Brody Curves Research Articles

On meromorphic functions which are Brody curves

We discuss meromorphic functions on the complex plane which are Brody curves regarded as holomorphic maps to P1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb {P}}}_1$$\\end{document}, i.e., which have bounded spherical derivative. For some special classes we gave explicit criteria which functions are Brody. We also discuss which divisors of very slow growth may occur as zero divisor of a Brody function and show that there are transcendental entire functions of arbitrarily slow growth which are not Brody.

On the energy of quasiconformal mappings and pseudoholomorphic curves in complex projective spaces

We prove that the energy density of uniformly continuous, quasiconformal mappings, omitting two points in $\mathbb{C} \mathbb{P}^1$, is equal to zero. We also prove the sharpness of this result, constructing a family of uniformly continuous, quasiconformal mappings, whose areas grow asymptotically quadratically. Finally, we prove that the energy density of pseudoholomorphic Brody curves, omitting three “complex lines” in general position in $\mathbb{C} \mathbb{P}^2$, is equal to zero.

A property of the spherical derivative of an entire curve in complex projective space

We establish a type of the Picard’s theorem for entire curves in $$P^n({\mathbb {C}})$$ whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union D of finite number of hypersurfaces in the complex projective space $$P^n({\mathbb {C}})$$ such that for every entire curve f in $$P^n({\mathbb {C}})$$ , if the spherical derivative $$f^{\#}$$ of f is bounded on $$ f^{-1}(D)$$ , then $$f^{\#}$$ is bounded on the entire complex plane, and hence, f is a Brody curve.

REMARK ON THE LOCAL NATURE OF METRIC MEAN DIMENSION

Metric mean dimension is a metric invariant of dynamical systems. It is a dynamical analogue of Minkowski dimension of metric spaces. We use old ideas of Bowen (Trans. Amer. Math. Soc. 164 (1972), 323-331) for clarifying the local nature of metric mean dimension. We also generalize the local formula to metric mean dimension of ℝD-action dynamics and apply the result to simplify a previous proof of the mean dimension estimate of the ℂ-action on the space of Brody curves.

Stochastic Diffusion Process Based on Generalized Brody Curve: Application to Real Data

A new stochastic diffusion process based on Generalized Brody curve is proposed. Such a process can be considered as an extension of the nonhomogeneous lognormal diffusion process. From the corresponding Itô’s stochastic differential equation (SDE), firstly we establish the probabilistic characteristics of the studied process, such as the solution to the SDE, the probability transition density function and their distribution, the moments function, in particular the conditional and non-conditional trend functions. Secondly, we treat the parameters estimation problem by using the maximum likelihood method in basis of the discrete sampling, thus we obtain nonlinear equations that can be solved by metaheuristic optimization algorithms such as simulated annealing and variable search neighborhood. Finally, we perform a simulation studies and we apply the model to the data of life expectancy at birth in Morocco.

Singular Directions of Brody Curves

In this paper, we give a version of “angular domain” type for the results of Da Costa and Duval (Math Ann 355(4):1593–1600, 2013) for Brody curves into a complex projective algebraic variety in $$P^N({\mathbb {C}})$$ intersecting hypersurfaces. Based on this version, we also establish the existence of singular directions of Brody curves into algebraic varieties.

Mean dimension of the dynamical system of Brody curves

Mean dimension is a topological invariant of dynamical systems counting the number of parameters averaged by dynamics. Brody curves are Lipschitz holomorphic maps $$f:\mathbb {C}\rightarrow \mathbb {C}P^N$$ , and they form an infinite dimensional dynamical system. Gromov started the problem of estimating its mean dimension in 1999. We solve this problem by proving the exact mean dimension formula. Our formula expresses the mean dimension by the energy density of Brody curves. As a key novel ingredient, we use an information theoretic approach to mean dimension introduced by Lindenstrauss and Weiss.

Generalized Hénon Mappings and Foliation by Injective Brody Curves

We consider a finite composition of generalized Henon mappings \({\mathfrak {f}}:{\mathbb {C}}^2\rightarrow {\mathbb {C}}^2\) and its Green function \({\mathfrak {g}}^+:{\mathbb {C}}^2\rightarrow {\mathbb {R}}_{\ge 0}\) (see Sect. 2). It is well known that each level set \(\{{\mathfrak {g}}^+=c\}\) for \(c>0\) is foliated by biholomorphic images of \({\mathbb {C}}\) and each leaf is dense. In this paper, we prove that each leaf is actually an injective Brody curve in \(\mathbb {P}^2\) (see Sect. 4). We also study the behavior of the level sets of \({\mathfrak {g}}^+\) near infinity.

Brody curves in complicated sets

For a hyperbolic generalized Hénon mapping (in the sense of [3]), J+, the boundary of the set of points with bounded orbit is known as a complicated set and also known to admit a lamination by biholomorphic images of C (see [3,6]). We prove that there exists a leaf, which is an injective Brody curve in P2, in the lamination of J+ for certain generalized Hénon mappings (for Brody curves and injective Brody curves, see Subsection 2.2).

Brody curves and mean dimension

We study the mean dimensions of the systems of Brody curves. In particular we give the formula of the mean dimension of the system of Brody curves in the Riemann sphere. A key notion is a non-degeneracy of Brody curves introduced by Yosida (1934). We develop a deformation theory of non-degenerate Brody curves and apply it to the calculation of the mean dimension. Moreover we show that there are sufficiently many non-degenerate Brody curves by using the method of gluing infinitely many rational curves.

Remark on energy density of Brody curves

We introduce several definitions of energy density of Brody curves and show that they give the same value in an appropriate situation.

Brody curves omitting hyperplanes

AbstractA Brody curve, a.k.a. normal curve, is a holomorphic map f fromthe complex line C to the complex projective space P n such that thefamily of its translations {z → f(z +a) : a ∈ C} is normal. We provethat Brody curves omitting n hyperplanes in general position havegrowth order at most one, normal type. This generalizes a result ofClunie and Hayman who proved it for n = 1. IntroductionWe consider holomorphic curves f : C → P n . The spherical derivativekf ′ k measures the length distortion from the Euclidean metric in C to theFubini–Study metric in P n . The explicit expression iskf ′ k 2 = kfk −4 X i6= j |f ′i f j −f i f ′j | 2 ,where (f 0 ,...,f n ) is a homogeneous representation of f (that is the f j areentire functions which never simultaneously vanish), andkfk 2 = X nj=0 |f j | 2 .A holomorphic curve is called a Brody curve if its spherical derivative isbounded. This is equivalent to normality of the family of translations {z →f(z +a) : a ∈ C}. ∗ Supported by NSF grant DMS-0555279.

CURVAS DE CRESCIMENTO NÃO-LINEARES PARA PESO E ALTURA EM QUATRO GRUPOS GENÉTICOS DE CAVALOS

Hanoverian, Brazilian Showjumper, English Thoroughbred and Crossbred horses reared by the Brazilian Army were weighed and measured from six months of age to adult. In total 4,860 measurements on 1,445 horses were available. General curves were estimated as a function of time by the Gompertz, Brody, Logistic, Weibull and Richards curves, using PROC NLIN procedures of SAS ®. The Richards Curve did not converge for weight or height of any of the genetic groups or sexes. The logistic curve did not converge for any of the weight traits while the Gompertz also did not converge for height in several groups. R2 varied between 0.55 for weight in females of the crossbred group to 0.92 for males of the same group. For the height traits the highest R2 (0.66) was found for female Hanovarian horses and lowest for males of the same breed (0.12). In general the curves estimated similar values for asymptotic height and weight, except for Logistic curve, which also showed lowest R2 and highest error. Results for the Weibull and Brody curves were similar in all cases so where possible the Brody curve was selected as the best curve as it had less parameters. The Gompertz curve tended to underestimate mature weights and height. Estimates for both weight and height were in general higher in males than for females. In most cases the b parameter was shown to account for < 0.0001% of the variation in the curve shape. The k parameters, which indicate maturity, were of similar magnitude for the Brody, Gompertz and Weibull curves, for both height and weight within breed. This parameter indicated that there is little difference in maturation rates between males and females.KEY WORDS: Brody, Gompertz, Logistic, Richards, Weibull.

Deformation of Brody curves and mean dimension

AbstractThe main purpose of this paper is to show that ideas of deformation theory can be applied to ‘infinite-dimensional geometry’. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.

A Packing Problem for Holomorphic Curves

AbstractWe propose a new approach to the value distribution theory of entire holomorphic curves. We define packing density of Brody curves, and show that it has various non-trivial properties. The packing density of Brody curves can be considered as an infinite dimensional version of characteristic number, and it has an application to Gromov’s mean dimension theory.

Moduli Space of Brody Curves, Energy and Mean Dimension

AbstractA Brody curve is a holomorphic map from the complex plane ℂ to a Hermitian manifold with bounded derivative. In this paper we study the value distribution of Brody curves from the viewpoint of moduli theory. The moduli space of Brody curves becomes infinite dimensional in general, and we study its “mean dimension”. We introduce the notion of “mean energy” and show that this can be used to estimate the mean dimension.

Logarithmic Surfaces and Hyperbolicity

In 1981 J. Noguchi proved that in a logarithmic algebraic manifold, having logarithmic irregularity strictly bigger than its dimension, any entire curve is algebraically degenerate. In the present paper we are interested in the case of manifolds having logarithmic irregularity equal to its dimension. We restrict our attention to Brody curves, for which we resolve the problem completely in dimension 2: in a logarithmic surface with logarithmic irregularity 2 and logarithmic Kodaira dimension 2, any Brody curve is algebraically degenerate. In the case of logarithmic Kodaira dimension 1, we still get the same result under a very mild condition on the Stein factorization map of the quasi-Albanese map of the log surface, but we show by giving a counter-example that the result is not true any more in general. Finally we prove that a logarithmic surface having logarithmic irregularity 2 admits certain types of algebraically non degenerate entire curves if and only if its logarithmic Kodaira dimension is zero, and we also give a characterization of this case in terms of the quasi-Albanese map.

On Brody and entire curves

We discuss an example of an open subset of a torus which admits a dense entire curve, but no dense Brody curve. Resume (Sur les courbes de Brody et les courbes entieres). — On presente un exemple de sous-ensemble de tore qui possede une courbe entiere dense mais pas de courbe de Brody.

Differences in food resource allocation in a long-term selection experiment for litter size in mice 1. Developmental trends in body weight and food intake against time

AbstractDifferences in the metabolic resource situation between non-reproductive male and female mice of a line selected for high litter size at birth (average of 22 born per litter) and a non-selected control line (average of 10 born per litter) were investigated in two replicates. Brody curves were fitted to individual data on body weight against age and linear regression lines were fitted to individual data on cumulative food intake against age. Mature body weight and mature daily food intake were higher in selected mice than in control mice and higher in males than in females. Selected males matured faster than selected females and control mice. In general, differences in growth and food intake curves between species or lines can mostly be explained by differences in mature size. Therefore, parameters were subsequently scaled by individual estimates of mature body weight. Differences that remain after scaling are a consequence of what have been called specific genetic factors. Scaled mature food intake was higher in selected mice than in control mice and higher in females than in males. Scaled maturation rate was higher in selected mice than in control mice and higher in selected males than in selected females. This shows that in the present study, specific genetic factors have been detected for both body weight and food intake, which suggests that selection for increased litter size has disproportionally changed the resource allocation pattern.

Algebraic families of smooth hyperbolic surfaces of low degree in ℙ ℂ 3

We reprove (after a paper of Y.T. Siu appeared in 1987) a simple vanishing theorem for the Wronskian of Brody curves under a suitable assumption on the existence of global meromorphic connections. Next we give a slight improvement of a result due to Y.T, Siu and A.M. Nadel (Duke Math. J., 1989) on the algebraic degeneracy of entire holomorphic curves contained in certain hypersurfaces of ℙ ℂ n . Especially, their result is generalized to a larger class of hypersurfaces. Our method produces algebraic families of smooth hyperbolic surfaces in ℙ ℂ 3 for all degreesd≥14; this brings us somewhat nearer than previously known from the expected ranged≥5.