Geometric Complex Analysis Research Articles

The use of control theory methods in neural networks’ trainingbased on a handwritten text

The paper shows the importance of reducing the neural networks’ training time at present stage and the role of new optimization methods in neural networks’ training. The paper researches a modification of stochastic gradient descent, which is based on the idea of gradient descent representation as a discrete dynamical system. The connection between the extreme points, to which the gradient descent iterations tend, and the stationary points of the corresponding discrete dynamical system is a consequence of this representation. The further applied stabilizing scheme with predictive control, for which a theoretical apparatus was developed bymeans of geometric complex analysis together with solving optimization tasks in a set of polynomials with real coefficients, was able to train a multilevel perceptron for recognizing handwritten numbers many times faster. The new algorithm software implementation used the PyTorch library, created for researches in the field of neural networks. All experiments were run on NVidia graphical processing unit to check the processing unit’s resource consumption. The numerical experiments did not reveal any deviation in training time. There was a slight increase in the used video memory, which was expected asthe new algorithm retains one additional copy of perceptron internal parameters. The importance of this result is associated with the growth in the useof deep neural network technology, which has grown three hundred thousand times from 2012 till 2018, and the associated resource consumption. This situation forces the industry to consider training optimization issues as well as their accuracy. Therefore, any training process acceleration that reduces the time or resources of the clusters is a desirable andimportant result, which was achieved in this article. The results obtained discover a new area of theoretical and practical research, since the stabilization usedis only one ofthe methods of stabilization and search for cycles in control theory. Such good practical results confirm the need to add the lagging control and the additional experiments with both predictive and lagging control elements.

Fredholm Eigenvalues and Quasiconformal Geometry of Polygons

An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory.This survey extends our previous survey of 2005 and presents the new approaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

Teichmüller spaces and coefficient problems for univalent holomorphic functions

We show that the features of Teichmuller spaces give rise to a completely new powerful tool for solving the classical coefficient problems in geometric function theory. Here we concern the univalent functions. Using the Bers isomorphism theorem for Teichmuller spaces of punctured Riemann surfaces and some of their other complex geometric features, we prove a general theorem on maximization of homogeneous polynomial (in fact, more general holomorphic) coefficient functionals $$J(f) = J(a_{m_1}, a_{m_2},\dots , a_{m_n}) $$ on some classes of univalent functions in the unit disk naturally connected with the canonical class S. The given functional J is lifted to the Teichmuller space $$\mathbf{T}_1$$ of the punctured disk $$\mathbb {D}_{*} = \{0< |z| < 1\}$$ which is biholomorphically equivalent to the Bers fiber space over the universal Teichmuller space. This generates a positive subharmonic function on the disk $$\{|t| < 4\}$$ with $$\sup _{|t|<4} u(t) = \max _{{\mathbf {T}}_1} |J|$$ attaining this maximal value only on the boundary circle, which correspond to rotations of the Koebe function. This theorem implies new sharp distortion estimates for univalent functions giving explicitly the extremal functions, and creates a new bridge between Teichmuller space theory and geometric complex analysis. In particular, it provides an alternate and direct proof of the Bieberbach conjecture.

Fredholm eigenvalues and quasiconformal geometry of polygons

An important open problem in geometric complex analysis is to establish algorithms for the explicit determination of the basic curvilinear and analytic functionals intrinsically connected with conformal and quasiconformal maps, such as their Teichmüller and Grunsky norms, Fredholm eigenvalues, and the quasireflection coefficient. This is important also for the potential theory but has not been solved even for convex polygons. This case has intrinsic interest in view of the connection of polygons with the geometry of the universal Teichmüller space and approximation theory. This survey extends our previous survey of 2005 and presents the newapproaches and recent essential progress in this field of geometric complex analysis and potential theory, having various important applications. Another new topic concerns quasireflections across finite collections of quasiintervals (to which the notion of Fredholm eigenvalues also can be extended).

On Fredholm Eigenvalues of Unbounded Polygons

An important open problem in geometric complex analysis is to establish some algorithms for explicit determination of the basic functionals intrinsically connected with conformal and quasiconformal mappings such as their Teichmuller and Grunsky norms, Fredholm eigenvalues and the quasireflection coefficient. This problem has not been solved even for generic quadrilaterals. We provide a restricted solution of the problem for unbounded rectilinear polygons.

On a family of extremal polynomials

For a pair of conjugate trigonometrical polynomials C(t)=∑j=1Najcos⁡jt, S(t)=∑j=1Najsin⁡jt with real coefficients and normalization a1=1 the following extremal value is found:supa2,…,aN⁡mint⁡{C(t):S(t)=0}=−14sec2⁡πN+2. An application of this result in geometric complex analysis is shown. Several conjectures for a number of extremal problems on classes of polynomials are suggested.

Robust discrete complex analysis: A toolbox

We prove a number of double-sided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: cross-ratios, extremal lengths and random walk partition functions. The results hold true for any simply connected discrete domain $\Omega$ with four marked boundary vertices and are uniform with respect to $\Omega$'s which can be very rough, having many fiords and bottlenecks of various widths. Moreover, due to results from [Boundaries of planar graphs, via circle packings (2013) Preprint], those estimates are fulfilled for domains drawn on any infinite "properly embedded" planar graph $\Gamma\subset \mathbb{C}$ (e.g., any parabolic circle packing) whose vertices have bounded degrees. This allows one to use classical methods of geometric complex analysis for discrete domains "staying on the microscopic level." Applications include a discrete version of the classical Ahlfors-Beurling-Carleman estimate and some "surgery technique" developed for discrete quadrilaterals.

A Question of Ahlfors

Abstract. In 1963, Ahlfors posed in [1] (and repeated in his book [2]) thefollowing question which gave rise to various investigations of quasiconformalextendibility of univalent functions. Question. Let � be a conformal map of the disk (or half-plane) ontoa domain with quasiconformal boundary (quasicircle). How can this map becharacterized? He conjectured that the characterization should be in analytic properties ofthe logarithmic derivative log� ′ = � ′′ /� ′ , and indeed, many results on quasicon-formal extensions of holomorphic maps have been established using � ′′ /� ′ andother invariants (see, e.g., the survey [9] and the references there).This question relates to another still not solved problem in geometric com-plex analysis: To what extent does the Riemann mapping function � of a Jordan domain �⊂ C determine the geometric and conformal invariants (characteristics) ofcomplementary domain � * = C ∖� ? The purpose of this paper is to provide a qualitative answer to these ques-tions, which discovers how the inner features of biholomorphy determine theadmissible bounds for quasiconformal dilatations and determine the Kobayashidistance for the corresponding points in the universal Teichmuller space.¨

Transfinite diameter, Chebyshev constants, and capacities in Cn

The famous result of geometric complex analysis, due to Fekete and Szegö, states that the transfinite diameter $d( K) $, characterizing the asymptotic size of $K$, the Chebyshev constant $\tau ( K) $, characterizing the minimal uniform deviation of a mo