An extremal metric approach to some problems on extremal decomposition

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. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 191–211.     

On symmetric configurations in some problems on extremal decomposition

Some problems on extremal decomposition in families of systems of nonoverlapping domains of various types with free parameters are studied. The functionals considered contain a sum of the reduced modules of biangles. For the problems under consideration, the existence of extremal configurations with appropriate symmetry is investigated. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 160–172.     

On symmetric configurations in some problems on extremal decomposition. II

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. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 158–179.     

On quadratic differentials on multiply connected domains that are perfect squares. II

Problems on extremal decomposition of a multiply connected domain are considered. Two theorems exhibiting the role in such problems of quadratic differentials that are perfect squares are proved. As an implication, it is shown that the extremal metric approach used in the paper naturally leads to the results recently obtained by Dubinin and Eyrikh (Zap. Nauchn. Semin. POMI, 314, 52–75 (2004)) with the use of the capacity theory for generalized condensers. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 40–51.     

Max-Min Problems on the Ranks and Inertias of the Matrix Expressions <Emphasis Type="Italic">A</Emphasis>−<Emphasis Type="Italic">BXC</Emphasis>±(<Emphasis Type="Italic">BXC</Emphasis>)<Superscript>∗</Superscript> with Applications

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ^∗), where A=±A ^∗ and (⋅)^∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)^∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)^∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ^∗ subject to Hermitian solutions of a consistent matrix equation AXA ^∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ^∗ A ^∼ B with respect to a Hermitian generalized inverse A ^∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.     

On extremal decomposition problems

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 M?bius 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. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 54–74.     

Conjugacy classes of non-connected semisimple algebraic groups

Consider a non-connected algebraic group G = G ⋉ Γ with semisimple identity component G and a subgroup of its diagram automorphisms Γ. The identity component G acts on a fixed exterior component Gτ, id ≠ τ ∈ Γ by conjugation. In this paper we will describe the conjugacy classes and the invariant theory of this action. Let T be a τ -stable maximal torus of G and its Weyl group W. Then the quotient space Gτ//G is isomorphic to (T/(1 − τ )(T))/W^τ. Furthermore, exploiting the Jordan decomposition, the reduced fibres of this quotient map are naturally associated bundles over semisimple G-orbits. Similar to Steinberg's connected and simply connected case [22] and under additional assumptions on the fundamental group of G, a global section to this quotient map exists. The material presented here is a synopsis of the Ph.D thesis of the author, cf. [15].     

The maximum of the product of conformal, radii of nonoverlapping domains in the disk

Two problems on the extremal decomposition of the unit disk are considered. It is shown that the associated quadratic differentials in these problems have at least two distinct zeros on the unit circle. Bibliography: 7 titles.Dedicated to the 90th anniversary of G. M. Goluzin's birthTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 105–118.     

Genus two extremal surfaces: Extremal discs, isometries and Weierstrass points

It is known that the largest disc that a compact hyperbolic surface of genusg may contain has radiusR=cosh⁻¹(1/2sin(π/(12g−6))). It is also known that the number of such (extremal) surfaces, although finite, grows exponentially withg. Elsewhere the authors have shown that for genusg>3 extremal surfaces contain only one extremal disc. Here we describe in full detail the situation in genus 2. Following results that go back to Fricke and Klein we first show that there are exactly nine different extremal surfaces. Then we proceed to locate the various extremal discs that each of these surfaces possesses as well as their set of Weierstrass points and group of isometries. Both authors partially supported by Grant BFM2000-0031 of the SGPI.MCYT.     

Convexes divisibles IV : Structure du bord en dimension 3

Divisible convex sets IV: Boundary structure in dimension 3 Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there exists a torsion free discrete group Γ of projective transformations preserving Ω such that the quotient M := Γ\Ω is compact. We study the structure of M and of ∂Ω, when Ω is not strictly convex: The union of the properly embedded triangles in Ω projects in M onto an union of finitely many disjoint tori and Klein bottles which induces an atoroidal decomposition of M. Every non extremal point of ∂Ω is on an edge of a unique properly embedded triangle in Ω and the set of vertices of these triangles is dense in the boundary of Ω (see Figs. 1 to 4). Moreover, we construct examples of such divisible convex open sets Ω.      

Single-point extremal functions in weighted bergman spaces

For weighted Berman spaces in the unit disk the extremal functions for invariant subspaces formed by functions vanishing at a fixed point are studied. In the case where the weight is radial and logarithmically subharmonic, it is shown that such extremal functions can serve for the separation of single zeros. It is also proved that the reproducing kernels of the Bergman spaces are univalent functions. Bibliography: 7 titles. Translated fromProblemy Matematicheskogo Analiza, No. 15, 1995, pp. 241–252.     

Geometric properties of extremal decompositions and estimates for the modulus of families of curves in an annulus

We study the geometric properties of the boundary for problems on extremal decomposition into two domains of an annulus or a rectangle with a barrier removed. Sharp estimates of moduli are proved for certain families of curves bending around the barrier in a canonical domain. Bibliography: 17 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 93–114. Translated by A. Yu. Solynin.     

Extremal barriers on cones with Phragmèn-Lindelöf theorems and other applications

Summary We give a detailed analysis of supersolutions of form rλf(θ) for the classL _α of uniformly elliptic operators in nondivergence form with ellipticity constant α. Fundamental to the analysis is the extremal supersolution rλFλ, α(θ) for each real λ, which is itself a solution of a certain extremal elliptic equation and which remains positive on a cone of wider aperature than any other supersolution of this form. These supersolutions are employed as “barriers” to yeild Phragmen-Lindel?f theorems (λ>0) for unbounded domains contained in a cone, and H?lder continuity (λ>0) and removeable singularity (λ<0) results at boundary points with an exterior cone property. In each case the growth condition 0(r_λ) involved, depending on the aperature of the cone, is best possible. Similar results carry through for operators with singular lower order terms and possibily positive zero order coefficient. This work was partially supported by AFOSR grant number 553–64. Entrata in Redazione il 29 gennaio 1971.     

Some extremal problems for functions of bounded boundary rotation

A variational method is developed within the class of functions of boundary rotation not exceedingkπ which is based on the fact that the set of representing measuresμ is convex. It shows that an extremal problem related to a functional with Gateaux derivative and some constraints leads to extremal measuresμ ₀ with finite support. The positive and negative part of aμ ₀ is located at points where a functionJ (depending onμ ₀) reaches its maximum and minimum respectively. The method is tested successfully on various problems.     

A class of constructions for turán’s (3, 4)-problem

Letf(n) denote the minimal number of edges of a 3-uniform hypergraphG=(V, E) onn vertices such that for every quadrupleY ⊂V there existsY ⊃e ∈E. Turán conjectured thatf(3k)=k(k−1)(2k−1). We prove that if Turán’s conjecture is correct then there exist at least 2 ^k−2 non-isomorphic extremal hypergraphs on 3k vertices.     

The maximum of the conformal radius in the families of domains satifying additional conditions

We solve the problems on the maximum of the conformal radius R(D,1) in the family D(R₀) of all simply connected domains D ⊃ ℂ containing the points 0 and 1 and having a fixed value of the conformal radius R(D,0)=R₀, and in the family D(R₀, ρ) of domains from D(R₀) with given hyperbolic distance ρ=ρ_D(0,1) between 0 and 1. Analogs of the mentioned problems for doubly-connected domains with given conformal module are considered. Solution of the above problems is based on results of general character in the theory of problems of extremal decomposition and related module problems. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 93–108.     

On local Torelli for a class of extremal varieties

We study the local behaviour of the period map for smooth non-degenerate extremal varieties sitting inside scrolls. We conclude with the following theorem:the period map for smooth non-degenerate extremal varieties of dimension k ≠ 2,3in n-projective space with n≥2k+4is generically locally injective for sufficiently high degree d. In particular, the following two conditions on d suffice: (1)d≥3.k.(n−k)+3; (2)d≥(n − k)²+1.     

On a Conjecture of E. A. Rakhmanov

It is shown that a conjecture of E. A. Rakhmanov is true concerning the zero distribution of orthogonal polynomials with respect to a measure having a discrete real support. We also discuss the case of extremal polynomials with respect to some discrete L _p -norm, 0 < p ≤∈fty , and give an extension to complex supports. Furthermore, we present properties of weighted Fekete points with respect to discrete complex sets, such as the weighted discrete transfinite diameter and a weighted discrete Bernstein—Walsh-like inequality. August 24, 1998. Date revised: March 26, 1999. Date accepted: April 27, 1999.     

The analytic fixed point function and its properties

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. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 238–252.