACT FROM AUTHOR]

Problems on extremal decomposition in which the associated quadratic differentials possess the property indicated in the title are solved. The results obtained supplement the classical results in problems under consideration. Bibliography: 15 titles. [ABSTRACT FROM AUTHOR]

A number of extremal problems of approximation theory of functions have been solved on the real line $$ \mathbb{R} $$ . Exact constants in the Jackson-type inequalities in the space L( $$ \mathbb{R} $$) are calculated. The exact values of average ν-widths are obtained for the classes of functions from L( $$ \mathbb{R} $$) defined by averaged moduli of continuity of the k-th order, as well as for the classes of functions defined by K-functionals. The quite complete analysis of the final results related to the solution of extremal problems of approximation theory in the periodic case and for the whole real axis is carried out in the chronological order. [ABSTRACT FROM AUTHOR]

We study the normal structure of the unipotent radical U of a Borel subgroup in a Lie type group over a field K. Thus, all maximal Abelian normal subgroups in U are described. This gives a new solution of C. Parker and P. Rowley's problem about extremal subgroups in U and the description in finite groups U of the large normal (and, as proved, also normal large) Abelian subgroups. [ABSTRACT FROM AUTHOR]

The paper consists of two parts, both related to the complex geometry of the universal Teichmüller space. We reprove that all contractible invariant metrics on this space coincide and apply this important fact to solving the general extremal problems for univalent functions with quasiconformal extensions. [ABSTRACT FROM AUTHOR]

In this paper, a description of the null-sets for the generalized Aikawa-Ohtsuka condenser module is obtained under the assumption that the module has the continuity property. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The problem of existence of an optimal control is solved on the basis of Weierstrass's classical theorem if the set of admissible controls belongs to the class of piecewise continuous functions. In the process of describing admissible controls, the main assumption is that the number of switchings (points of discontinuity) is uniformly bounded and not just finite, as in the main problem of optimal control theory. On the one hand, this assumption does not restrict the spectrum of optimal control applications. On the other hand, it fits the Weierstrass's theorem owing to the convenience in characterizing the sequential compactness. The formulation of Weierstrass's theorem, which asserts the existence of continuous function extrema on sequentially compact sets, is customary, and its proof complies with the traditional scheme, whereas the concepts (convergent sequences and some others) are adapted to the peculiarity of optimal problems. [ABSTRACT FROM AUTHOR]

An optimal control problem with two phase constraints and a two-dimensional control is considered. We prove that its optimal trajectories have a countable number of points of contact with the phase constraint bound on a finite time interval. The optimal synthesis of the problem after applying the quotient mapping by the action of the scale group has a complicated topological structure and, in some neighborhood of singular extremals, it is the Reeb foliation. [ABSTRACT FROM AUTHOR]

Some problems on extremal decomposition in families of non overlapping domains containing systems of biangles with distinguished vertices are considered. The present work continues earlier author’s investigations. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

This paper studies a class of optimal control problems and the Hamiltonian systems generated by them in the space l2. The authors prove the existence of extremals with countably many switchings on a finite interval of time that compose a bundle with piecewise fibers over the base of singular extremals of condimension 4. [ABSTRACT FROM AUTHOR]

Some problems on extremal decomposition in families of nonoverlapping domains containing systems of biangles with free vertices on a circle are considered. Simultaneously, some progress in solving the classical problem on the maximum of a well-known conformal invariant is achieved. This exhibits the role of symmetric configurations in extremal problems under consideration. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

The capacity approach and symmetrization method arc adapted to some extremal decomposition problems on the unit disk or an annulus. The problems on the maximum product of the interior radii of pairwise nonoverlapping domains and the maximum product of the Robin radii of such domains are considered. New invariants with respect to the Möbius transformations of the Riemann sphere are introduced. In particular, for these invariants problems on extremal decomposition with free poles on the unit circle are investigated. Bibliography: 19 titles. [ABSTRACT FROM AUTHOR]

The paper considers optimal control problems linearly depending on the scalar control parameter in which there exist first-order singular extremals. The author proves a theorem on the structure of a generic Lagrange manifold (field of extremals) in a neighborhood of first-order singular extremals. As a consequence of this theorem, the author proves the optimality of singular extremals and nonsingular extremals in problems with fixed endpoints on small intervals of time. As an illustration, the paper presents constructions of Lagrange manifolds for the general linear-quadratic control problem with completely integrable linear system of differential constraints and for a certain problem of mathematical economics, a two-factor economic growth model with production function of the Cobb-Douglas type. [ABSTRACT FROM AUTHOR]

Some properties of the analytic fixed point function, introduced recently by D. Mejía and Ch. Pommerenke, are discussed. In particular, several extremal properties of such functions, related to mappings from the unit disk onto symmetric Riemann surfaces, are established. Bibliography: 17 titles. [ABSTRACT FROM AUTHOR]

The paper presents a simple extremal metric approach to problems on extremal decomposition in families of systems of nonoverlapping domains of different types. The approach is based on using general properties of trajectories of the associated quadratic differentials for simpler extremal decomposition problems. Bibliography: 8 titles. [ABSTRACT FROM AUTHOR]

New definitions of normal (in the sense of Grötzsch) annular and circular domains with radial slits are considered. Their geometric and functional properties are established. Arguments based on the method of extremal metric in the sense of Fuglede allow one to extend these properties to the n-dimensional case, n ≥ 2. The properties established provide a means for simplifying the proofs of the related results by Strebel and Reich and for bringing them to a complete form. Bibliography: 12 titles. [ABSTRACT FROM AUTHOR]

Principles of symmetry and composition for the conformal capacity of generalized condensers are considered. Interrelations among these principles, the classical Grötzsch lemmas, and certain types of symmetrization are discussed. Applications to estimating the logarithmic and hyperbolic capacities of compacts and to extremal decomposition problems are presented. Bibliography: 27 titles. [ABSTRACT FROM AUTHOR]

The paper considers the problem on extremal decomposition of a multiply connected domain G ⊂ ℂ in the case where the associated quadratic differential is a perfect square. It is shown that in the case considered, the value of the functional for this extremal decomposition is the least one in a certain class of decompositions. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

The present paper continues the series of articles published in “Zapiski Nauchnykh Seminarov POMI” and devoted to the problems indicated in the title and related questions. Bibliography: 7 titles. [ABSTRACT FROM AUTHOR]

A new approach to the classical Lagrange problem on the form of the strongest circular column of fixed volume and height clamped at both ends is proposed. The existence of the optimal column is proved and an algorithm for finding its design is given. [ABSTRACT FROM AUTHOR]

Examines the quadratic extremality conditions for broken extremals in the general problem of the calculus of variations. Abstract scheme for obtaining higher-order conditions in smooth extremal problems with constraints; Creation of the quadratic theory of conditions for Pontryagin minimum in the general problem of the calculus of variations that does not contain the local relation g(x,u,t) = 0; Decoding of the basic constant for the set of so-called "Pontryagin sequences".

Studies extremal limit laws for discrete random variables. Non-negative integers with probability mass function and cumulative distribution function; Findings that limiting distribution is not nondegenerate.

Discusses a functional extremal criterion. Description of the model; Use of the notion increasing in the sense of nondecreasing; Use of an unboundedly increasing in n sequence of mappings; Formation of associated processes with additive increments.

Some properties of extremal functions in the inequality $\backslash inf\_\{u\backslash in\; W^1\_p(\backslash Omega),\; u\backslash ne\; 0\}\backslash frac\{\backslash |u\backslash |\_\{W^1\_p(\backslash Omega)\}\}\{\backslash |u\backslash |\_\{L\_q(\backslash Omega)\}\}\; \backslash gt\; 0$ are studied. Bibliography: 10 titles. [ABSTRACT FROM AUTHOR]

In this paper, we study the regularity properties of sub-Riemannian distances and their interrelations with abnormal extremals. The existence of singular minimizers makes the study of sub-Riemannian distances rather complicated. It is known that these distances are not C"- in a neighborhood of a point reached by an abnormal minimizer [8] and may also fail to be subanalytic [2]. [ABSTRACT FROM AUTHOR]

We study, in a unified way, the following questions related to the properties of Pont ryagin extremals for optimal control problems with unrestricted controls: (i) How do the transformations defining the equivalence of two problems transform the extremals? (ii) How does one obtain quantities which are conserved along any extremal? (iii) How does one verify that the set of extremals includes the minimizers predicted by the existence theory? These questions are closely related with (i) the Carathéodory method which establishes a correspondence between the minimizing curves of equivalent problems; (ii) the interrelation between the concept of invariance and the theory of optimality conditions in optimal control, which are the concern of the Noether theorems; (iii) the regularity conditions for the minimizers and the work pioneered by Tonelli. [ABSTRACT FROM AUTHOR]

Discrete analogs of extrema of curvature and generalizations of the four-vertex theorem to the case of polygons and polyhedra are suggested and developed. Several interrelated approaches are considered. For smooth curves and polygonal lines in the plane, a formula relating the number of extrema of curvature to the winding numbers of the curves (polygonal lines) and their caustics is obtained. One of the main results in the higher-dimensional case says that any regular triangulation of a d-ball containing at least d simplices has at least d so-called “ears.” Bibliography: 25 titles. [ABSTRACT FROM AUTHOR]

One-dimensional branching extremals of Lagrangian-type functionals are considered. Such extremals appear as solutions to the classical Steiner problem on a shortest network, i.e., a connected system of paths that has the smallest total length among all the networks spanning a given finite set of terminal points in the plane. In the present paper, the Manhattan-length functional is investigated, with Lagrangian equal to the sum of the absolute values of projections of the velocity vector onto the coordinate axes. Such functionals are useful in problems arising in electronics, robotics, chip design, etc. In this case, in contrast to the case of the Steiner problem, local minimality does not imply extremality (however, each extreme network is locally minimal). A criterion of extremality is presented, which shows that the extremality with respect to the Manhattan-length functional is a global topological property of networks. Bibliography: 95 titles. [ABSTRACT FROM AUTHOR]

We consider some conventional problems of the theory of functions of a complex variable such that their extremal configurations have the n-fold symmetry. We discuss two-point distortion theorems corresponding to the two-fold symmetry. New estimates are obtained for the module of a doubly connected domain. These estimates generalize known results by Rengel, Grötzsch, and Teichmüller to the case of rings with the n-fold symmetry, where $n\backslash ge2$. New distortion theorems are proved for functions meromorphic and univalent in a disk or in a ring. In these theorems, the extremal function also has the corresponding symmetry. All of the problems mentioned above are unified by the method applied; this method is based on properties of the conformal capacity and on symmetrization. Bibliography: 27 titles. [ABSTRACT FROM AUTHOR]

In terms of polar sets, we complement some Dubinin's inequalities expressing the monotonicity of the generalized reduced module for open subsets of $\backslash Bbb\; C^n$ and $\backslash Bbb\; R^n$ by equality cases. These results are applied to extremal problems of geometric function theory. Bibliography: 19 titles. [ABSTRACT FROM AUTHOR]

We apply a variant of the method of the extremal metric to some problems concerning extremal decompositions and related problems. Let $\backslash bold\; a=\backslash \{a\_1,\backslash ldots\; ,a\_n\backslash \}$ be a system of distinct points on $\backslash overline\{\backslash Bbb\; C\}$ and let $\backslash Cal\; D(\backslash bold\; a)$ be the family of all systems $\backslash Bbb\; D=\backslash \{D\_1,\backslash ldots\; ,D\_n\backslash \}$ of nonoverlapping simply connected domains on $\backslash overline\{\backslash Bbb\; C\}$ such that $a\_k\backslash in\; D\_k$, $k=1,\backslash ldots,n$. Let $J(\backslash bold\; a)=\backslash max\_\{\backslash Bbb\; D\backslash subset\; \backslash Cal\; D(\backslash bold\; a)\}\; \backslash Big\backslash \{\; 2\backslash pi\; \backslash sum^n\_\{k=1\}M(D\_k,a\_k)-\backslash frac2\{n-1\}\backslash sum\_\{1\backslash le\; k\; \backslash lt\; l\backslash le\; n\}\backslash log\; |a\_k-a\_l|\; \backslash Big\backslash \},$ where $M(D\_k,a\_k)$ is the reduced module of the domain $D\_k$ with respect to the point $a\_k$. At present, the problem concerning the value $\backslash max\backslash limits\_\{\backslash bold\; a\}J(\backslash bold\; a)$ was solved completely for $n=2,3,4$. In this work, we continue the previous author's investigations and consider the case $n=5$. In addition, we consider the problem concerning the maximum of the sum $\backslash alpha^2\backslash big\backslash \{\; M(D\_0,0)+M(D\_\{n+1\},\backslash infty)\backslash big\backslash \}\; +\backslash sum^n\_\{k=1\}M(D\_k,a\_k)$ in the family $\backslash Cal\; D(\backslash bold\; a)$ introduced above, where $\backslash bold\; a=\backslash \{0,a\_1,\backslash ldots,a\_n,\backslash infty\backslash \}$, $a\_k$, $k=1,\backslash ldots,n$, are arbitrary points of the circle $|z|=1$, and $\backslash alpha$ is a positive number. We prove that if $\backslash alpha/n\backslash le\; 1/\backslash sqrt\; 8$, then the maximum is attained only for systems of equidistant points of the circle $|z|=1$. For $\backslash alpha/n=1/\backslash sqrt8$, this result was obtained earlier by Dubinin who applied the method of symmetrization. It is shown that if $n\backslash ge2$, where $\backslash alpha/n\backslash ge1/2$ is an even number, then equidistant points of the circle $|z|=1$ do not realize the indicated maximum. Bibliography: 11 titles. [ABSTRACT FROM AUTHOR]

An inequality for the Vandermonde determinants is proved. In particular, the results confirm the conjecture suggested in [J. Math. Sci. 117 (2003), No. 3] about the extremal property of the so-called Euler integrated processes. Bibliography: 2 titles. [ABSTRACT FROM AUTHOR]