In this paper, we study right half-plane harmonic mappings $f_0$ and $f$, where $f_0$ is fixed an... more In this paper, we study right half-plane harmonic mappings $f_0$ and $f$, where $f_0$ is fixed and $f$ is a special dilation of a conformal mapping. We obtain a sufficient condition for the convolution of such mappings to be convex in the direction of the real axis. The result of the paper is a generalization of the result of by Li and Ponnusamy, which itself originates from a problem posed by M. Dorff M. Nowak and M. Woloszkiewicz.
For λ ≥ 0 and 0 ≤ α < 1 < β, we denote by K (λ; α, β) the class of normalized analytic functions ... more For λ ≥ 0 and 0 ≤ α < 1 < β, we denote by K (λ; α, β) the class of normalized analytic functions satisfying the two sided-inequality
In this paper, we establish a three circles type theorem, involving the harmonic area function, f... more In this paper, we establish a three circles type theorem, involving the harmonic area function, for harmonic mappings. Also, we give bounds for length and area distortion for harmonic quasiconformal mappings. Finally, we will study certain Lipschitz-type spaces on harmonic mappings.
We study the connection between multiplicities of the zeros and boundary behavior of bounded harm... more We study the connection between multiplicities of the zeros and boundary behavior of bounded harmonic and quasiregular mappings of the plane.
Bridging Math-Gaps with the Learning Environment MUMIE. The European Project S3M2 and the German Math-Bridge Course OMB+ for the Enhancement of Student Mobility}
Diagnosing nursing students' errors in medication calculation. Designing a method based on the 4 Cs teaching model for analysing mathematical proficiency}
We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. ... more We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. Our main result is that in uniformly smooth Banach spaces a quasihyperbolic ball of a convex domain is C 1 -smooth. The question about the smoothness of quasihyperbolic balls is old, originating back to the discussions of F.W. Gehring and M. Vuorinen in 1970's. To our belief, the result is new also in the Euclidean setting. We also address some other issues involving the smoothness of quasihyperbolic balls.
In this paper, we continue our investigation of function spaces on certain classes of complex-val... more In this paper, we continue our investigation of function spaces on certain classes of complex-valued functions. In particular, we give characterizations on Hardy-type, Bergman-type and Dirichlet-type spaces. Furthermore, we present applications of our results to certain nonlinear PDEs.
In this paper, we present a generalization of the algorithm from our earlier work, for numerical ... more In this paper, we present a generalization of the algorithm from our earlier work, for numerical computation of conformal mappings, on multiply connected domains. An implementation of the algorithm, along with several examples and illustrations are given. 1. Introduction. Conformal mappings play an important role in both theoret-1 ical complex analysis and in certain engineering applications, such as electrostatics, 2 aerodynamics and fluid mechanics. Existence of conformal mappings of simply con-3 nected domains onto the upper-half plane or the unit disk follows from the Riemann 4 mapping theorem, a well-known result in complex analysis [2], and there are gener-5 alizations of this result for doubly and multiply connected domains. However, con-6 structing such mappings analytically is usually very difficult, and use of numerical 7 methods is required. 8 There exists an extensive literature on numerical construction of conformal map-9 pings for simply and doubly connected domains [23]. One popular method is based 10 on the Schwarz-Christoffel formula [11], and its implementation SC Toolbox is due to 11 Driscoll [9, 10]. SC Toolbox itself is based on earlier FORTRAN package by Trefethen 12 [26]. A new algorithm involving a finite element method and the conjugate harmonic
In this paper, we first establish the Schwarz-Pick lemma of higherorder and apply it to obtain a ... more In this paper, we first establish the Schwarz-Pick lemma of higherorder and apply it to obtain a univalency criteria for planar harmonic mappings. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.
Characterizations of free quasiconformality in metric spaces
ABSTRACT In this paper, we investigate the concept of (dimension) free quasiconformality in metri... more ABSTRACT In this paper, we investigate the concept of (dimension) free quasiconformality in metric spaces. We establish three results demonstrating that this concept is useful in a very general metric setting. First, we show several sufficient conditions for a homeomorphism to be fully semisolid in suitable metric spaces. These conditions indicate that the quasihyperbolic metrics are quasi-invariant under several different kinds of mappings, for example, quasisymmetric mappings, weakly quasisymmetric mappings etc. One of these sufficient conditions is a generalization of the main result, Theorem 1.6, in [X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, to appear in Trans. Amer. Math. Soc.]. Second, as the main result of this paper, we prove that, in suitable Boman metric spaces, all the sufficient conditions obtained for full semisolidity are also necessary, and then, as a direct corollary, we obtain six alternative characterizations for free quasiconformality of a homeomorphism. Finally, as an application of our main result, we prove that the composition of two locally weakly quasisymmetric mappings in a large class of metric spaces is locally quasisymmetric, and also it is quasiconformal.
We study the problem of computing the conformal modulus of rings and quadrilaterals with strong s... more We study the problem of computing the conformal modulus of rings and quadrilaterals with strong singularities and cusps on their boundary. We reduce this problem to the numerical solution of the associated Dirichlet and Dirichlet-Neumann type boundary values problems for the Laplace equation. Several experimental results, with error estimates, are reported. In particular, we consider domains with dendrite like boundaries, in such cases where an analytic formula for the conformal modulus can be derived. Our numerical method makes use of an hp-FEM algorithm, written for this very complicated geometry with strong singularities.
In this paper, we will give Schwarz-Pick type estimates of arbitrary order partial derivatives fo... more In this paper, we will give Schwarz-Pick type estimates of arbitrary order partial derivatives for bounded pluriharmonic mappings defined in the unit polydisk. Our main results are generalizations of results of Colonna for planar harmonic mappings in [Indiana Univ. Math. J. 38: 829-840, 1989].
In this paper, we study right half-plane harmonic mappings $f_0$ and $f$, where $f_0$ is fixed an... more In this paper, we study right half-plane harmonic mappings $f_0$ and $f$, where $f_0$ is fixed and $f$ is a special dilation of a conformal mapping. We obtain a sufficient condition for the convolution of such mappings to be convex in the direction of the real axis. The result of the paper is a generalization of the result of by Li and Ponnusamy, which itself originates from a problem posed by M. Dorff M. Nowak and M. Woloszkiewicz.
For λ ≥ 0 and 0 ≤ α < 1 < β, we denote by K (λ; α, β) the class of normalized analytic functions ... more For λ ≥ 0 and 0 ≤ α < 1 < β, we denote by K (λ; α, β) the class of normalized analytic functions satisfying the two sided-inequality
In this paper, we establish a three circles type theorem, involving the harmonic area function, f... more In this paper, we establish a three circles type theorem, involving the harmonic area function, for harmonic mappings. Also, we give bounds for length and area distortion for harmonic quasiconformal mappings. Finally, we will study certain Lipschitz-type spaces on harmonic mappings.
We study the connection between multiplicities of the zeros and boundary behavior of bounded harm... more We study the connection between multiplicities of the zeros and boundary behavior of bounded harmonic and quasiregular mappings of the plane.
Bridging Math-Gaps with the Learning Environment MUMIE. The European Project S3M2 and the German Math-Bridge Course OMB+ for the Enhancement of Student Mobility}
Diagnosing nursing students' errors in medication calculation. Designing a method based on the 4 Cs teaching model for analysing mathematical proficiency}
We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. ... more We investigate properties of quasihyperbolic balls and geodesics in Euclidean and Banach spaces. Our main result is that in uniformly smooth Banach spaces a quasihyperbolic ball of a convex domain is C 1 -smooth. The question about the smoothness of quasihyperbolic balls is old, originating back to the discussions of F.W. Gehring and M. Vuorinen in 1970's. To our belief, the result is new also in the Euclidean setting. We also address some other issues involving the smoothness of quasihyperbolic balls.
In this paper, we continue our investigation of function spaces on certain classes of complex-val... more In this paper, we continue our investigation of function spaces on certain classes of complex-valued functions. In particular, we give characterizations on Hardy-type, Bergman-type and Dirichlet-type spaces. Furthermore, we present applications of our results to certain nonlinear PDEs.
In this paper, we present a generalization of the algorithm from our earlier work, for numerical ... more In this paper, we present a generalization of the algorithm from our earlier work, for numerical computation of conformal mappings, on multiply connected domains. An implementation of the algorithm, along with several examples and illustrations are given. 1. Introduction. Conformal mappings play an important role in both theoret-1 ical complex analysis and in certain engineering applications, such as electrostatics, 2 aerodynamics and fluid mechanics. Existence of conformal mappings of simply con-3 nected domains onto the upper-half plane or the unit disk follows from the Riemann 4 mapping theorem, a well-known result in complex analysis [2], and there are gener-5 alizations of this result for doubly and multiply connected domains. However, con-6 structing such mappings analytically is usually very difficult, and use of numerical 7 methods is required. 8 There exists an extensive literature on numerical construction of conformal map-9 pings for simply and doubly connected domains [23]. One popular method is based 10 on the Schwarz-Christoffel formula [11], and its implementation SC Toolbox is due to 11 Driscoll [9, 10]. SC Toolbox itself is based on earlier FORTRAN package by Trefethen 12 [26]. A new algorithm involving a finite element method and the conjugate harmonic
In this paper, we first establish the Schwarz-Pick lemma of higherorder and apply it to obtain a ... more In this paper, we first establish the Schwarz-Pick lemma of higherorder and apply it to obtain a univalency criteria for planar harmonic mappings. Then we discuss distortion theorems, Lipschitz continuity and univalency of planar harmonic mappings defined in the unit disk with linearly connected images.
Characterizations of free quasiconformality in metric spaces
ABSTRACT In this paper, we investigate the concept of (dimension) free quasiconformality in metri... more ABSTRACT In this paper, we investigate the concept of (dimension) free quasiconformality in metric spaces. We establish three results demonstrating that this concept is useful in a very general metric setting. First, we show several sufficient conditions for a homeomorphism to be fully semisolid in suitable metric spaces. These conditions indicate that the quasihyperbolic metrics are quasi-invariant under several different kinds of mappings, for example, quasisymmetric mappings, weakly quasisymmetric mappings etc. One of these sufficient conditions is a generalization of the main result, Theorem 1.6, in [X. Huang and J. Liu, Quasihyperbolic metric and quasisymmetric mappings in metric spaces, to appear in Trans. Amer. Math. Soc.]. Second, as the main result of this paper, we prove that, in suitable Boman metric spaces, all the sufficient conditions obtained for full semisolidity are also necessary, and then, as a direct corollary, we obtain six alternative characterizations for free quasiconformality of a homeomorphism. Finally, as an application of our main result, we prove that the composition of two locally weakly quasisymmetric mappings in a large class of metric spaces is locally quasisymmetric, and also it is quasiconformal.
We study the problem of computing the conformal modulus of rings and quadrilaterals with strong s... more We study the problem of computing the conformal modulus of rings and quadrilaterals with strong singularities and cusps on their boundary. We reduce this problem to the numerical solution of the associated Dirichlet and Dirichlet-Neumann type boundary values problems for the Laplace equation. Several experimental results, with error estimates, are reported. In particular, we consider domains with dendrite like boundaries, in such cases where an analytic formula for the conformal modulus can be derived. Our numerical method makes use of an hp-FEM algorithm, written for this very complicated geometry with strong singularities.
In this paper, we will give Schwarz-Pick type estimates of arbitrary order partial derivatives fo... more In this paper, we will give Schwarz-Pick type estimates of arbitrary order partial derivatives for bounded pluriharmonic mappings defined in the unit polydisk. Our main results are generalizations of results of Colonna for planar harmonic mappings in [Indiana Univ. Math. J. 38: 829-840, 1989].
Maailman kartta: johdatus matemaattiseen geodesiaan
Lähes kaikki ihmisen toimet vaativat liikkumista paikasta toiseen ja elinympäristön tuntemista. K... more Lähes kaikki ihmisen toimet vaativat liikkumista paikasta toiseen ja elinympäristön tuntemista. Kartta on tällaisen keskeisen tiedon tallentamisen ja välittämisen väline.
Teknologinen kehitys on tuonut perinteisten graafisten karttojen rinnalle satelliittipaikannustekniikan ja mobiiliteknologian. Ne ovat jo lähes välttämättömiä apuvälineitä kartta-aineistojen käytössä. Tämä kirja esittää keskeisimmät karttoihin, kartoitukseen ja paikkatiedon laadintaan liittyvät matemaattiset perusteet.
Kartat ovat keskeisiä elementtejä paikantamispalveluissa, navigoinnissa, yritystoiminnassa ja visualisoinnissa. Karttoihin perustuvat järjestelmät ylläpitävät yhteiskunnan häiriötöntä toimintaa, sillä tieto niin valtioiden kuin tonttimaankin rajoista on tarpeen tallentaa tarkasti ja luotettavasti. Paikkatietoa tarvitaan myös esimerkiksi sääennusteita varten.
Karttojen laatiminen on myös osa ihmisen pyrkimystä ymmärtää paikkaamme maailmankaikkeudessa. Maapallon kartoitus on kulkenut käsi kädessä tieteen kehityksen kanssa kautta historian ja liittyy moniin luonnontieteiden peruskysymyksiin.
This is a high school level open text book intended for the course MAA11: Number theory and logic... more This is a high school level open text book intended for the course MAA11: Number theory and logic (Finnish curriculum). The emphasis of the book is mathematical thinking and in teaching the students to prove elementary mathematical results. In Finnish only, at the moment.
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Papers by Antti Rasila
Teknologinen kehitys on tuonut perinteisten graafisten karttojen rinnalle satelliittipaikannustekniikan ja mobiiliteknologian. Ne ovat jo lähes välttämättömiä apuvälineitä kartta-aineistojen käytössä. Tämä kirja esittää keskeisimmät karttoihin, kartoitukseen ja paikkatiedon laadintaan liittyvät matemaattiset perusteet.
Kartat ovat keskeisiä elementtejä paikantamispalveluissa, navigoinnissa, yritystoiminnassa ja visualisoinnissa. Karttoihin perustuvat järjestelmät ylläpitävät yhteiskunnan häiriötöntä toimintaa, sillä tieto niin valtioiden kuin tonttimaankin rajoista on tarpeen tallentaa tarkasti ja luotettavasti. Paikkatietoa tarvitaan myös esimerkiksi sääennusteita varten.
Karttojen laatiminen on myös osa ihmisen pyrkimystä ymmärtää paikkaamme maailmankaikkeudessa. Maapallon kartoitus on kulkenut käsi kädessä tieteen kehityksen kanssa kautta historian ja liittyy moniin luonnontieteiden peruskysymyksiin.