Picard Theorem Research Articles

NEVANLINNA THEORY FOR HOLOMORPHIC CURVES FROM PUNCTURED DISKS INTO SEMI-ABELIAN VARIETIES

The purpose of this paper is twofold. The first is to show a second main theorem with truncated counting function for holomorphic curves from punctured disks into semi-Abelian varieties. The second is to give an alternative proof of a Big Picard's theorem for algebraically non-degenerate holomorphic curves by Nevanlinna theory.

A Difference Picard Theorem for Meromorphic Functions of Several Variables

It is shown that if n ∈ ℕ, c ∈ ℂn, and three distinct values of a meromorphic function f: ℂn sr 1 of hyper-order gV(f) strictly less than 2/3 have forward invariant pre-images with respect to a translation τ: ℂn sr ℂn, τ (z) = z + c, then f is a periodic function with period c. This result can be seen as a generalization of M. Green’s Picard-Type Theorem in the special case where gV(f) < 2/3, since the empty pre-images of the usual Picard exceptional values are by definition always forward invariant. In addition, difference analogues of the Lemma on the Logarithmic Derivative and of the Second Main Theorem of Nevanlinna theory for meromorphic functions ℂn → ℙ P1 are given, and their applications to partial difference equations are discussed.

Nonlinear equations with essentially infinite-dimensional differential operators

We consider nonlinear differential equations and boundary-value problems with essentially infinite-dimensional operators (of the Laplace–Levy type). An analog of the Picard theorem is proved.

Http://logicandanalysis.org/index.php/jla/article/viewFile/114/41

We study Picard's Theorem and Peano's Theorem from a constructive reverse perspective. This means that we have to change our focus from global properties to local properties. We also extend the theory of pointwise continuously differentiable functions to include Rolle's Theorem, the Mean Value Theorem, and the full Fundamental Theorem of Calculus. 2000 Mathematics Subject Classification 03F60, 26E40 (primary); 03F55 (sec- ondary)

On Gundersen's solution to the Fermat-type functional equation of degree 6

In this note we observe the relation between theta functions and known examples for Fermat-type functional equations , especially the one given by Gundersen [Meromorphic solutions of f 6 + g 6 + h 6 ≡ 1, Analysis (Munich) 18(3) (1998), pp. 285–290] when (n, k) = (3, 6), say here. We recall how Gundersen constructs his solutions, and then observe the concrete expression of by means of the theta functions. Its image in ℙ3(ℂ) is observed. We discuss the relationship between an entire map given by or by Green [Some Picard theorems for holomorphic maps to algebraic varieties, Amer. J. Math. 97 (1975), pp. 43–75] and the second main theorems due to Cartan, Gundersen and Hayman or Eremenko and Sodin. In this purpose, the Wronskian for is calculated. Remarks A–C are observations concerning other known examples for (n, k) = (3, 4), (4, 8), (2, 3), respectively. We give some more remarks on this topic through and these examples. An optimistic conjecture is also posed concerning ‘truncations’.

The theory of shell-based -mappings in geometric function theory

Open, discrete -mappings in , , , are proved to be absolutely continuous on lines, to belong to the Sobolev class , to be differentiable almost everywhere and to have the -property (converse to the Luzin -property). It is shown that a family of open, discrete shell-based -mappings leaving out a subset of positive capacity is normal, provided that either has finite mean oscillation at each point or has only logarithmic singularities of order at most . Under the same assumptions on it is proved that an isolated singularity of an open discrete shell-based -map is removable; moreover, the extended map is open and discrete. On the basis of these results analogues of the well-known Liouville, Sokhotskii-Weierstrass and Picard theorems are obtained.Bibliography: 34 titles.

On the Oscillations Appearing in Numerical Solutions of Solvable and Nonsolvable Integral Equations for Thin-Wire Antennas

Differences between certain solvable and nonsolvable ill-posed integral equations, with the same nonsingular kernel, are discussed. The main results come from constructing a solvable equation in the context of straight thin-wire antennas. The kernel of this equation is the usual approximate (also called reduced) kernel, while its exact solution is the familiar sinusoidal current. Numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual-Halle¿n and Pocklington-equations with the approximate kernel; it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large. It is found that the difficulties encountered in the former (solvable) equation are much less important compared to those of the nonsolvable ones. The same conclusion is brought out from other integral equations, arising in different contexts (thin-wire circular-loop antenna, Method of Auxiliary Sources, and straight wire antenna of infinite length). We discuss the consistency of our results with Picard's theorem. The results in this paper supplement previous publications regarding the difficulties of numerically solving thin-wire integral equations with the approximate kernel.

Towards a theory of removable singularities for maps with unbounded characteristic of quasi-conformity

We prove that sets of zero modulus with weight (in particular, isolated singularities) are removable for discrete open -maps if the function has finite mean oscillation or a logarithmic singularity of order not exceeding on the corresponding set. We obtain analogues of the well-known Sokhotskii-Weierstrass theorem and also of Picard's theorem. In particular, we show that in the neighbourhood of an essential singularity, every discrete open -map takes any value infinitely many times, except possibly for a set of values of zero capacity.

Open discrete mappings having local ACL n inverses

We consider open, discrete mappings between domains from R n satisfying condition (N), having local ACL n inverses on D∖B f , so that μ n (B f ) = 0, H*(·, f) < ∞ on B f and . For this class of mappings (or even for larger classes of open, discrete mappings) we generalize the important modular inequality of Poleckii, also using the modular estimates of the spherical rings from Cristea (Local homeomorphism having local ACL n inverses, Complex Var. Elliptic Equ. 53(1) (2008), 77–99). We continue the work from the same paper by generalizing some basic facts from the theory of quasiregular mappings. We give equicontinuity results, Picard, Montel and Liouville type theorems, estimates of the modulus of continuity, analogues of Schwarz's lemma, eliminability results and boundary extensions theorems. Together with the multiple extensions of Zoric's theorem from the paper, we establish strong generalizations of one of the most important theorems from the theory of quasiregular mappings. We also extend similar results given in some recent classes of functions larger than the class of quasiregular functions, as the class of mappings of finite distortion and satisfying condition (𝒜), or the class of mappings of finite dilatation with dilatation in the bounded mean oscillation class.

Generalizations of Siegel’s and Picard’s theorems

We prove new theorems that are higher-dimensional generalizations of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from C to affine curves. These include results on integral points over varying number fields of bounded degree and results on Kobayashi hyperbolicity. We give a number of new conjectures describing, from our point of view, how we expect Siegel's and Picard's theorems to optimally generalize to higher dimensions. In this article we prove new theorems that are higher-dimensional generaliza tions of the classical theorems of Siegel on integral points on affine curves and of Picard on holomorphic maps from C to affine curves. In Section 2, we will give the statements of Siegel's and Picard's theorems, and we will recall how these two theorems from such seemingly different areas of mathematics are related. We will then proceed to give a number of new conjectures describing, from our point of view, how we expect Siegel's and Picard's theorems to optimally generalize to higher dimensions. These include conjectures on integral points over varying number fields of bounded degree and conjectures addressing hyperbolic questions. These conjectures appear to be fundamentally new. We will then summarize our progress on these conjectures. We have been able to get results in all dimensions, with best-possible results in many cases for surfaces.

Extension and convergence theorems of pseudoholomorphic maps

First, we generalize the big Picard theorem for pseudoholomorphic maps. Next, we prove Noguchi-type extension-convergence theorems for pseudoholomorphic maps. Finally, we show that the $J$-automorphism group of an almost complex submanifold hyperbolically embedded in a compact, almost complex manifold of real dimension 4 is finite when it is the complement of an immersed pseudoholomorphic curve.

Generalizations of rigid analytic Picard theorems

Berkovich's Picard theorem states that there are no non-constant analytic maps from the affine line to the complement of two points on a non-singular projective curve. The purpose of this article is to find generalizations of this result in higher dimensional varieties.

The Big Picard Theorem and other results on Riemann surfaces

In this paper we provide a new proof of the Big Picard Theorem, based on some simple observations about mappings between Riemann surfaces.

An extension of Picard's theorem for meromorphic functions of small hyper-order

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if three distinct values of a meromorphic function f of hyper-order less than 1/n^2 have forward invariant pre-images with respect to a fixed branch of the algebraic function t(z)=z+a_{n-1} z^{1-1/n}+...+a_1 z^{1/n}+a_0 with constant coefficients, then f(t(z)) = f(z) for all z. This is a generalization of Picard's theorem for meromorphic functions of small hyper-order, since the (empty) pre-images of the usual Picard exceptional values are special cases of forward invariant pre-images.

A Big Picard Theorem for Holomorphic Maps into Complex Projective Space

AbstractWe prove a big Picard type extension theoremfor holomorphic maps f : X–A → M, where X is a complex manifold, A is an analytic subvariety of X, and M is the complement of the union of a set of hyperplanes in ℙn but is not necessarily hyperbolically imbedded in ℙn.

A generalization of the little Picard theorem

We show the little Picard theorem for a map of a quasicompact manifold to a relatively compact domain of P n whose degeneracy locus of the Kobayashi pseudodistance is contained in some algebraic hypersurface.

A Picard Theorem for iterative differential equations

AbstractA Picard type existence and uniqueness theorem is established for iterative differential equations of the form

On a complex differential Riccati equation

We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schrödinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation such as the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical ‘one-dimensional’ results, we discuss new features of the considered equation including an analogue of the Cauchy integral theorem.

Functional equations related to family

By Nevanlinna's value distribution theory, we give a more precise form of the meromorphic function f provided that f m can be expressed as a summation of n + 1 (n!≤! m) functions in family The results in this article generalize some known results obtained by Green, M., 1974, On the functional equation a new Picard theorem. Transactions of the American Mathematical Society, 195, 223–230; Noda, Y., 1993, On the fuctional equation and rigidity theorems for holomorphic curves. Kodai Mathematical Journal, 16, 90–117.

Nevanlinna theory for the $q$-difference operator and meromorphic solutions of $q$-difference equations

It is shown that, if f is a meromorphic function of order zero and q is an element of C, then[GRAPHICS]for all r on a set of logarithmic density 1. The remainder of the paper consists of applications of identity (double dagger) to the study of value distribution of zero-order meromorphic functions, and, in particular, zero-order meromorphic solutions of q-difference equations. The results obtained include q-shift analogues of the second main theorem of Nevanlinna theory, Picard's theorem, and Clunie and Mohon'ko lemmas.