Picard Theorem Research Articles

Comment on “Multiple scattering: The key to unravel the subwavelength world from the far-field pattern of a scattered wave”

Contrary to the main conclusion of Simonetti [Phys. Rev. E 73, 036619 (2006)], we maintain that multiple scattering (MS) is not the "key" for subwavelength detection. Indeed, even with no MS between subwavelength structures, subwavelength detection is still possible. Our statement is numerically confirmed. A simple mathematical argument explains this result. From our point of view, the incorrect conclusion of Simonetti comes from a misinterpretation of the Picard's theorem.

On a Kobayashi hyperbolic manifold N modulo a closed subset ΔN and its applications

We show that the degeneration locus of the Kobayashi pseudodistance on a complex manifold is always a pseudoconcave set of order 1. We give some results cocerning the degeneration locus of the Kobayashi pseudodistance. Next we prove a generalization of the little Picard theorem relevantly. Finally, we consider the case N = ΔN.

A Domain-Theoretic Account of Picard's Theorem

AbstractWe present a domain-theoretic version of Picard's theorem for solving classical initial value problems in ℝn. For the case of vector fields that satisfy a Lipschitz condition, we construct an iterative algorithm that gives two sequences of piecewise linear maps with rational coefficients, which converge, from below and above respectively, exponentially fast, to the unique solution of the initial value problem. We provide a detailed analysis of the speed of convergence and the complexity of computing the iterates. The algorithm uses proper data types based on rational arithmetic, where no rounding of real numbers is required. Thus we obtain a sound implementation framework to solve initial value problems. In particular, the use of rational arithmetic guarantees that implementations of our technique will adhere to the bounds on convergence speed and algebraic complexity.

Domain Theoretic Solutions of Initial Value Problems for Unbounded Vector Fields

This paper extends the domain theoretic method for solving initial value problems, described in [A. Edalat and D. Pattinson. A domain theoretic account of picard's theorem. In Proc. ICALP 2004, number 3142 in Lect. Notes in Comp. Sci., pages 494–505, 2004], to unbounded vector fields. Based on a sequence of approximations of the vector field, we construct two sequences of piecewise linear functions that converge exponentially fast from above and below to the classical solution of the initial value problem. We then show how to construct approximations of the vector field. First, we show that fast convergence is preserved under composition of approximations, if the approximated functions satisfy an additional property, which we call “Hausdorff Lipschitz from below”. In particular, this frees us from the need to work with maximal extensions of classical functions. In a second step, we show how to construct approximations that satisfy this condition from a given computable vector field.

Super-exponential decay and holomorphic extensions for semilinear equations with polynomial coefficients

We show that all eigenfunctions of linear partial differential operators in R n with polynomial coefficients of Shubin type are extended to entire functions in C n of finite exponential type 2 and decay like exp ( − | z | 2 ) for | z | → ∞ in conic neighbourhoods of the form | Im z | ⩽ γ | Re z | . We also show that under semilinear polynomial perturbations all nonzero homoclinics keep the super-exponential decay of the above type, whereas a loss of the holomorphicity occurs, namely we show holomorphic extension into a strip { z ∈ C n | | Im z | ⩽ T } for some T > 0 . The proofs are based on geometrical and perturbative methods in Gelfand–Shilov spaces. The results apply in particular to semilinear Schrödinger equations of the form (∗) − Δ u + | x | 2 u − λ u = F ( x , u , ∇ u ) . Our estimates on homoclinics are sharp. In fact, we exhibit examples of solutions of (∗) with super-exponential decay, which are meromorphic functions, the key point of our argument being the celebrated great Picard theorem in complex analysis.

Big Picard's theorems for holomorphic mappings of several complex variables into [formula omitted] with moving hyperplanes

Motivated by the accomplishment of the second main theorem with moving targets, many authors studied the moving target problems in value distribution theory and related topics. In this paper, we prove some big Picard's theorems for holomorphic mappings of several complex variables into P N ( C ) with moving hyperplanes, related to Nochka's little Picard-type theorems. As its application, we give a new quasi-normal criterion for families of meromorphic mappings of several complex variables into P N ( C ) with moving hyperplanes. The new results in this paper greatly improve some earlier related results.

Generalized Big Picard Theorem for pseudo-holomorphic maps

We show how an appropriate choice for affine connections in the target manifold allows the pseudo-holomorphic curves to be realized as harmonic maps. As an application, we present a generalized Big Picard Theorem for pseudo-holomorphic maps between manifolds with almost complex structures.

On linear extremal problems in the class of schlicht functions

The form of the Schiffer differential equation puts severe restrictions on the class of functions that can occur as extremal functions for arbitrary coefficient-functionals of finite degree. Theorem 1 characterizes the algebraic extremal functions for coefficient-functionals of finite degree. Furthermore it is shown that the extremal function either is an algebraic function or it must possess a non-isolated singularity, or must have a transcendental branch-point. The results are closely related to the Malmquist-Yosida theorems. However Nevanlinna's Theory of Value Distribution is not the mainly used tool but the special form of the Schiffer differential equation and the multiplicity of certain values together with the Great Picard Theorem are exploited.

Approximation Methods for Solving the Cauchy Problem

In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.

Domain theory and differential calculus (functions of one variable)

We introduce a domain-theoretic framework for differential calculus. We define the set of primitive maps as well as the derivative of an interval-valued Scott continuous function on the domain of intervals, and show that they are dually related, providing an extension of the classical duality of differentiation and integration as in the fundamental theorem of calculus. It is shown that, for locally Lipschitz functions of a real variable, the domain-theoretic derivative coincides with the Clarke's derivative. We then construct a domain for differentiable real-valued functions of a real variable by pairing consistent information about the function and information about its derivative. The set of classical functions, locally Lipschitz functions and more generally of all continuous functions. We prove that consistency of function information and derivative information is decidable on rational step functions, which shows that our domain can be given an effective structure. We thus obtain a data type for differential calculus. As an immediate application, we present a domain-theoretic formulation of Picard's theorem, which provides a data type for solving differential equations.

A big Picard theorem for quasiregular mappings into manifolds with many ends

We study quasiregular mappings from a punctured Euclidean ball intonn-manifolds with many ends and prove, by using Harnack’s inequality, a version of the big Picard theorem.

Rigid analytic Picard theorems

We prove a geometric logarithmic derivative lemma for rigid analytic mappings to algebraic varieties in characteristic zero. We use the lemma to give a new and simpler proof (at least in characteristic zero) of Berkovich's little Picard theorem, which says there are no nonconstant rigid analytic maps from the affine line to nonsingular projective curves of positive genus, and of Cherry's result that there are no nonconstant rigid analytic maps from the affine line to Abelian varieties. Furthermore, we use the lemma to prove new theorems of little and big Picard type for dominant mappings, in close analogy with Griffiths and King. For the little Picard type theorem, we prove that if X is a smooth projective variety with a simple normal crossings divisor D such that ( X, D ) has nonnegative logarithmic Kodaira dimension, then there are no dominant rigid analytic maps f from A m to X \ D . For the big Picard type theorem, we prove that if Y is a nonsingular rigid analytic space, E is an effective simple normal crossings divisor on Y , and if X is a smooth projective variety with a simple normal crossings divisor D such that ( X, D ) is of log-general type, then any dominant rigid analytic map f : Y \ E →. X \ D extends to an analytic map from Y to X .

A proof of the Great Picard Theorem

In this paper we present a simple and self contained proof of the great Picard Theorem based on certain Harnack-type inequalities due to J. Lewis.

Some Picard theorems for minimal surfaces

This paper deals with the study of those closed subsets F ⊂ R 3 F \subset \mathbb {R}^3 for which the following statement holds: If S S is a properly immersed minimal surface in R 3 \mathbb {R}^3 of finite topology that is eventually disjoint from F , F, then S S has finite total curvature. The same question is also considered when the conclusion is finite type or parabolicity.

CURVES, DOMAINS AND PICARD'S THEOREM

Based on computing spherical lengths of polygonal curves and spherical areas of domains bounded by polygonal curves on the complex plane, this paper proves a property of holomorphic functions and, as an application, gives a very brief proof of a famous inequality obtained by L. V. Ahlfors which easily implies Picard's theorem. Taken together with the arguments given by Alhfors, therefore, this paper in fact gives an elementary, brief and geometrical proof of Picard's theorem.

A generalization of picard's theorem with moving targets

For given 2n+1 hyperplanes with moving targets in pointwise general position and any holomorphic map f from into a n-dimensional complex projective space omitting those 2n+1 hyperplanes, we show that there exist finitely many (n+1)×(n+1) everywhere invertible matrices with entire holomorphic functions as entries such that the holomorphic map f multiplied by one of the matrices becomes constant. In addition, this set of matrices depends only on the hyperplanes, and can be determined effectively.

Logarithmic jet bundles and applications

We generalize Demailly's construction of projective jet bundles and strictly negatively curved pseudometrics on them to the logarithmic case. We establish this logarithmic generalization explicitly via coordinates, just as Noguchi's generalization of the jets used by Green-Griffiths. As a first application, we give a metric proof for the logarithmic version of Lang's conjecture concerning the hyperbolicity of complements of divisors in a semi-abelian variety as well as for the corresponding big Picard theorem.

Covering properties of derivatives of meromorphic functions

The Ahlfors theory of covering surfaces yields a striking generahzation of Picard's theorem. We discuss whether Picard type theorems where certain derivatives omit values admit similar generahzations.

Quasiconformal immersions of Riemannian manifolds and a picard type theorem

We study singularities of quasiconformal immersions of Riemannian manifolds and show that the phenomenon of compulsory continuation holds in dimensionn≥3. In particlar, this result in a stronger version of the Picard theorem—one without omitted values.

An inverse problem in electrostatics

We reconstruct the shape of a conductor from knowledge of its Green function on a large sphere. To this end, we consider an integral operator F having as kernel the difference between the Green function of the conductor and the Green function of free space. We use the representation F = (1/2)GSG* with the single-layer potential S and a suitable operator G to show that the range of F1/2 and the range of GS1/2 coincide. This allows us to characterize the range of GS1/2 with the help of the singular system of F via Picard's theorem. The characterization, in turn, provides a possibility of deciding whether a point is an element of D by computing a series involving the singular system of F and the free space Green function.