Matti Vuorinen

University of Turku | UTU · Department of Mathematics and Statistics

Let Ω be the multiply connected domain in the extended complex plane \(\overline {\mathbb {C}}\) obtained by removing m non-overlapping rectilinear segments from the infinite strip \(S=\{z : \left |\text {Im} z\right |<\pi /2\}\). In this paper, we present an iterative method for numerical computation of a conformally equivalent bounded multiply co...

We study computational methods for the approximation of special functions recurrent in geometric function theory and quasiconformal mapping theory. The functions studied can be expressed as quotients of complete elliptic integrals and as inverses of such quotients. In particular, we consider the distortion function $\varphi_K(r)$ which gives a majo...

We study certain points significant for the hyperbolic geometry of the unit disk. We give explicit formulas for the intersection points of the Euclidean lines and the stereographic projections of great circles of the Riemann sphere passing through these points. We prove several results related to collinearity of these intersection points, offer new...

For augmentation of the square-shaped image data of a convolutional neural network (CNN), we introduce a new method, in which the original images are mapped onto a disk with a conformal mapping, rotated around the center of this disk and mapped under such a M\"obius transformation that preserves the disk, and then mapped back onto their original sq...

We study the mixed Dirichlet–Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane. The Dirichlet / Neumann conditions at opposite pairs of sides are \(\{0,1\}\) and \(\{0,0\},\) resp. The solution to this problem is a harmonic function in the unbounded complement of the...

We study numerical conformal mappings of planar Jordan domains with boundaries consisting of finitely many circular arcs, also called polycircular domains, and compute the moduli of quadrilaterals for these domains. Experimental error estimates are provided and, when possible, comparison to exact values or other methods are given. We also analyze t...

We study numerical conformal mapping of multiply connected planar domains with boundaries consisting of unions of finitely many circular arcs, so called polycircular domains. We compute the conformal capacities of condensers defined by polycircular domains. Experimental error estimates are provided for the computed capacity and, when possible, the...

We study the point pair function in subdomains G of $${\mathbb {R}}^n$$ R n . We prove that, for every domain $$G\subsetneq {\mathbb {R}}^n$$ G ⊊ R n , this function is a quasi-metric with the constant less than or equal to $$\sqrt{5}/2$$ 5 / 2 . Moreover, we show that it is a metric in the domain $$G={\mathbb {R}}^n{\setminus }\{0\}$$ G = R n \ {...

We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as the Alhazen problem. For the spherical mirror problem, we focus on the reflection property of an ellipse and show that the catacaustic curve of the unit circle follows naturally from the equation obt...

We study inequalities between the hyperbolic metric and intrinsic metrics in convex polygonal domains in the complex plane. Special attention is paid to the triangular ratio metric in rectangles. A local study leads to an investigation of the relationship between the conformal radius at an arbitrary point of a planar domain and the distance of the...

We study the mixed Dirichlet-Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane. The Dirichlet\,/\,Neumann conditions at opposite pairs of sides are $\{0,1\}$ and $\{0,0\},$ resp. The solution to this problem is a harmonic function in the unbounded complement of the p...

For a domain G in the one-point compactification $\overline{\mathbb{R}}^n = {\mathbb{R}}^n \cup \{ \infty\}$ of ${\mathbb{R}}^n, n \geqslant 2$ , we characterise the completeness of the modulus metric $\mu_G$ in terms of a potential-theoretic thickness condition of $\partial G\,,$ Martio’s M -condition [ 35 ]. Next, we prove that $\partial G$ is un...

In this paper we discuss problems concerning the conformal condenser capacity of "hedgehogs", which are compact sets $E$ in the unit disk $\mathbb{D}=\{z:\,|z|<1\}$ consisting of a central body $E_0$ that is typically a smaller disk $\overline{\mathbb{D}}_r=\{z:\,|z|\le r\}$, $0<r<1$, and several spikes $E_k$ that are compact sets lying on radial i...

A new intrinsic metric called the t-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains G⊊Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepacka...

Let $S$ be a strip in ${\mathbb{C}}$ and let $E\subset S$ be a union of disjoint segments. For the domain $S \setminus E$, we construct a numerical conformal mapping onto a domain bordered by smooth Jordan curves. To this aim, we use the boundary integral equation method from [19]. In particular, we apply this method to study the conformal capacity...

The hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under quasiconformal maps from one sector onto another one.

We study numerical conformal mapping of multiply connected planar domains with boundaries consisting of unions of finitely many circular arcs, so called polycircular domains. We compute the conformal capacities of condensers defined by polycircular domains. Experimental error estimates are provided for the computed capacity and, when possible, the...

We study the point pair function in subdomains $G$ of $\mathbb{R}^n$. We prove that, for every domain $G\subsetneq\mathbb{R}^n$, the this function is a quasi-metric with the constant less than or equal to $\sqrt{5}\slash2$. Moreover, we show that it is a metric in the domain $G=\mathbb{R}^n\setminus\{0\}$ with $n\geq1$. We also consider generalized...

We study the conformal capacity by using novel computational algorithms based on implementations of the fast multipole method, and analytic techniques. Especially, we apply domain functionals to study the capacities of condensers (G,E) where G is a simply connected domain in the complex plane and E is a compact subset of G. Due to conformal invaria...

We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in...

We discuss the problem of the reflection of light on spherical and quadric surface mirrors. In the case of spherical mirrors, this problem is known as Alhazen's problem. Among a number of equations that solve the spherical mirror problem, we focus on the equation that uses the reflection property of an ellipse and discuss the relationship between t...

Given a compact connected set E in the unit disk B2, we give a new upper bound for the conformal capacity of the condenser (B2,E) in terms of the hyperbolic diameter t of E. Moreover, for t>0 we construct a set of hyperbolic diameter t and apply novel numerical methods to show that it has larger capacity than a hyperbolic disk with the same diamete...

We study numerical conformal mappings of planar Jordan domains with boundaries consisting of finitely many circular arcs and compute the moduli of quadrilaterals for these domains. Experimental error estimates are provided and, when possible, comparison to exact values or other methods are given. The main ingredients of the computation are boundary...

Given a nonempty compact set \( E \) in a proper subdomain \( \Omega \) of the complex plane, we denote the diameter of \( E \) and the distance from \( E \) to the boundary of \( \Omega \) by \( d(E) \) and \( d(E,\partial\Omega) \), respectively. The quantity \( d(E)/d(E,\partial\Omega) \) is invariant under similarities and plays an important ro...

The M\"obius metric $\delta_G$ is studied in the cases where its domain $G$ is an open sector of the complex plane. We introduce upper and lower bounds for this metric in terms of the hyperbolic metric and the angle of the sector, and then use these results to find bounds for the distortion of the M\"obius metric under quasiregular mappings defined...

We apply domain functionals to study the conformal capacity of condensers $(G,E)$ where $G$ is a simply connected domain in the complex plane and $E$ is a compact subset of $G$. Due to conformal invariance, our main tools are the hyperbolic geometry and functionals such as the hyperbolic perimeter of $E$. Novel computational algorithms based on imp...

We study moduli of planar ring domains whose complements are linear segments and establish formulas for their moduli in terms of the Weierstrass elliptic functions. Numerical tests are carried out to illustrate our results.

We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics. We also prove distortion results for this metric under quasiregular maps.

For compact subsets E of the unit disk D we study the capacity of the condenser cap(D,E) by means of set functionals defined in terms of hyperbolic geometry. In particular, we study experimentally the case of a hyperbolic triangle and arrive at the conjecture that of all triangles with the same hyperbolic area, the equilateral triangle has the leas...

We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies to a wide variety of geometries: domains are assumed to have smooth or piecewise smooth boundaries. The method we use is based on the boundary integ...

The triangular ratio metric is studied in a domain G ⊊ R n , n ≥ 2 . Several sharp bounds are proven for this metric, especially in the case where the domain is the unit disk of the complex plane. The results are applied to study the Hölder continuity of quasiconformal mappings.

Given a compact connected set $E$ in the unit disk $\mathbb{B}^2$, we give a new upper bound for the conformal capacity of the condenser $(\mathbb{B}^2, E)\,$ in terms of the hyperbolic diameter $t$ of $E$. Moreover, for $t>0$ we construct a set of diameter $t$ and show by numerical computation that it has larger capacity than a hyperbolic disk wit...

For a given ring (domain) in \(\overline{\mathbb {R}}^n\), we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all \(n\ge 3\), the standard definition of uniformly perfect sets in terms of the Euclidean metric is equivalent to the bounde...

This paper studies the numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The used method is based on the boundary integral equation with the generalized Neumann kernel. Several numer...

For compact subsets $E$ of the unit disk $ \mathbb{D}$ we study the capacity of the condenser ${\rm cap}( \mathbb{D},E)$ by means of set functionals defined in terms of hyperbolic geometry. In particular, we study experimentally the case of a hyperbolic triangle and arrive at the conjecture that of all triangles with the same hyperbolic area, the e...

We consider Mityuk's function and radius which have been proposed in [17] as generalizations of the reduced modulus and conformal radius to the cases of multiply connected domains. We present a numerical method to compute Mityuk's function and radius for canonical domains that consist of the unit disk with circular/radial slits. Our method is based...

A new intrinsic metric called $t$-metric is introduced. Several sharp inequalities between this metric and the most common hyperbolic type metrics are proven for various domains $G\subsetneq\mathbb{R}^n$. The behaviour of the new metric is also studied under a few examples of conformal and quasiconformal mappings, and the differences between the ba...

Important geometric or analytic properties of domains in the Euclidean space $\mathbb{R}^n$ or its one-point compactification (the M\"obius space) $\overline{\mathbb{R}}^n$ $(n\ge 2)$ are often characterized by comparison inequalities between two intrinsic metrics on a domain. For instance, a proper subdomain $G$ of $\mathbb{R}^n$ is {\it uniform}...

The triangular ratio metric is studied in a domain $G\subsetneq\mathbb{R}^n$, $n\geq2$. Several sharp bounds are proven for this metric, especially, in the case where the domain is the unit disk of the complex plane. The results are applied to study the H\"older continuity of quasiconformal mappings.

We study the geometry of the scale-invariant Cassinian metric and prove sharp comparison inequalities between this metric and the hyperbolic metric in the case when the domain is either the unit ball or the upper half space. We also prove sharp distortion inequalities for the scale-invariant Cassinian metric under Möbius transformations.

For a non-empty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega),$ respectively. The quantity $d(E)/d(E,\partial\Omega)$ is invariant under similarities and plays an important role in Geometric Function Theory....

We investigate moduli of planar circular quadrilaterals symmetric with respect to both the coordinate axes. First, making use the Schwarz ODE, we develop an analytic method to determine a conformal mapping the unit disk onto a given circular quadrilateral. We devise a numerical method to find the accessory parameters, solve the equation and to comp...

For a given ring (domain) in $\overline{\mathbb{R}}^n$ we discuss whether its boundary components can be separated by an annular ring with modulus nearly equal to that of the given ring. In particular, we show that, for all $n\ge 3\,,$ the standard definition of uniformly perfect sets in terms of Euclidean metric is equivalent to the boundedness of...

We investigate the structure and stability of the steady states for a bacterial colony model with density-suppressed motility. We treat the growth rate of bacteria as a bifurcation parameter to explore the local and global structure of the steady states. Relying on asymptotic analysis and the theory of Fredholm solvability, we derive the second-ord...

We introduce a new intrinsic metric in subdomains of a metric space and give upper and lower bounds for it in terms of well-known metrics. We also prove distortion results for this metric under quasiregular maps.

Hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under quasiconformal maps from one sector onto another one.

Answering a question about triangle inequality suggested by R. Li, Barrlund [The p-relative distance is a metric. SIAM J Matrix Anal Appl. 1999;21:699.702] introduced a distance function which is a metric on a subdomain of Rn. We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sha...

In the present chapter we shall introduce, as a special case of curve families and their moduli, the notion of a condenser and its capacity, and we shall examine various properties of condensers.

In geometric function theory, invariance properties of metrics are important. In our work below, two notions of invariance are most important; invariance with respect to the group of Möbius transformations and invariance with respect to the group of similarity transformations.

The parallel postulate of the euclidean geometry says that given a line and a point not on it, there exists only one line through this point and not intersecting the given line. Since the days of Euclid, it had been an open problem studied by many generations of mathematicians, whether the parallel postulate follows from the other axioms of geometr...

In the preceding chapters we have studied some properties of the conformal invariant M( Δ(E, F;G)). In this chapter we shall introduce two other conformal invariants, the modulus metric \(\mu _{G}^{}(x,y)\) and its “dual” quantity \(\lambda _{G}^{}(x,y)\), where G is a domain in \(\overline {\mathbb {R}}^n\) and x, y ∈ G.

We start the comparison of metrics with those ones we have considered in the earlier chapters, namely the chordal metric q, ( 3.5), the hyperbolic metric ρ, ( 4.8), ( 4.16), the distance ratio metric j, ( 4.27) and the quasihyperbolic metric k, ( 5.2).

In this chapter we shall consider some geometric issues related to the hyperbolic or quasihyperbolic metric. We begin with several comparison results for the quasihyperbolic metric. Here an important fact is that various metrics may be comparable in some but not in all domains.

The present chapter is devoted to the study of uniform continuity properties of a quasiregular mapping f : G → fG as a mapping between the metric spaces \((G,k_{G}^{})\) and \((fG,k_{fG}^{})\) and to the study of its restrictions.

(2) By orthogonality r² = |c|² − 1 .

For the sake of easy reference and for the reader’s convenience we shall give in this chapter the basic properties of the modulus of a curve family. The proofs of several well-known results are omitted.

The goal of this survey is to give the reader a brief overview of the theory of quasiconformal (qc) and quasiregular (qr) mappings and of some related topics.

For x∈ℝn and r > 0 let Bn(x,r)={z∈ℝn:|x−z

In this chapter we shall study the behavior of several hyperbolic type metrics under Möbius transformations and quasiconformal maps. In what follows, we need results involving the functions η and η∗.

In this chapter we introduce quasiregular and quasiconformal mappings and their basic properties.

In this chapter we study some geometric properties of metric balls for small radii. It is natural to expect that balls of small radii are like euclidean balls whereas the geometric structure of ∂G strongly influences on the shape of balls for large radii.

For a domain \(G \subset \mathbb {R}^n\), n ≥ 2 , with \( \mathop {\text{card}} \nolimits ( \overline {\mathbb {R}}^n \setminus G) \ge 3\,,\) we consider the following class of homeomorphisms.

Let E and F be compact disjoint non-empty sets in \(\overline {\mathbb {R}}^n\) and \(\mathsf {M}( \Delta _{EF})=\mathsf {M}\bigl (\Delta (E,F)\bigr )\).

In Chaps. 4 and 5 we studied the quasihyperbolic metric kD ( 5.2) and the distance ratio metric jD ( 4.27). These two metrics are generalizations of the hyperbolic metric (Chap. 4) and in this chapter we introduce other such generalizations.

In this chapter we consider inclusion of balls defined by several metrics.

In the present chapter we shall put into effective use the transformation formulae 15.36(1) and (2) for the conformal invariants \(\lambda _{G}^{}\) and \(\mu _{G}^{}\).

We present a boundary integral method for numerical computation of the capacity of generalized condensers. The presented method applies to a wide variety of generalized condenser geometry including the cases when the plates of the generalized condenser are bordered by piecewise smooth Jordan curves or are rectilinear slits. The presented method is...

We study numerical computation of several conformal invariants of simply connected domains in the complex plane including, the hyperbolic distance, the reduced modulus, the harmonic measure, and the modulus of a quadrilateral. The method we use is based on the boundary integral equation with the generalized Neumann kernel. Several numerical example...

This book is an introduction to the theory of quasiconformal and quasiregular mappings in the euclidean n-dimensional space, (where n is greater than 2). There are many ways to develop this theory as the literature shows. The authors' approach is based on the use of metrics, in particular conformally invariant metrics, which will have a key role th...

Making use of two different analytical–numerical methods for capacity computation, we obtain matching to a very high precision numerical values for capacities of a wide family of planar condensers. These two methods are based respectively on the use of the Lauricella function (Bezrodnykh and Vlasov, 2002; Bezrodnykh, 2016 [64,65]) and Riemann theta...

Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are euclidean or hyperbolic lines. In the case of hyperbolic lines, this type of quadrilaterals are called ideal qua...

We study numerical computation of conformal invariants of domains in the complex plane. In particular, we provide an algorithm for computing the conformal capacity of a condenser. The algorithm applies for wide kind of geometries: domains are assumed to have smooth or piecewise smooth boundaries. The method we use is based on the boundary integral...

We present a boundary integral method for numerical computation of the capacity of generalized condensers. The presented method applies to a wide variety of generalized condenser geometry including the cases when the plates of the generalized condenser are bordered by piecewise smooth Jordan curves or are rectilinear slits. The presented method is...

We consider Mityuk's function and radius which have been proposed in \cite{Mit} as generalizations of the reduced modulus and conformal radius to the cases of multiply connected domains. We present a numerical method to compute Mityuk's function and radius for canonical domains that consist of the unit disk with circular/radial slits. Our method is...

We study moduli of planar ring domains whose complements are linear segments and establish formulas for their moduli in terms of the Weierstrass elliptic functions. Numerical tests are carried out to illuminate our results.

The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschitz) of its domain. We prove that under these mappings the uniformity, ϕ-uniformity and δ-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic met...