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Machining accuracy of the electrochemical micro machining with pulse voltage mainly depends on pulse width of the voltage signals. To give nanometer level machining resolution, an expensive picoseconds pulse power supply is required. In this paper, a quadratic differential feedback method for electrochemical micro machining is proposed. With the method, nanometer level machining resolution can be obtained easily by tuning feedback loop gain under micro second pulse width. Here, the circuits of the machining system are designed and analyzed, and the circuit equation and transfer function of the system are deduced. By the equation, the basic principle of the machining method is revealed and the equation of the machining resolution for the method is also given. By micro hole machining experiments, effects of the feedback gain on the resolution are investigated and compared with the calculated ones. Good agreement is obtained which illustrates the proposed machining method. Besides it, some micro structures are produced and nanometer level machining resolution is obtained. [ABSTRACT FROM AUTHOR]

Let S denote the class of analytic and univalent (i.e., one-to-one) functions f (z) = z + ∑ n = 2 ∞ a n z n in the unit disk D = { z ∈ C : | z | < 1 } . For f ∈ S , In 1999, Ma proposed the generalized Zalcman conjecture that | a n a m - a n + m - 1 | ≤ (n - 1) (m - 1) , for n ≥ 2 , m ≥ 2 , with equality only for the Koebe function k (z) = z / (1 - z) 2 and its rotations. In the same paper, Ma (J Math Anal Appl 234:328–339, 1999) asked for what positive real values of λ does the following inequality hold? 0.1 | λ a n a m - a n + m - 1 | ≤ λ n m - n - m + 1 (n ≥ 2 , m ≥ 3). Clearly equality holds for the Koebe function k (z) = z / (1 - z) 2 and its rotations. In this paper, we prove the inequality (0.1) for λ = 3 , n = 2 , m = 3 . Further, we provide a geometric condition on extremal function maximizing (0.1) for λ = 2 , n = 2 , m = 3 . [ABSTRACT FROM AUTHOR]

We study Hill’s differential equation with potential expressed by elliptic functions which arises in some problems of physics and mathematics. Analytical method can be applied to study the local properties of the potential in asymptotic regions of the parameter space. The locations of the saddle points of the potential are determined, the locations of turning points can be determined too when they are close to a saddle point. Combined with the quadratic differential associated with the differential equation, these local data give a qualitative explanation for the asymptotic eigensolutions obtained recently. A relevant topic is about the generalisation of Floquet theorem for ODE with doubly-periodic elliptic function coefficient which bears some new features compared to the case of ODE with real valued singly-periodic coefficient. Beyond the local asymptotic regions, global properties of the elliptic potential are studied using numerical method. [ABSTRACT FROM AUTHOR]

In this paper, we show Liouville-type theorems for exponentially harmonic maps and p-harmonic maps between Kähler manifolds. We then study the quadratic form of an exponentially harmonic map of Kähler manifolds and its applications. We also prove that if f : M → N is an exponentially harmonic map from a compact Kähler manifold M into a Kähler manifold N with constant energy density such that the curvature of N is strongly semi-negative, then f is holomorphic or antiholomorphic. [ABSTRACT FROM AUTHOR]

Orthogonal families of bicircular quartics are naturally viewed as pairs of singular foliations of C ^ by vertical and horizontal trajectories of a non-vanishing quadratic differential. Yet the identification of these trajectories with real quartics in C P 2 is subtle. Here, we give an efficient, geometric argument in the course of updating the classical theory of confocal families in the modern language of quadratic differentials and the Edwards normal form for elliptic curves. In particular, we define a parameterized Edwards transformation, providing explicit birational equivalence between each curve in a confocal family and a fixed curve in normal form. [ABSTRACT FROM AUTHOR]

We apply the theory of quadratic differentials, to present a classification of orthogonal pairs of foliations of the hyperbolic plane by hyperbolic conics. Light rays are represented by trajectories of meromorphic differentials, and mirrors are represented by trajectories of the quadratic differential that represents the geometric mean of two such differentials. Using the notion of a hyperbolic conic as a mirror, we classify the types of orthogonal pairs of foliations of the hyperbolic plane by confocal conics. Up to diffeomorphism, there are nine types: three of these types admit one parameter up to isometry; the remaining six types are unique up to isometry. The families include all possible hyperbolic conics. [ABSTRACT FROM AUTHOR]

We describe a graph parametrization of rational quadratic differentials with presence of a simple pole, whose critical trajectories form a network depending on parameters focusing on the network topological jumps. Obtained bifurcation diagrams are associated with the Stasheff polytopes. [ABSTRACT FROM AUTHOR]

In the paper, we consider an open problem of finding the maximum of product of inner radii of mutually non-overlapping domains with respect to the points of the unit circle on a certain positive degree 𝛾 of the inner radius of the domain with respect to the origin, moreover, the domain containing origin does not intersect with other domains.

The upper estimate for the functional composed of the product of the inner radii of domains has been considered for spaces with a dimension larger than 2. [ABSTRACT FROM AUTHOR]

In this paper, the problem concerning the maximum product of the inner radii of n mutually non-overlapping domains symmetric with respect to a unit circle and the inner radius of some domain with respect to the zero raised to a positive degree γ is studied. The problem has been solved for n = 4 , 7 ¯ and some γ > 1. [ABSTRACT FROM AUTHOR]

In this paper we introduce flat grafting as a deformation of quadratic differentials on a surface of finite type that is analogous to the grafting map on hyperbolic surfaces. Flat grafting maps are generic in the strata structure and preserve parallel measured foliations. The 1-parameter family obtained by flat grafting allows us to explicitly describe a path connecting any pair of quadratic differentials. The slices of quadratic differentials closed under flat grafting maps with a fixed direction arise naturally and we prove rigidity properties with respect to the lengths of closed curves. [ABSTRACT FROM AUTHOR]

In geometric function theory of a complex variable problems on extremal decomposition with free poles on the unit circle are well known. One of such problem is the problem on maximum of the functional

In the paper we study a generalization of the extremal problem of geometric theory of functions of a complex variable on non-overlapping domains with free poles: Fix any γ ∈ R + and find the maximum (and describe all extremals) of the functional [r (B 0 , 0) r (B ∞ , ∞)] γ Π n k=1 r (B k , a k ) , where n ∈ N, n >= 2, a 0 = 0, |a k | = 1, B 0 , B ∞ , {B k } n k=1 is a system of mutually non-overlapping domains, a k ∈ B k ⊂ C, k = 0, n, ∞ ∈ B ∞ ⊂ C, (r(B, a) is an inner radius of the domain B ⊂ C at a ∈ B). Instead of the classical condition that the poles are on the unit circle, we require that the system of free poles is an n-radial system of points normalized by some "control" functional. A partial solution of this problem is obtained.

The paper is devoted to an extremal problem of geometric function theory of complex variable related with estimate of functional defined on system of mutually non-overlapping domains. An improved method is proposed for solving problems on extremal decomposition of the complex plane. The main results of the paper generalize and strengthening all known results in this problem for n-radial systems of points. In particular, the problem on extremal decomposition of the complex plane is solved for two free poles. [ABSTRACT FROM AUTHOR]

Problems on extremal decomposition of the complex plane with free poles located on an (n;m)-ray system of points are studied. A method that allowed us to obtain new upper bounds for the maximum of the products of the inner radii of mutually non-overlapping domains is proposed. [ABSTRACT FROM AUTHOR]

Although much research (f. e. [1], [3], [5], [7-15]) has been devoted to the extremal problems of a geometric function theory associated with estimates of functionals defined on systems of non-overlapping domains, however, in the general case the problems remain unsolved. The paper describes the problem of finding the maximum of a functional. This problem is to find a maximum of the product of inner radii of mutually non-overlapping symmetric domains with respect to a unit circle and the inner radius in some positive certain degree of the domain with respect to zero and description of extreme configurations. The topic of the paper is devoted to the study of the problem of the classical direction of the geometric theory of complex variable functions, namely, the extremal problems for non-overlapping domains. [ABSTRACT FROM AUTHOR]

The paper is devoted to one open extremal problem in the geometric function theory of complex variables associated with estimates of a functional defined on the systems of non-overlapping domains. We consider the problem of the maximum of a product of inner radii of n non-overlapping domains containing points of a unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in Dubinin's paper in the journal "Russian Mathematical Surveys" in the list of unsolved problems and then repeated in his monograph in 2014. Currently, it is not solved in general. In this paper, we obtained a solution of the problem for five simply connected domains and power γ ∈ (1; 2:57] and generalized this result to the case of multidimensional complex space. [ABSTRACT FROM AUTHOR]

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

Two problems of description of extremal configurations maximizing a product of the inner radii of mutually nonoverlapping domains are studied. One of the problems is analyzed in a more general situation: instead of nonoverlapping domains, the domains under the condition of partial disjointness are considered. The well-known problem posed in the work by V. N. Dubinin in 1988 is solved by Theorems 1 and 2. We study also the problem of maximum of a functional with the additional condition of symmetry defined by the domain G0. Theorems 3 and 4 give its partial solution. [ABSTRACT FROM AUTHOR]

We consider the problem of estimation of the functional In γ = r γ B o 0 ∏ k = 1 n r B k a k , where r(Bk; ak) is the inner radius of a domain Bk relative to the point ak, under the condition a 0 = 0 , ∣ a k ∣ = 1 , k = 1 , n ¯ , a k ∈ B k ⊂ ℂ ¯ , where the domains Bk ∩ Bp = ∅ Bk ∩ Bp = ∅, k ≠ p, k, p = 0 , n ¯ , and the domains Bk; k = 1 , n ¯ , possess a symmetry relative to a unit circle. In some partial cases, this problem was solved in [2–5]. The present work is devoted to the study of the problem for γ ∈ 1 n 1 3 and n ≥ 14. [ABSTRACT FROM AUTHOR]

In this work derivation of accurate estimate the production inner radius non-overlapping domains and open set. The problems arise such type in the first time in work [1]. It is late result this work generalize and strengthen in works [2 - 13]. In works [7, 8, 10] introduce the general systems points, the name n-radial systems points. In this work a success the draw generalize some results the work [7].

The results of this work are referred to the well-known trend of the geometric theory of functions of complex variable, namely, to the extreme problems on the classes of nonoverlapping domains. It was started by Lavrent’ev’s classical work [1], where, in particular, the problem of the product of conformal radii of two nonoverlapping domains was first solved. Now, this trend is intensively developed. The main results can be found in [2-8] and [9-13]. Our results present a generalization of some results in [7]. [ABSTRACT FROM AUTHOR]

We study special "discriminant" circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by Q0ℝ (−7) and Q0ℝ ([−3]2). The space Q0ℝ (−7) is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order seven with real periods; it appears naturally in the study of a neighborhood of the Witten cycle W5 in the combinatorial model based on Jenkins-Strebel quadratic differentials of Mg,n. The space Q0ℝ ([−3]2) is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most three with real periods; it appears in the description of a neighborhood of Kontsevich's boundary W1,1 of the combinatorial model. Applying the formalism of the Bergman tau function to the combinatorial model (with the goal of analytically computing cycles Poincaré dual to certain combinations of tautological classes) requires studying special sections of circle bundles over Q0ℝ (−7) and Q0ℝ ([−3]2). In the Q0ℝ (−7) case, a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces Q0ℝ (−7) and Q0ℝ ([−3]2), also called the spaces of Boutroux curves, in detail together with the corresponding circle bundles. [ABSTRACT FROM AUTHOR]

Abstract We describe the relationship between two spaces associated to a quiver with potential. The first is a complex manifold parametrizing Bridgeland stability conditions on a triangulated category, and the second is a cluster variety with a natural Poisson structure. For quivers of type A , we construct a local biholomorphism from the space of stability conditions to the cluster variety. The existence of this map follows from results of Sibuya in the classical theory of ordinary differential equations. [ABSTRACT FROM AUTHOR]

We consider one of the classical problems of the geometric theory of functions of a complex variable on a maximum of the functionalrB0.0rB∞∞γ∏k=1nrBkak,where n ∈ ℕ, n ≥ 2, γ ∈ ℝ+, An=akk=1n is a system of points such that |ak| = 1, a0 = 0, B0, B∞, Bkk=1n is a system of pairwise nonoverlapping domains, ak∈Bk⊂ℂ¯, k=0,n¯, ∞∈B∞⊂ℂ¯, r(B, a) is the inner radius of the domain B⊂ℂ¯ with respect to the point a ∈ B. We have analyzed this problem under some weaker restrictions on pairwise nonoverlapping domains. [ABSTRACT FROM AUTHOR]

We consider the well-known problem of maximum of the functionalInγ=rγB0.0∏k=1nrBkak,where B0, …, Bn are pairwise disjoint domains in ℂ¯, a0 = 0, |ak| = 1, k=1,n¯, are different points of the circle, γ ∈ (0, n], and r(B, a) is the inner radius of the domain B⊂ℂ¯ relative to the point a. In the case of simply connected domains for n=2, 3, and 4, we have obtained the solution of this problem for the maximum interval of values of the parameter γ. [ABSTRACT FROM AUTHOR]

Flat surfaces that correspond to k-differentials on compact Riemann surfaces are of finite area provided there is no pole of order k or higher. We denote by flat surfaces with poles of higher order those surfaces with flat structures defined by a k-differential with at least one pole of order at least k. Flat surfaces with poles of higher order have different geometrical and dynamical properties than usual flat surfaces of finite area. In particular, they can have a finite number of saddle connections. We give lower and upper bounds for the number of saddle connections and related quantities. In the case k=1or2, we provide a combinatorial characterization of the strata for which there can be an infinite number of saddle connections. [ABSTRACT FROM AUTHOR]

The paper is devoted to one extremal problem in geometric function theory of complex variables associated with estimates of functionals defined on the systems of non-overlapping domains. We consider Dubinin's problem of the maximum of product of inner radii of n non-overlapping domains containing points of the unit circle and the power γ of the inner radius of a domain containing the origin. The problem was formulated in 1994 in the work of Dubinin and then repeated in his monograph in 2014. Currently it is not solved in general. In this paper we generalized it to the case of the more general system of points and obtained a solution of this problem for some concrete values of n and γ. [ABSTRACT FROM AUTHOR]

The article is devoted to the study of a quite general problem of the geometric theory of functions on an extreme decomposition of the complex plane. The problem of maximum of the functionalInγ=rγB00∏k=1nrBkak,where γ ∈ (0, 1], n ≥ 2, a0 = 0,ak=1,k=1,n¯,ak∈Bk⊂C¯,k=0,n¯,Bkk=0n are pairwise disjoint domains, Bkk=0n are symmetric domains with respect to the unit circle, and r(B, a) is the inner radius of the domain B ⊂ C¯ relative to the point a ∈ B, is considered. [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]