Computational Methods and Function Theory最新文献

We determine the exact Borel class of escaping sets in the exponential family (exp (z)+a). We also prove that the sets of non-escaping Julia points for many of these functions are topologically equivalent.

Abstract

This paper is devoted to the development of Beckenbach’s theory of meromorphic minimal surfaces. We get an estimate of the sum of Petrenko’s deviations of the meromorphic minimal surface of finite lower order in term of Valiron’s defect (Delta ({textbf {0}}, S_u)) . We also give an example showing that the estimate is sharp.

Consider a compact surface (mathscr {R}) with distinguished points (z_1,ldots ,z_n) and conformal maps (f_k) from the unit disk into non-overlapping quasidisks on (mathscr {R}) taking 0 to (z_k). Let (Sigma ) be the Riemann surface obtained by removing the closures of the images of (f_k) from (mathscr {R}). We define forms which are meromorphic on (mathscr {R}) with poles only at (z_1,ldots ,z_n), which we call Faber–Tietz forms. These are analogous to Faber polynomials in the sphere. We show that any (L^2) holomorphic one-form on (Sigma ) is uniquely expressible as a series of Faber–Tietz forms. This series converges both in (L^2(Sigma )) and uniformly on compact subsets of (Sigma ).

In this paper, we explore a generalization of one-dimensional tropical Nevanlinna theory developed by Halburd & Southall and Laine & Toghe for a scheme on general locally finite graphs. We first give a probabilistic interpretation of a fundamental observation in one-dimensional tropical Nevanlinna theory on the graph with countably infinitely many vertices of degree two, aiming at its extension in terms of one-dimensional Brownian motion. A counterpart of Lemma on the logarithmic derivative in the classical Nevanlinna theory was proved by Halburd and Southall (cf. Int. Math. Res. Not. 2009:887–911, 2009, https://doi.org/10.1093/imrn/rnn150). Taking advantage of the stochastic analytical interpretation, we prove an analogous result to their lemma on the logarithmic derivative on infinite graphs admitting tree structure.

If f is a power series with radius R of convergence, (R > 1), it is well-known that the method of Carathéodory–Fejér constructs polynomial approximations of f on the closed unit disk which show the typical phenomenon of near-circularity on the unit circle. Let E be compact and connected and let f be holomorphic on E. If (left{ p_nright} _{nin mathbb {N}}) is a sequence of polynomials converging maximally to f on E, it is shown that the modulus of the error functions (f-p_n) is asymptotically constant in capacity on level lines of the Green’s function (g_Omega (z,infty )) of the complement (Omega ) of E in (overline{mathbb {C}}) with pole at infinity, thereby reflecting a type of near-circularity, but without gaining knowledge of the winding numbers of the error curves with respect to the point 0.

In his recent work, Bengt Fornberg describes the construction of finite difference schemes (FDS) for accurate numerical computation of higher order derivatives of analytic functions. In this note we introduce the characteristic function of these schemes and explore how it encodes properties of the FDS. Visualizations of the characteristic function and their modifications allow one to read off these properties by visual inspection of phase portraits. The cover of this volume shows a phase portrait of a function which is related to a FDS with nine nodes that approximates the 4th derivative with an error of order (h^8).

The paper presents general criteria for the uniqueness of a non-constant meromorphic function having finitely many poles and a non-constant L-function in the Selberg class when they share two sets. Our results provide the best cardinalities ever obtained in the literature improving all the existing results Li et al. (Lith. Math. J. 58(2), 249–262 (2018)), Kundu and Banerjee (Rend. Circ. Mat. Palermo (2) 70(3), 1227–1244 (2021), Banerjee and Kundu (Lith. Math. J. 61(2), 161–179 (2021) with regard to the most general setting. Further, we have exhibited a number of examples throughout the paper showing the far reaching applications of our results.

We prove that if E is a compact subset of the unit disk ({{mathbb {D}}}) in the complex plane, if E contains a sequence of distinct points (a_nnot = 0) for (nge 1) such that (lim _{nrightarrow infty } a_n=0) and for all n we have ( |a_{n+1}| ge |a_n|/2 ), and if (G={{mathbb {D}}} {setminus } E) is connected and (0in partial G), then there is a constant (c>0) such that for all (zin G) we have ( lambda _{G } (z) ge c/|z| ) where (lambda _{G } (z)) is the density of the hyperbolic metric in G.

Abstract

In this paper, a special second-order Briot–Bouquet differential equation is studied. We construct explicit meromorphic solutions by the Kowalevski–Gambier method and a careful discussion. How we take into account the corresponding series at zeros, as opposed to the Laurent series at poles. This method is also useful for the study of many other non-linear differential equations.

The second eigenvalue of the Robin Laplacian is shown to be maximal for a spherical cap among simply connected Jordan domains on the 2-sphere, for substantial intervals of positive and negative Robin parameters and areas. Geodesic disks in the hyperbolic plane similarly maximize the eigenvalue on a natural interval of negative Robin parameters. These theorems extend work of Freitas and Laugesen from the Euclidean case (zero curvature) and the authors’ hyperbolic and spherical results for Neumann eigenvalues (zero Robin parameter). Complicating the picture is the numerically observed fact that the second Robin eigenfunction on a large spherical cap is purely radial, with no angular dependence, when the Robin parameter lies in a certain negative interval depending on the cap aperture.