Showing papers on "Entire function published in 2011"
01 Jan 2011
TL;DR: In this paper, the authors give an introduction to the theory of functions of several complex variables, and give an elementary introduction which essentially follows Weierstrass, and the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions.
Abstract: The Jacobi inversion theorem leads to functions of several variables with many periods. So, we are led to the problem of developing a theory of them which is analogous to the theory of elliptic functions. First of all, we need to give an introduction to the theory of functions of several complex variables. In this chapter, we give an elementary introduction which essentially follows Weierstrass. One of the main topics will be the proof of the theorem that any meromorphic function on ℂn can be written as a quotient of two entire functions. Weierstrass called this a very difficult problem. The first proof was given by Poincare. In the case n = 1, it is not difficult to show this by means of the theory of Weierstrass products, even for arbitrary domains D ⊂ ℂ instead of ℂ. The case n > 1 is more involved, since the zero sets and pole sets of analytic functions of several complex variables are not discrete.
312 citations
TL;DR: In this paper, an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components was constructed, which gave a partial positive answer to the aforementioned question.
Abstract: We construct an entire function in the Eremenko-Lyubich class B whose Julia set has only bounded path-components. This answers a question of Eremenko from 1989 in the negative. On the other hand, we show that for many functions in B, in particular those of nite order, every escaping point can be connected to 1 by a curve of escaping points. This gives a partial positive answer to the aforementioned question of Eremenko, and answers a question of Fatou from 1926.
160 citations
[...]
TL;DR: In this paper, a system of Bethe-Ansatz type equations, which specify a unique array of Young tableau responsible for the leading contribution to the Nekrasov partition function in the ϵ2 → 0 limit is derived.
Abstract: A system of Bethe-Ansatz type equations, which specify a unique array of Young tableau responsible for the leading contribution to the Nekrasov partition function in the ϵ2 → 0 limit is derived. It is shown that the prepotential with generic ϵ1 is directly related to the (rescaled by ϵ2) number of total boxes of these Young tableau. Moreover, all the expectation values of the chiral fields \( \left\langle {{\text{tr}}{\phi^J}} \right\rangle \) are simple symmetric functions of their column lengths. An entire function whose zeros are determined by the column lengths is introduced. It is shown that this function satisfies a functional equation, closely resembling Baxter’s equation in 2d integrable models. This functional relation directly leads to a nice generalization of the equation defining Seiberg-Witten curve.
130 citations
Book•
06 Oct 2011
TL;DR: The Riemann-Stieltjes Integral Integral Theorem of T(r,f) as discussed by the authors is a fundamental theorem of Nevanlinna theory.
Abstract: * Introduction * The Riemann-Stieltjes Integral * Jensen's Theorem and Applications * The First Fundamental Theorem of Nevanlinna Theory * Elementary Properties of T(r,f) * The Cartan Formulation of the Characteristic * The Poisson-Jensen Formula * Applications of T(r) * A Lemma of Borel and Some Applications * The Maximum Term of an Entire Function * Relation Between the Growth of an Entire Function and the Size of Its Taylor Coefficients * Carleman's Theorem * A Fourier Series Method * The Miles-Rubel-Taylor Theorem on Quotient Representations of Meromorphic Functions * Canonical Products * Formal Power Series * Picard's Theorem and the Second Fundamental Theorem * A Proof of the Second Fundamental Theorem * 'Two Constant' Theorems and the Phragmen-Lindelof Theorems * The Polya Representation Theorem * Integer-Valued Entire Functions * On Small Entire Functions of Exponential-Type with Given Zeros * The First-Order Theory of the Ring of All Entire Functions * Identities of Exponential Functions * References * Index
82 citations
TL;DR: There is a unique radius that minimizes the loss of accuracy caused by round-off errors, and for large classes of functions, this radius actually gives about full accuracy; a remarkable fact that is explained by the theory of Hardy spaces, by the Wiman–Valiron and Levin–Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis.
Abstract: High-order derivatives of analytic functions are expressible as Cauchy integrals over circular contours, which can very effectively be approximated, e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius of convergence is equal, numerical stability strongly depends on r. We give a comprehensive study of this effect; in particular, we show that there is a unique radius that minimizes the loss of accuracy caused by round-off errors. For large classes of functions, though not for all, this radius actually gives about full accuracy; a remarkable fact that we explain by the theory of Hardy spaces, by the Wiman–Valiron and Levin–Pfluger theory of entire functions, and by the saddle-point method of asymptotic analysis. Many examples and nontrivial applications are discussed in detail.
74 citations
TL;DR: In this paper, the value distribution of difference polynomials has been studied in the context of shift of a well-known conjecture by Bruck, and some results on two difference products of entire functions sharing one value with the same multiplicities have been obtained.
64 citations
TL;DR: In this paper, the authors consider a particle system of the squared Bessel processes with index ν>−1 conditioned never to collide with each other, in which if −1<ν<0 the origin is assumed to be reflecting.
Abstract: We consider a particle system of the squared Bessel processes with index ν>−1 conditioned never to collide with each other, in which if −1<ν<0 the origin is assumed to be reflecting. When the number of particles is finite, we prove for any fixed initial configuration that this noncolliding diffusion process is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel called the correlation kernel. When the number of particles is infinite, we give sufficient conditions for initial configurations so that the system is well defined. There the process with an infinite number of particles is determinantal and the correlation kernel is expressed using an entire function represented by the Weierstrass canonical product, whose zeros on the positive part of the real axis are given by the particle-positions in the initial configuration. From the class of infinite-particle initial configurations satisfying our conditions, we report one example in detail, which is a fixed configuration such that every point of the square of positive zero of the Bessel function J
ν is occupied by one particle. The process starting from this initial configuration shows a relaxation phenomenon converging to the stationary process, which is determinantal with the extended Bessel kernel, in the long-term limit.
60 citations
TL;DR: In this article, the variance of linear statistics of random complex zeroes has been studied and the optimal conditions on a test function that yield asymptotic normality of fluctuations of the corresponding linear statistics have been derived.
Abstract: By random complex zeroes we mean the zero set of a random entire function whose Taylor coefficients are independent complex-valued Gaussian variables, and the variance of the k-th coefficient is 1/k!. This zero set is distribution invariant with respect to isometries of the complex plane. Extending the previous results of Sodin and Tsirelson, we compute the variance of linear statistics of random complex zeroes, and find close to optimal conditions on a test-function that yield asymptotic normality of fluctuations of the corresponding linear statistics. We also provide examples of test-functions with abnormal fluctuations of linear statistics.
54 citations
TL;DR: In this article, the authors studied the dynamical behavior of a transcendental entire function in any multiply connected wandering domain of the Fatou set, and showed that the union of these annuli acts as an absorbing set for the iterates of the function in the domain.
Abstract: The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function $f$ in any multiply connected wandering domain $U$ of $f$. By introducing a certain positive harmonic function $h$ in $U$, related to harmonic measure, we are able to give the first detailed description of this dynamical behaviour. Using this new technique, we show that, for sufficiently large $n$, the image domains $U_n=f^n(U)$ contain large annuli, $C_n$, and that the union of these annuli acts as an absorbing set for the iterates of $f$ in $U$. Moreover, $f$ behaves like a monomial within each of these annuli and the orbits of points in $U$ settle in the long term at particular `levels' within the annuli, determined by the function $h$. We also discuss the proximity of $\partial U_n$ and $\partial C_n$ for large $n$, and the connectivity properties of the components of $U_n \setminus \bar{C_n}$. These properties are deduced from new results about the behaviour of an entire function which omits certain values in an annulus.
51 citations
TL;DR: In this article, it was shown that the limit of these minima when i tends to infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class.
Abstract: In 1907 M.Petrovitch initiated the study of a class of entire functions all whose finite sections are real-rooted polynomials. An explicit description of this class in terms of the coefficients of a series is impossible since it is determined by an infinite number of discriminantal inequalities one for each degree. However, interesting necessary or sufficient conditions can be formulated. In particular, J.I.Hutchinson has shown that an entire function p(x)=a_0+a_1x+...+a_nx^n+... with strictly positive coefficients has the property that any its finite segment a_ix^i+...+a_jx^j has all real roots if and only if for all i=1,2,... one has a_i^2/a_{i-1}a_{i+1} is greater than or equal to 4. In the present paper we give sharp lower bounds on the ratios a_i^2/a_{i-1}a_{i+1} for the class considered by M.Petrovitch. In particular, we show that the limit of these minima when i tends to infinity equals the inverse of the maximal positive value of the parameter for which the classical partial theta function belongs to the Laguerre-Polya class.
46 citations
02 Mar 2011
TL;DR: In this article, it was shown that if enough boundary points of a Fatou component are escaping, then the component is an escaping wandering domain of the entire function of the component.
Abstract: Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of $U$ are escaping, then $U$ is an escaping Fatou component. Some applications of these results are given; for example, if $I(f)$ is the escaping set of $f$, then $I(f)\cup\{\infty\}$ is connected.
01 Nov 2011
TL;DR: In this paper, the authors construct new classes of transcendental entire functions, f, such that both the escaping set and the fast escaping set have a structure known as a spider's web and show that some of these classes have a degree of stability under changes in the function.
Abstract: We construct several new classes of transcendental entire functions, f , such that both the escaping set, I ( f ), and the fast escaping set, A( f ), have a structure known as a spider’s web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I ( f ) and A( f ) are spiders’ webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders’ webs to give new results concerning functions with no unbounded Fatou components.
TL;DR: In this article, the authors modify a construction of Kisaka and Shishikura to show that there exists an entire function f which has both a simply connected and a multiply connected wandering domain, and these domains are contained in the set A(f) consisting of the points where the iterates of f tend to infinity fast.
Abstract: We modify a construction of Kisaka and Shishikura to show that there exists an entire function f which has both a simply connected and a multiply connected wandering domain. Moreover, these domains are contained in the set A(f) consisting of the points where the iterates of f tend to infinity fast. The results answer questions by Rippon and Stallard.
TL;DR: In this paper, value distribution and uniqueness problems for difference polynomials of entire functions such as f(f − 1)f(z + c) were studied and the existence of entire solutions of a difference equation of the form f + P (z)(∆cf) m = Q(z) was shown.
Abstract: This paper is devoted to value distribution and uniqueness problems for difference polynomials of entire functions such as f(f − 1)f(z + c). We also consider sharing value problems for f(z) and its shifts f(z + c), and improve some recent results of Heittokangas et al. [J. Math. Anal. Appl. 355 (2009), 352–363]. Finally, we obtain some results on the existence of entire solutions of a difference equation of the form f + P (z)(∆cf) m = Q(z).
TL;DR: In this paper, it was shown that the Julia set of the complex exponential map f(z)=e^z is the entire complex plane, where a is a real constant and λ is positive.
Abstract: Let f be a real entire function whose set S(f) of singular values is real and bounded. We show that, if f satisfies a certain function-theoretic condition (the "sector condition"), then $f$ has no wandering domains. Our result includes all maps of the form f(z)=\lambda sinh(z)/z + a, where a is a real constant and {\lambda} is positive.
We also show the absence of wandering domains for certain non-real entire functions for which S(f) is bounded and the iterates of f tend to infinity uniformly on S(f).
As a special case of our theorem, we give a short, elementary and non-technical proof that the Julia set of the complex exponential map f(z)=e^z is the entire complex plane.
Furthermore, we apply similar methods to extend a result of Bergweiler, concerning Baker domains of entire functions and their relation to the postsingular set, to the case of meromorphic functions.
TL;DR: In this article, the uniqueness of functions sharing one or two finite values with their shifts or difference operators was proved for the Bruck conjecture with respect to a function sharing a fixed number of shifts and difference operators.
Abstract: We prove some results on the uniqueness of entire functions sharing one or two finite values CM with their shifts or difference operators. Our results include shifted and difference analogues of the Bruck conjecture.
TL;DR: In this paper, a generalization of the notion of functions associated to a space is introduced, which is used to characterize whether there exists a real and zerofree function N-associated to a de Branges Pontryagin space in terms of Kreĭn's Q-function associated with the multiplication operator.
Abstract: In the theory of de Branges Hilbert spaces of entire functions, so-called ‘functions associated to a space’ play an important role. In the present paper we deal with a generalization of this notion in two directions, namely with functions N-associated \(({N \in\mathbb {Z}})\) to a de Branges Pontryagin space. Let a de Branges Pontryagin space \({\mathcal {P}}\) and \({N \in \mathbb {Z}}\) be given. Our aim is to characterize whether there exists a real and zerofree function N-associated to \({\mathcal {P}}\) in terms of Kreĭn’s Q-function associated with the multiplication operator in \({\mathcal {P}}\) . The conditions which appear in this characterization involve the asymptotic distribution of the poles of the Q-function plus a summability condition. Although this question may seem rather abstract, its answer has a variety of nontrivial consequences. We use it to answer two questions arising in the theory of general (indefinite) canonical systems. Namely, to characterize whether a given generalized Nevanlinna function is the intermediate Weyl-coefficient of some system in terms of its poles and residues, and to characterize whether a given general Hamiltonian ends with a specified number of indivisible intervals in terms of the Weyl-coefficient associated to the system. In addition, we present some applications, e.g., dealing with admissible majorants in de Branges spaces or the continuation problem for hermitian indefinite functions.
Posted Content•
TL;DR: In this paper, it was shown that the noncolliding Brownian motion (the Dyson model) and the squared Bessel process are determinantal diffusion processes for any deterministic initial configuration, in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel.
Abstract: When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process are determinantal diffusion processes for any deterministic initial configuration $\xi=\sum_{j \in \Lambda} \delta_{x_j}$, in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel, which is specified by an entire function $\Phi$ having zeros in $\supp \xi$. Using such entire functions $\Phi$, we define new topologies called the $\Phi$-moderate topologies. Then we construct three infinite-dimensional determinantal processes, as the limits of sequences of determinantal diffusion processes with finite numbers of particles in the sense of finite dimensional distributions in the $\Phi$-moderate topologies, so that the probability distributions are continuous with respect to initial configurations $\xi$ with $\xi(\R)=\infty$. We show that our three infinite particle systems are versions of the determinantal processes with the extended sine, Bessel, and Airy kernels, respectively, which are reversible with respect to the determinantal point processes obtained in the bulk scaling limit and the soft-edge scaling limit of the eigenvalue distributions of the Gaussian unitary ensemble, and the hard-edge scaling limit of that of the chiral Gaussian unitary ensemble studied in the random matrix theory. Then Markovianity is proved for the three infinite-dimensional determinantal processes.
TL;DR: In this article, the authors give necessary and sufficient conditions for totally real sets in Stein manifolds to admit Carleman approximation of class k ≥ 1, by entire functions, in the form of functions.
Abstract: We give necessary and sufficient conditions for totally real sets in Stein manifolds to admit Carleman approximation of class \({\mathcal C^k}\), k ≥ 1, by entire functions.
TL;DR: In this article, the Schatten class membership of the canonical solution operator was characterized in terms of Δϕ, where ϕ is a subharmonic function with Δ ϕ a doubling measure.
TL;DR: In this article, the authors consider the case where A(z) has a finite deficient value and give some conditions on B(z), which can guarantee that every solution f ≠ 0 of the equation has infinite order.
Abstract: In this paper, we consider the differential equation f″ + Af′ + Bf = 0, where A(z) and B(z) ≠ 0 are entire functions. Assume that A(z) has a finite deficient value, then we will give some conditions on B(z) which can guarantee that every solution f ≠ 0 of the equation has infinite order.
Posted Content•
TL;DR: In this paper, Nachbin's holomorphy types are used to generalize some recent results concerning hypercyclic convolution operators on Fr\'echet spaces of entire functions of bounded type of infinitely many complex variables.
Abstract: In this paper we use Nachbin's holomorphy types to generalize some recent results concerning hypercyclic convolution operators on Fr\'echet spaces of entire functions of bounded type of infinitely many complex variables.
01 Jun 2011
TL;DR: In this article, the authors describe the recent progress in understanding the zero sets of two remarkable Gaussian random functions: the Gaussian entire function with invariant distribution of zeroes with respect to isometries of the complex plane, and Gaussian spherical harmonics on the two-dimensional sphere.
Abstract: In these notes, we describe the recent progress in understanding the zero sets of two remarkable Gaussian random functions: the Gaussian entire function with invariant distribution of zeroes with respect to isometries of the complex plane, and Gaussian spherical harmonics on the two-dimensional sphere.
01 Jan 2011
TL;DR: In this paper, the uniqueness problem on functions sharing xed points with the same multiplicities was studied, and the authors generalize some previous results to the case of functions sharing the same number of points.
Abstract: In this paper, we study the uniqueness problem on entire functions sharing xed points with the same multiplicities. We generalize some previous results.
TL;DR: In this paper, the authors discuss and outline proofs of some recent results on application of singular perturbation techniques for solutions in entire space of the Allen-Cahn equation Δu + u − u ≥ 3 = 0.
Abstract: We discuss and outline proofs of some recent results on application of singular perturbation techniques for solutions in entire space of the Allen-Cahn equation Δu + u − u
3 = 0. In particular, we consider a minimal surface Γ in $${\mathbb {R}^9}$$
which is the graph of a nonlinear entire function x
9 = F(x
1, . . . , x
8), found by Bombieri, De Giorgi and Giusti, the BDG surface. We sketch a construction of a solution to the Allen Cahn equation in $${\mathbb {R}^9}$$
which is monotone in the x9 direction whose zero level set lies close to a large dilation of Γ, recently obtained by M. Kowalczyk and the authors. This answers a long standing question by De Giorgi in large dimensions (1978), whether a bounded solution should have planar level sets. We sketch two more applications of the BDG surface to related questions, respectively in overdetermined problems and in eternal solutions to the flow by mean curvature for graphs.
01 Jan 2011
TL;DR: In this article, different properties of relative order of entire functions defined on the basis of their maximum terms are studied. But they do not consider the relation between functions defined in the same order.
Abstract: In the paper we study different properties of relative order of entire functions defined on the basis of their maximum terms. Mathematics Subject Classification: 30D30, 30D35
TL;DR: In this paper, Erdos Turan estimates for discrepancy of point set distributions in the multi-dimensional torus have been obtained for approximating by eigenfunctions of differential operators and discrepancy on compact manifolds.
Abstract: There exists a positive function psi(t) on t >= 0 with fast decay infinity such that for every measurable set Omega in the Euclidean space and R > 0 there exist entire functions A (x) and B (x) of exponential type R satisfying A(x) <= (chi Omega)(x) <= B(x) and |B(x) - A(x)| <= psi(R dist (x, boundary (Omega))). This leads to Erdos Turan estimates for discrepancy of point set distributions in the multi-dimensional torus. Analogous results hold for approximations by eigenfunctions of differential operators and discrepancy on compact manifolds.
TL;DR: In this paper, the Laplace transformation of analytic functionals is used to describe a pre-dual to the function algebra A − ∞(D) (D being either a bounded C 2-smooth convex domain in N (N > 1), or a bounded convex space in C 2 -smooth domain in C, where each function has certain growth properties.
Abstract: In this paper we describe, via the Laplace transformation of analytic functionals, a pre-dual to the function algebra A
−∞(D) (D being either a bounded C
2-smooth convex domain in $${\mathbb{C}^N (N > 1)}$$
, or a bounded convex domain in $${\mathbb{C}}$$
) as a space of entire functions with certain growth. A possibility of representation of functions from the pre-dual space in a form of Dirichlet series with frequencies from D, is also studied.
TL;DR: In this article, the greatest lower bound for the type of an entire function of order whose sequence of zeros lies on one ray and has prescribed lower and upper -densities was found.
Abstract: We find the greatest lower bound for the type of an entire function of order whose sequence of zeros lies on one ray and has prescribed lower and upper -densities. We make a thorough study of the dependence of this extremal quantity on and on properties of the distribution of zeros. The results are applied to an extremal problem on the radii of completeness of systems of exponentials.