arXiv subjects

Matti Vuorinen

Publications and source records attributed to Matti Vuorinen.

At least 55 records · Page 3Linked to original sources

arXiv2021

Conformal Moduli of Symmetric Circular Quadrilaterals With Cusps

We investigate moduli of planar circular quadrilaterals symmetric with respect to both the coordinate axes. First we develop an analytic approach which reduces this problem to ODEs and devise a numeric method to find out the accessory parameters. This method uses the Schwarz equation to determine conformal mapping of the unit disk onto a given circular quadrilateral. We also give an example of a circular quadrilateral for which the value of the conformal modulus can be found in the analytic form; this example is used to validate the numeric calculations. We also use another method, so called hpFEM, for the numeric calculation of the moduli. These two different approaches provide results agreeing with high accuracy.

math.NA↗

arXiv2021

Intrinsic geometry and boundary structure of plane domains

For a non-empty compact set $E$ in a proper subdomain $Ω$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $Ω$ by $d(E)$ and $d(E,\partialΩ),$ respectively. The quantity $d(E)/d(E,\partialΩ)$ is invariant under similarities and plays an important role in Geometric Function Theory. In the present paper, when $Ω$ has the hyperbolic distance $h_Ω(z,w),$ we consider the infimum $κ(Ω)$ of the quantity $h_Ω(E)/\log(1+d(E)/d(E,\partialΩ))$ over compact subsets $E$ of $Ω$ with at least two points, where $h_Ω(E)$ stands for the hyperbolic diameter of the set $E.$ We denote the upper half-plane by $\mathbb{H}$. Our main results claim that $κ(Ω)$ is positive if and only if the boundary of $Ω$ is uniformly perfect and that the inequality $κ(Ω)\leqκ(\mathbb{H})$ holds for all $Ω,$ where equality holds precisely when $Ω$ is convex.

math.CV↗

arXiv2021

Triangular ratio metric in the unit disk

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 Hölder continuity of quasiconformal mappings.

math.MG↗

arXiv2021

Isoperimetric properties of condenser capacity

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 equilateral triangle has the least capacity.

math.CV↗

arXiv2021

A new intrinsic metric and quasiregular maps

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.

math.CV↗

arXiv2020

Introducing a new intrinsic metric

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 balls drawn with all the metrics considered are compared by both graphical and analytical means.

math.MG↗

arXiv2020

Triangular ratio metric under quasiconformal mappings in sector domains

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.

math.MG↗

arXiv2020

Computation of conformal invariants

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 equation method developed and implemented in [30]. A characteristic feature of this method is that, with small changes in the code, a wide spectrum of problems can be treated and we include code snippets within the text to indicate implementation details. We compare the performance and accuracy to previous results in the cases when numerical data is available and also in the case of several model problems where exact results are available.

math.CV↗

arXiv2020

Remarks on the scale invariant Cassinian metric

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.

math.CV↗

arXiv2020

Numerical computation of Mityuk's function and radius for circular-radial slit domains

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 based on the boundary integral equation with the generalized Neumann kernel. Special attention is given to the validation of the theoretical results on the existence of critical points and the boundary behavior of Mityuk's radius.

math.CV↗

arXiv2020

On cyclic quadrilaterals in euclidean and hyperbolic geometries

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 quadrilaterals. Our main result gives a euclidean counterpart of an earlier result on the hyperbolic distances between the opposite sides of ideal quadrilaterals. The proof is based on computations involving hyperbolic geometry. We also found a new formula for the hyperbolic midpoint of a hyperbolic geodesic segment in the unit disk. As an application of some geometric properties, we provided a euclidean construction of the symmetrization of random four points on the unit circle with respect to a diameter which preserves the absolute cross ratio of quadruples.

math.MG↗

arXiv2020

Teichmüller's theorem in higher dimensions and its applications

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 moduli of separating rings. We also establish separation theorems for a "half" of a ring. As applications of those results, we will prove boundary Hölder continuity of quasiconformal mappings of the ball or the half space in $\mathbb{R}^n.$

math.CV↗

arXiv2020

Global bifurcation and stability of steady states for a bacterial colony model with density-suppressed motility

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-order approximate expression of the steady states. We analytically establish the stability criterion of the bifurcation solutions, and show that sufficiently large growth rate of bacteria leads to a stable uniform steady state. While the growth rate of bacteria is less than some certain value, there is pattern formation with the admissible wave mode. All the analytical results are corroborated by numerical simulations from different stages.

math.AP↗

arXiv2020

Barrlund's distance function and quasiconformal maps

Answering a question about triangle inequality suggested by R. Li, A. Barrlund introduced a distance function which is a metric on a subdomain of ${\mathbb R}^n\,.$ We study this Barrlund metric and give sharp bounds for it in terms of other metrics of current interest. We also prove sharp distortion results for the Barrlund metric under quasiconformal maps.

math.CV↗

arXiv2020

Conformal Invariants in Simply Connected Domains

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 examples are presented. We validate the performance and accuracy of our method by considering several model problems with known analytic solutions.

math.CV↗

arXiv2019

Numerical computation of the capacity of generalized condensers

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 used also to compute the harmonic measure in multiply connected domains.

math.CV↗

arXiv2019

On capacity computation for symmetric polygonal condensers

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 and Riemann theta functions. We apply these results to benchmark the performance of numerical algorithms, which are based on adaptive $hp$--finite element method and boundary integral method.

math.NA↗

arXiv2018

Weakly quasisymmetric maps and uniform spaces

Suppose that $X$ and $Y$ are quasiconvex and complete metric spaces, that $G\subset X$ and $G'\subset Y$ are domains, and that $f: G\to G'$ is a homeomorphism. In this paper, we first give some basic properties of short arcs, and then we show that: if $f$ is a weakly quasisymmetric mapping and $G'$ is a quasiconvex domain, then the image $f(D)$ of every uniform subdomain $D$ in $G$ is uniform. As an application, we get that if $f$ is a weakly quasisymmetric mapping and $G'$ is an uniform domain, then the images of the short arcs in $G$ under $f$ are uniform arcs in the sense of diameter.

math.CV↗