Frederick Gardiner | Graduate Center of the City University of New York (original) (raw)
Papers by Frederick Gardiner
Research Journal of Mathematics and Statistics, 2014
This study presents a mathematical model that simulates oxygen and carbon-dioxide concentrations ... more This study presents a mathematical model that simulates oxygen and carbon-dioxide concentrations in a sealed environment without the actual measurement of the presence of these substances. Using Mathematical software for simulation, the model was tested using data (volume in cubic meters) of a sealed environment whiles varying the number of people in that environment. The results revealed that, the number of people in the sealed environment affected the concentrations of these substances in the air. Also the model revealed that there is a time (threshold) beyond which it will be unsafe to stay in such an environment.
arXiv (Cornell University), 2010
In this article we give an expository account of the holomorphic motion theorem based on work of ... more In this article we give an expository account of the holomorphic motion theorem based on work of Mãne-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have |ǫ log ǫ| moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz's lemma and integration over the holomorphic variable to produce Hölder continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashi's and Teichmüller's metrics on the Teichmüller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions.
Mathematical surveys, Dec 7, 1999
Mathematical surveys, Dec 7, 1999
American Mathematical Society eBooks, Dec 7, 1999
American Mathematical Society eBooks, Dec 7, 1999
Mathematical surveys, Dec 7, 1999
arXiv (Cornell University), Oct 31, 2017
Contemporary mathematics, 2002
American Mathematical Society eBooks, 2012
University Microfilms eBooks, Feb 1, 1967
Abstract : An investigation was made to determine whether or not a certain map from universal Tei... more Abstract : An investigation was made to determine whether or not a certain map from universal Teichmuller space, T, to bounded operators in a certain Banach space was a homomorphism. It is found that this map is not homomorphic but, at the same time, precisely the extent to which it fails to be homomorphic. The other chief results of the paper are the theorems 1 through 5. Theorems 1 and 2 give two descriptions of right translation which are, in a sense, dual to each other. Theorem 3 is a local description of the derivative of right translation on the tangent bundle of Teichmuller space. Its importance is that it yields a corollary which gives new reproducing formulas and it makes possible theorem 5 which gives an explicit form of the operator 1/L(2, C to the mu power). Theorem 4 gives a geometric method of constructing inverses in Teichmuller space when points in T are viewed as quasicircles passing through 0, 1, and infinity. Namely, the inverse is given simply by complex conjugation. (Author)
Transactions of the American Mathematical Society, Sep 18, 2012
The Teichmüller theory of any hyperbolic Riemann surface R induces two closely related metrics on... more The Teichmüller theory of any hyperbolic Riemann surface R induces two closely related metrics on R in the following way. From a theorem of Bers, the fiber K = Ψ −1 ([identity]) of the forgetful map Ψ from the Teichmüller space T eich(R − p) onto the Teichmüller space T eich(R) is conformal to a disc and the evaluation map K [f ] → f (p) ∈ R is a universal covering of R. There are two infinitesimal metrics on K coming from Kobayashi's construction: (1) T eich K is the restriction of the Teichmüller infinitesimal metric on T eich(R − p) to the submanifold K, and (2) Kob K is the Kobayashi metric on K. We show these metrics, respectively, are the lifts via the evaluation map of infinitesimal forms λ and ρ on R, where λ and ρ are the Teichmüller and Poincaré densities. λ and ρ have very different descriptions. For plane domains λ(p) = inf{||∂V || ∞ }, where the infimum is taken over all continuous functions V for which V (p) = 1 and V vanishes on the boundary of R, and ρ(p) = inf{1/|f (0)|}, where the infimum is taken over all holomorphic functions f mapping the unit disc into R with f (0) = p. We also show (1/2)Kob K ≤ T eich K ≤ Kob K and (1/2)ρ ≤ λ ≤ ρ, and λ/ρ = 1/2 when R is simply connected, λ/ρ = 1 when R is a thrice punctured sphere, and in all other cases these inequalities are strict.
Proceedings of The London Mathematical Society, Feb 20, 2006
数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, Jun 1, 2010
Cambridge University Press eBooks, Feb 11, 2010
Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, comp... more Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, complex analysis, fractal geometry, conformal dynamics, discrete groups, geometric group theory, algebraic curves and their moduli, various kinds of deformation theory, coding, thermodynamic formalism, and topology of three-dimensional manifolds. This collection of articles, authored by leading authorities in the field, comprises 16 expository essays presenting original research and expert surveys of important topics related to Riemann surfaces and their geometry. It complements the body of recorded research presented in the primary literature by broadening, re-working and extending it in a more focused and less formal framework, and provides a valuable commentary on contemporary work in the subject. An introductory section sets the scene and provides sufficient background to allow graduate students and research workers from other related areas access to the field.
数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, Jun 1, 2010
We present the minimum norm or Dirichlet principle for measured foliations on a Riemann surface o... more We present the minimum norm or Dirichlet principle for measured foliations on a Riemann surface of finite type. In this setting the principle says that if you minimize total energy in a given measure class, you will find a unique representative which is harmonic and represented by the imaginary part of a holomorphic quadratic differential. We include as part of this principle the notion of extremal length of a measured foliation and the extremal length functional on Teichmüller space. We show that this functional is differentiable and that its derivative is represented by the unique holomorphic quadratic differential whose heights are equal to the heights of the initially given measured foliation. In previous publications, [6], [7], [8], [9], [10], [12], [13], [14], some of them more than twenty years old, a Dirichlet principle for measured foliations has been developed. But nowhere has the principle been fully stated in a single theorem. Moreover, its analogy to the solution of the classical Dirichlet problem for finding a harmonic function with given boundary values is not explained. In this largely expository paper we take the opportunity to emphasize these points. There is also a part which is not expository and involves the introduction of the new concept of a partial measured foliation. A partial measured foliation does not necessarily determine leaves. Roughly speaking, it is only a family of real valued functions v_{j} defined on open subsets U_{j} of a Riemann surface R with the property that (1) v_{j}=\pm v_{k}+c_{jk} 2000 Mathematics Subject Classification(s): Primary 30\mathrm{F}60 ; Secondary 32\mathrm{G}15, 30\mathrm{C}70, 30\mathrm{C}75. Partially supported by NSF grant 0700052.
Research Journal of Mathematics and Statistics, 2014
This study presents a mathematical model that simulates oxygen and carbon-dioxide concentrations ... more This study presents a mathematical model that simulates oxygen and carbon-dioxide concentrations in a sealed environment without the actual measurement of the presence of these substances. Using Mathematical software for simulation, the model was tested using data (volume in cubic meters) of a sealed environment whiles varying the number of people in that environment. The results revealed that, the number of people in the sealed environment affected the concentrations of these substances in the air. Also the model revealed that there is a time (threshold) beyond which it will be unsafe to stay in such an environment.
arXiv (Cornell University), 2010
In this article we give an expository account of the holomorphic motion theorem based on work of ... more In this article we give an expository account of the holomorphic motion theorem based on work of Mãne-Sad-Sullivan, Bers-Royden, and Chirka. After proving this theorem, we show that tangent vectors to holomorphic motions have |ǫ log ǫ| moduli of continuity and then show how this type of continuity for tangent vectors can be combined with Schwarz's lemma and integration over the holomorphic variable to produce Hölder continuity on the mappings. We also prove, by using holomorphic motions, that Kobayashi's and Teichmüller's metrics on the Teichmüller space of a Riemann surface coincide. Finally, we present an application of holomorphic motions to complex dynamics, that is, we prove the Fatou linearization theorem for parabolic germs by involving holomorphic motions.
Mathematical surveys, Dec 7, 1999
Mathematical surveys, Dec 7, 1999
American Mathematical Society eBooks, Dec 7, 1999
American Mathematical Society eBooks, Dec 7, 1999
Mathematical surveys, Dec 7, 1999
arXiv (Cornell University), Oct 31, 2017
Contemporary mathematics, 2002
American Mathematical Society eBooks, 2012
University Microfilms eBooks, Feb 1, 1967
Abstract : An investigation was made to determine whether or not a certain map from universal Tei... more Abstract : An investigation was made to determine whether or not a certain map from universal Teichmuller space, T, to bounded operators in a certain Banach space was a homomorphism. It is found that this map is not homomorphic but, at the same time, precisely the extent to which it fails to be homomorphic. The other chief results of the paper are the theorems 1 through 5. Theorems 1 and 2 give two descriptions of right translation which are, in a sense, dual to each other. Theorem 3 is a local description of the derivative of right translation on the tangent bundle of Teichmuller space. Its importance is that it yields a corollary which gives new reproducing formulas and it makes possible theorem 5 which gives an explicit form of the operator 1/L(2, C to the mu power). Theorem 4 gives a geometric method of constructing inverses in Teichmuller space when points in T are viewed as quasicircles passing through 0, 1, and infinity. Namely, the inverse is given simply by complex conjugation. (Author)
Transactions of the American Mathematical Society, Sep 18, 2012
The Teichmüller theory of any hyperbolic Riemann surface R induces two closely related metrics on... more The Teichmüller theory of any hyperbolic Riemann surface R induces two closely related metrics on R in the following way. From a theorem of Bers, the fiber K = Ψ −1 ([identity]) of the forgetful map Ψ from the Teichmüller space T eich(R − p) onto the Teichmüller space T eich(R) is conformal to a disc and the evaluation map K [f ] → f (p) ∈ R is a universal covering of R. There are two infinitesimal metrics on K coming from Kobayashi's construction: (1) T eich K is the restriction of the Teichmüller infinitesimal metric on T eich(R − p) to the submanifold K, and (2) Kob K is the Kobayashi metric on K. We show these metrics, respectively, are the lifts via the evaluation map of infinitesimal forms λ and ρ on R, where λ and ρ are the Teichmüller and Poincaré densities. λ and ρ have very different descriptions. For plane domains λ(p) = inf{||∂V || ∞ }, where the infimum is taken over all continuous functions V for which V (p) = 1 and V vanishes on the boundary of R, and ρ(p) = inf{1/|f (0)|}, where the infimum is taken over all holomorphic functions f mapping the unit disc into R with f (0) = p. We also show (1/2)Kob K ≤ T eich K ≤ Kob K and (1/2)ρ ≤ λ ≤ ρ, and λ/ρ = 1/2 when R is simply connected, λ/ρ = 1 when R is a thrice punctured sphere, and in all other cases these inequalities are strict.
Proceedings of The London Mathematical Society, Feb 20, 2006
数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, Jun 1, 2010
Cambridge University Press eBooks, Feb 11, 2010
Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, comp... more Riemann surfaces is a thriving area of mathematics with applications to hyperbolic geometry, complex analysis, fractal geometry, conformal dynamics, discrete groups, geometric group theory, algebraic curves and their moduli, various kinds of deformation theory, coding, thermodynamic formalism, and topology of three-dimensional manifolds. This collection of articles, authored by leading authorities in the field, comprises 16 expository essays presenting original research and expert surveys of important topics related to Riemann surfaces and their geometry. It complements the body of recorded research presented in the primary literature by broadening, re-working and extending it in a more focused and less formal framework, and provides a valuable commentary on contemporary work in the subject. An introductory section sets the scene and provides sufficient background to allow graduate students and research workers from other related areas access to the field.
数理解析研究所講究録別冊 = RIMS Kokyuroku Bessatsu, Jun 1, 2010
We present the minimum norm or Dirichlet principle for measured foliations on a Riemann surface o... more We present the minimum norm or Dirichlet principle for measured foliations on a Riemann surface of finite type. In this setting the principle says that if you minimize total energy in a given measure class, you will find a unique representative which is harmonic and represented by the imaginary part of a holomorphic quadratic differential. We include as part of this principle the notion of extremal length of a measured foliation and the extremal length functional on Teichmüller space. We show that this functional is differentiable and that its derivative is represented by the unique holomorphic quadratic differential whose heights are equal to the heights of the initially given measured foliation. In previous publications, [6], [7], [8], [9], [10], [12], [13], [14], some of them more than twenty years old, a Dirichlet principle for measured foliations has been developed. But nowhere has the principle been fully stated in a single theorem. Moreover, its analogy to the solution of the classical Dirichlet problem for finding a harmonic function with given boundary values is not explained. In this largely expository paper we take the opportunity to emphasize these points. There is also a part which is not expository and involves the introduction of the new concept of a partial measured foliation. A partial measured foliation does not necessarily determine leaves. Roughly speaking, it is only a family of real valued functions v_{j} defined on open subsets U_{j} of a Riemann surface R with the property that (1) v_{j}=\pm v_{k}+c_{jk} 2000 Mathematics Subject Classification(s): Primary 30\mathrm{F}60 ; Secondary 32\mathrm{G}15, 30\mathrm{C}70, 30\mathrm{C}75. Partially supported by NSF grant 0700052.