TY - JOUR A1 - Weishäupl, Sebastian T1 - The weak Gram law for Hecke \(L\)-functions JF - The Ramanujan Journal N2 - We generalize a theorem by Titchmarsh about the mean value of Hardy’s \(Z\)-function at the Gram points to the Hecke \(L\)-functions, which in turn implies the weak Gram law for them. Instead of proceeding analogously to Titchmarsh with an approximate functional equation we employ a different method using contour integration. KW - Gram’s law KW - Gram points KW - Hecke eigenforms KW - Hecke L-functions Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324404 SN - 1382-4090 VL - 60 IS - 4 ER - TY - JOUR A1 - Lu, Lu A1 - Mönius, Katja T1 - Algebraic degree of Cayley graphs over abelian groups and dihedral groups JF - Journal of Algebraic Combinatorics N2 - For a graph \(\Gamma\) , let K be the smallest field containing all eigenvalues of the adjacency matrix of \(\Gamma\) . The algebraic degree \(\deg (\Gamma )\) is the extension degree \([K:\mathbb {Q}]\). In this paper, we completely determine the algebraic degrees of Cayley graphs over abelian groups and dihedral groups. KW - Cayley graph KW - integral graph KW - algebraic degree Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324380 SN - 0925-9899 VL - 57 IS - 3 ER - TY - JOUR A1 - Greefrath, Gilbert A1 - Oldenburg, Reinhard A1 - Siller, Hans-Stefan A1 - Ulm, Volker A1 - Weigand, Hans-Georg T1 - Mathematics students’ characteristics of basic mental models of the derivative JF - Journal für Mathematik-Didaktik N2 - The concept of derivative is characterised with reference to four basic mental models. These are described as theoretical constructs based on theoretical considerations. The four basic mental models—local rate of change, tangent slope, local linearity and amplification factor—are not only quantified empirically but are also validated. To this end, a test instrument for measuring students’ characteristics of basic mental models is presented and analysed regarding quality criteria. Mathematics students (n = 266) were tested with this instrument. The test results show that the four basic mental models of the derivative can be reconstructed among the students with different characteristics. The tangent slope has the highest agreement values across all tasks. The agreement on explanations based on the basic mental model of rate of change is not as strongly established among students as one would expect due to framework settings in the school system by means of curricula and educational standards. The basic mental model of local linearity plays a rather subordinate role. The amplification factor achieves the lowest agreement values. In addition, cluster analysis was conducted to identify different subgroups of the student population. Moreover, the test results can be attributed to characteristics of the task types as well as to the students’ previous experiences from mathematics classes by means of qualitative interpretation. These and other results of students’ basic mental models of the derivative are presented and discussed in detail. N2 - Der Begriff der Ableitung wird anhand von vier Grundvorstellungen charakterisiert. Diese werden als theoretische Konstrukte beschrieben, die auf theoretischen Überlegungen beruhen. Die vier Grundvorstellungen – lokale Änderungsrate, Tangentensteigung, lokale Linearität und Verstärkungsfaktor – werden empirisch quantifiziert und validiert. Zu diesem Zweck wird ein Testinstrument zur Messung der Charakteristika dieser Grundvorstellungen von Lernenden erstellt, bzgl. Gütekriterien ausgewertet und an Mathematikstudierenden (n = 266) getestet. Die Ergebnisse zeigen, dass die vier Grundvorstellungen der Ableitung bei den Lernenden mit unterschiedlichen Merkmalen rekonstruiert werden können. Die Tangentensteigung weist über alle Aufgaben hinweg die höchsten Übereinstimmungswerte auf. Die Übereinstimmung bei Erklärungen, die auf der Grundvorstellung der lokalen Änderungsrate beruhen, ist bei den Studierenden nicht so stark ausgeprägt, wie man es aufgrund der Rahmenbedingungen im Schulsystem durch Lehrpläne und Bildungsstandards erwarten würde. Die Grundvorstellung der lokalen Linearität spielt eine eher untergeordnete Rolle. Der Verstärkungsfaktor erzielt die geringsten Übereinstimmungswerte. Darüber hinaus wurde eine Clusteranalyse durchgeführt, um verschiedene Untergruppen der Schülerpopulation zu identifizieren. Die Testergebnisse können mittels qualitativer Interpretation auf Merkmale der Aufgabentypen sowie auf die Vorerfahrungen der Studierenden aus dem Mathematikunterricht zurückgeführt werden. Diese und weitere Ergebnisse zu den grundlegenden mentalen Modellen der Studierenden zur Ableitung werden ausführlich dargestellt und diskutiert. KW - derivative KW - basic mental models KW - structure KW - test instrument KW - Ableitung KW - Grundvorstellung KW - Struktur KW - Testinstrument Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324317 SN - 0173-5322 VL - 44 IS - 1 ER - TY - JOUR A1 - Siller, Hans-Stefan A1 - Elschenbroich, Hans-Jürgen A1 - Greefrath, Gilbert A1 - Vorhölter, Katrin T1 - Mathematical modelling of exponential growth as a rich learning environment for mathematics classrooms JF - ZDM Mathematics Education N2 - Mathematical concepts are regularly used in media reports concerning the Covid-19 pandemic. These include growth models, which attempt to explain or predict the effectiveness of interventions and developments, as well as the reproductive factor. Our contribution has the aim of showing that basic mental models about exponential growth are important for understanding media reports of Covid-19. Furthermore, we highlight how the coronavirus pandemic can be used as a context in mathematics classrooms to help students understand that they can and should question media reports on their own, using their mathematical knowledge. Therefore, we first present the role of mathematical modelling in achieving these goals in general. The same relevance applies to the necessary basic mental models of exponential growth. Following this description, based on three topics, namely, investigating the type of growth, questioning given course models, and determining exponential factors at different times, we show how the presented theoretical aspects manifest themselves in teaching examples when students are given the task of reflecting critically on existing media reports. Finally, the value of the three topics regarding the intended goals is discussed and conclusions concerning the possibilities and limits of their use in schools are drawn. KW - basic mental models KW - exponential growth KW - mathematics classrooms KW - mathematical modelling KW - growth models Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324393 SN - 1863-9690 VL - 55 IS - 1 ER - TY - JOUR A1 - Steuding, Jörn A1 - Tongsomporn, Janyarak T1 - On the order of growth of Lerch zeta functions JF - Mathematics N2 - We extend Bourgain’s bound for the order of growth of the Riemann zeta function on the critical line to Lerch zeta functions. More precisely, we prove L(λ, α, 1/2 + it) ≪ t\(^{13/84+ϵ}\) as t → ∞. For both, the Riemann zeta function as well as for the more general Lerch zeta function, it is conjectured that the right-hand side can be replaced by t\(^ϵ\) (which is the so-called Lindelöf hypothesis). The growth of an analytic function is closely related to the distribution of its zeros. KW - Lerch zeta function KW - Hurwitz zeta function KW - (approximate) functional equation KW - order of growth KW - exponent pairs KW - MSC 11M35 Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-303981 SN - 2227-7390 VL - 11 IS - 3 ER - TY - JOUR A1 - Heins, Michael A1 - Roth, Oliver A1 - Waldmann, Stefan T1 - Convergent star products on cotangent bundles of Lie groups JF - Mathematische Annalen N2 - For a connected real Lie group G we consider the canonical standard-ordered star product arising from the canonical global symbol calculus based on the half-commutator connection of G. This star product trivially converges on polynomial functions on T\(^*\)G thanks to its homogeneity. We define a nuclear Fréchet algebra of certain analytic functions on T\(^*\)G, for which the standard-ordered star product is shown to be a well-defined continuous multiplication, depending holomorphically on the deformation parameter \(\hbar\). This nuclear Fréchet algebra is realized as the completed (projective) tensor product of a nuclear Fréchet algebra of entire functions on G with an appropriate nuclear Fréchet algebra of functions on \({\mathfrak {g}}^*\). The passage to the Weyl-ordered star product, i.e. the Gutt star product on T\(^*\)G, is shown to preserve this function space, yielding the continuity of the Gutt star product with holomorphic dependence on \(\hbar\). KW - Lie groups KW - star products Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324324 SN - 0025-5831 VL - 386 IS - 1-2 ER - TY - JOUR A1 - Jotz, M. A1 - Mehta, R. A. A1 - Papantonis, T. T1 - Modules and representations up to homotopy of Lie n-algebroids JF - Journal of Homotopy and Related Structures N2 - This paper studies differential graded modules and representations up to homotopy of Lie n-algebroids, for general \(n\in {\mathbb {N}}\). The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie n-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-degeneracy of higher Poisson structures. Moreover, the Weil algebra of a Lie n-algebroid is computed explicitly in terms of splittings, and representations up to homotopy of Lie n-algebroids are used to encode decomposed VB-Lie n-algebroid structures on double vector bundles. KW - Lie n-algebroids KW - representations up to homotopy KW - differential graded modules KW - Poisson algebras KW - adjoint and coadjoint representations Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324333 SN - 2193-8407 VL - 18 IS - 1 ER - TY - JOUR A1 - Gerber, Sebastian A1 - Quarder, Jascha A1 - Greefrath, Gilbert A1 - Siller, Hans-Stefan T1 - Promoting adaptive intervention competence for teaching simulations and mathematical modelling with digital tools BT - theoretical background and empirical analysis of a university course in teacher education JF - Frontiers in Education N2 - Providing adaptive, independence-preserving and theory-guided support to students in dealing with real-world problems in mathematics lessons is a major challenge for teachers in their professional practice. This paper examines this challenge in the context of simulations and mathematical modelling with digital tools: in addition to mathematical difficulties when autonomously working out individual solutions, students may also experience challenges when using digital tools. These challenges need to be closely examined and diagnosed, and might – if necessary – have to be overcome by intervention in such a way that the students can subsequently continue working independently. Thus, if a difficulty arises in the working process, two knowledge dimensions are necessary in order to provide adapted support to students. For teaching simulations and mathematical modelling with digital tools, more specifically, these knowledge dimensions are: pedagogical content knowledge about simulation and modelling processes supported by digital tools (this includes knowledge about phases and difficulties in the working process) and pedagogical content knowledge about interventions during the mentioned processes (focussing on characteristics of suitable interventions as well as their implementation and effects on the students’ working process). The two knowledge dimensions represent cognitive dispositions as the basis for the conceptualisation and operationalisation of a so-called adaptive intervention competence for teaching simulations and mathematical modelling with digital tools. In our article, we present a domain-specific process model and distinguish different types of teacher interventions. Then we describe the design and content of a university course at two German universities aiming to promote this domain-specific professional adaptive intervention competence, among others. In a study using a quasi-experimental pre-post design (N = 146), we confirm that the structure of cognitive dispositions of adaptive intervention competence for teaching simulations and mathematical modelling with digital tools can be described empirically by a two-dimensional model. In addition, the effectiveness of the course is examined and confirmed quantitatively. Finally, the results are discussed, especially against the background of the sample and the research design, and conclusions are derived for possibilities of promoting professional adaptive intervention competence in university courses. KW - adaptive intervention competence KW - diagnosis KW - simulation KW - mathematical modelling KW - digital tools KW - teacher education KW - pedagogical content knowledge KW - technology Y1 - 2023 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-323701 SN - 2504-284X VL - 8 ER - TY - JOUR A1 - Kakaroumpas, Spyridon A1 - Soler i Gibert, Odí T1 - Dyadic product BMO in the Bloom setting JF - Journal of the London Mathematical Society N2 - Ó. Blasco and S. Pott showed that the supremum of operator norms over L\(^{2}\) of all bicommutators (with the same symbol) of one-parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent 1 < p < ∞. The main tool is a new characterization in terms of paraproducts and two-weight John–Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multiparameter setting, with the appropriate iterated commutator in each case. KW - dyadic product BMO KW - bicommutators KW - Bloom setting Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-318602 SN - 0024-6107 VL - 106 IS - 2 SP - 899 EP - 935 ER - TY - JOUR A1 - Dashkovskiy, Sergey A1 - Slynko, Vitalii T1 - Stability conditions for impulsive dynamical systems JF - Mathematics of Control, Signals, and Systems N2 - In this work, we consider impulsive dynamical systems evolving on an infinite-dimensional space and subjected to external perturbations. We look for stability conditions that guarantee the input-to-state stability for such systems. Our new dwell-time conditions allow the situation, where both continuous and discrete dynamics can be unstable simultaneously. Lyapunov like methods are developed for this purpose. Illustrative finite and infinite dimensional examples are provided to demonstrate the application of the main results. These examples cannot be treated by any other published approach and demonstrate the effectiveness of our results. KW - lyapunov methods KW - stability KW - robustness KW - impulsive systems KW - infinite-dimensional systems KW - nonlinear systems KW - input-to-state stability Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-268390 SN - 1435-568X VL - 34 IS - 1 ER - TY - JOUR A1 - Helin, Tapio A1 - Kretschmann, Remo T1 - Non-asymptotic error estimates for the Laplace approximation in Bayesian inverse problems JF - Numerische Mathematik N2 - In this paper we study properties of the Laplace approximation of the posterior distribution arising in nonlinear Bayesian inverse problems. Our work is motivated by Schillings et al. (Numer Math 145:915–971, 2020. https://doi.org/10.1007/s00211-020-01131-1), where it is shown that in such a setting the Laplace approximation error in Hellinger distance converges to zero in the order of the noise level. Here, we prove novel error estimates for a given noise level that also quantify the effect due to the nonlinearity of the forward mapping and the dimension of the problem. In particular, we are interested in settings in which a linear forward mapping is perturbed by a small nonlinear mapping. Our results indicate that in this case, the Laplace approximation error is of the size of the perturbation. The paper provides insight into Bayesian inference in nonlinear inverse problems, where linearization of the forward mapping has suitable approximation properties. KW - Bayesian inverse problems KW - Laplace approximation KW - nonlinear inverse problems Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-265399 VL - 150 IS - 2 ER - TY - JOUR A1 - Hellmuth, Kathrin A1 - Klingenberg, Christian T1 - Computing Black Scholes with uncertain volatility — a machine learning approach JF - Mathematics N2 - In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples. KW - numerical finance KW - Black Scholes equation KW - uncertainty quantification KW - uncertain volatility KW - polynomial chaos KW - Bi-Fidelity method Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-262280 SN - 2227-7390 VL - 10 IS - 3 ER - TY - JOUR A1 - Fall, Mouhammed Moustapha A1 - Minlend, Ignace Aristide A1 - Ratzkin, Jesse T1 - Foliation of an Asymptotically Flat End by Critical Capacitors JF - The Journal of Geometric Analysis N2 - We construct a foliation of an asymptotically flat end of a Riemannian manifold by hypersurfaces which are critical points of a natural functional arising in potential theory. These hypersurfaces are perturbations of large coordinate spheres, and they admit solutions of a certain over-determined boundary value problem involving the Laplace–Beltrami operator. In a key step we must invert the Dirichlet-to-Neumann operator, highlighting the nonlocal nature of our problem. KW - asymptotically flat ends KW - foliation KW - over-determined problem Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269997 SN - 1559-002X VL - 32 IS - 2 ER - TY - JOUR A1 - Easton, Andrew A1 - van Dalen, Okki A1 - Goeb, Rainer A1 - Di Bucchianico, Alessandro T1 - Bivariate copula monitoring JF - Quality and Reliability Engineering International N2 - The assumption of multivariate normality underlying the Hotelling T\(^{2}\) chart is often violated for process data. The multivariate dependency structure can be separated from marginals with the help of copula theory, which permits to model association structures beyond the covariance matrix. Copula‐based estimation and testing routines have reached maturity regarding a variety of practical applications. We have constructed a rich design matrix for the comparison of the Hotelling T\(^{2}\) chart with the copula test by Verdier and the copula test by Vuong, which allows for weighting the observations adaptively. Based on the design matrix, we have conducted a large and computationally intensive simulation study. The results show that the copula test by Verdier performs better than Hotelling T\(^{2}\) in a large variety of out‐of‐control cases, whereas the weighted Vuong scheme often fails to provide an improvement. KW - copula KW - multivariate Gaussian distribution KW - multivariate statistical process control (SPC) KW - phase I KW - phase II Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-276501 VL - 38 IS - 3 SP - 1272 EP - 1288 ER - TY - JOUR A1 - Campana, Francesca Calà A1 - Borzì, Alfio T1 - On the SQH Method for Solving Differential Nash Games JF - Journal of Dynamical and Control Systems N2 - A sequentialquadratic Hamiltonian schemefor solving open-loop differential Nash games is proposed and investigated. This method is formulated in the framework of the Pontryagin maximum principle and represents an efficient and robust extension of the successive approximations strategy for solving optimal control problems. Theoretical results are presented that prove the well-posedness of the proposed scheme, and results of numerical experiments are reported that successfully validate its computational performance. KW - successive approximations strategy KW - sequential quadratic hamiltonian method KW - differential nash games KW - pontryagin maximum principle Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-269111 SN - 1573-8698 VL - 28 ER - TY - JOUR A1 - Kraus, Daniela A1 - Moucha, Annika A1 - Roth, Oliver T1 - A sharp Bernstein–type inequality and application to the Carleson embedding theorem with matrix weights JF - Analysis and Mathematical Physics N2 - We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil (Int. Math. Res. Not. 2019: 3301–3312, 2019) on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal. KW - Bernstein-type inequality KW - complex polynomials KW - Carleson embedding theorem Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-270485 SN - 1664-235X VL - 12 IS - 1 ER - TY - JOUR A1 - Just, Janina A1 - Siller, Hans-Stefan T1 - The role of mathematics in STEM secondary classrooms: a systematic literature review JF - Education Sciences N2 - Nowadays, science, technology, engineering, and mathematics (STEM) play a critical role in a nation’s global competitiveness and prosperity. Thus, there is a need to educate students in these subjects to meet the current and future demands of personal life and society. While applications, especially in science, engineering, and technology, are directly obvious, mathematics underpins the other STEM disciplines. It is recognized that mathematics is the foundation for all other STEM disciplines; the role of mathematics in classrooms is not clear yet. Therefore, the question arises: What is the current role of mathematics in secondary STEM classrooms? To answer this question, we conducted a systematic literature review based on three publication databases (Web of Science, ERIC, and EBSCO Teacher Referral Center). This literature review paper is intended to contribute to the current state of the role of mathematics in STEM education in secondary classrooms. Through the search, starting with 1910 documents, only 14 eligible documents were found. In these, mathematics is often seen as a minor matter and a means to an end in the eyes of science educators. From this, we conclude that the role of mathematics in the STEM classroom should be further strengthened. Overall, the paper highlights a major research gap, and proposes possible initial solutions to close it. KW - STEM education KW - role of mathematics in STEM KW - literature review KW - STEM integration KW - STEM classroom KW - secondary education Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-288075 SN - 2227-7102 VL - 12 IS - 9 ER - TY - JOUR A1 - Appell, Jürgen A1 - Brito, Belén López A1 - Reinwand, Simon T1 - Counterexamples on compositions JF - Mathematische Semesterberichte N2 - We give a collection of 16 examples which show that compositions \(g\) \(\circ\) \(f\) of well-behaved functions \(f\) and \(g\) can be badly behaved. Remarkably, in 10 of the 16 examples it suffices to take as outer function \(g\) simply a power-type or characteristic function. Such a collection of examples may serve as a source of exercises for a calculus course. KW - composition of functions KW - examples and counterexamples Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324306 SN - 0720-728X VL - 70 IS - 1 ER - TY - JOUR A1 - Kanzow, Christian A1 - Mehlitz, Patrick T1 - Convergence properties of monotone and nonmonotone proximal gradient methods revisited JF - Journal of Optimization Theory and Applications N2 - Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis. KW - non-Lipschitz optimization KW - nonsmooth optimization KW - proximal gradient method Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324351 SN - 0022-3239 VL - 195 IS - 2 ER - TY - JOUR A1 - Kourou, Maria A1 - Zarvalis, Konstantinos T1 - Compact sets in petals and their backward orbits under semigroups of holomorphic functions JF - Potential Analysis N2 - Let (ϕ\(_t\))\(_{t≥0}\) be a semigroup of holomorphic functions in the unit disk \(\mathbb {D}\) and K a compact subset of \(\mathbb {D}\). We investigate the conditions under which the backward orbit of K under the semigroup exists. Subsequently, the geometric characteristics, as well as, potential theoretic quantities for the backward orbit of K are examined. More specifically, results are obtained concerning the asymptotic behavior of its hyperbolic area and diameter, the harmonic measure and the capacity of the condenser that K forms with the unit disk. KW - semigroup of holomorphic functions KW - backward orbit KW - petal KW - harmonic measure KW - condenser capacity KW - Koenigs function KW - green energy KW - hyperbolic area Y1 - 2022 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:bvb:20-opus-324368 SN - 0926-2601 VL - 59 IS - 4 ER -