We construct solutions of the two-dimensional Toda-Hirota equation (2dTHE) expressed by the Gelfand hypergeometric function (HGF) on the Grassmannian GM(2, N) of confluent or non-confluent type, which is labeled by a partition of N. A system of hyperbolic equations in N complex variables is obtained from the differential equations which form the main body of the Gelfand hypergeometric system and characterize the image of the Radon transform. We use the Laplace sequence of the system of hyperbolic operators to find an elementary seed solution of the 2dTHE and then use the B & auml;cklund transformation to obtain higher solutions expressed in terms of the Gelfand HGF. In constructing the Laplace sequence, the contiguity relations (operators) for the Gelfand HGF play an important role.
We present a formula for computing the Milnor number of certain weighted L & ecirc;- Yomdin hypersurface singularities. This formula partially extends to hypersurfaces of arbitrary dimension a similar result previously established for surface singularities by Artal-Bartolo et al (Int. Math. Res. Not. IMRN 2010(22) (2010), 4301-4318).
Our aim in this paper is to establish some weighted second main theorems in Nevanlinna theory for meromorphic mappings from parabolic manifolds intersecting moving targets in general position with truncated counting functions, which generalize and improve upon the previous results of Yan (J. Math. Anal. Appl. 422 (2015), 456-477) and Thoan et al (Kyushu J. Math. 77 (2023), 203-220) by removing the condition that the codimension of the intersection of inverse images of two moving targets by the mappings is at least two. An algebraic dependence theorem for meromorphic mappings on parabolic manifolds sharing moving targets regardless of multiplicity is also given as an application of the main results.
In this paper we investigate the continuity property of several invariant sets J, J+ and J* of the H & eacute;non map Hc,a(x, y) = (x2 + c + ay, ax) as the parameters (c, a) E C2 vary. More precisely, we show that, if a sequence of parameters (cn, an) converges horocyclically to (c*, a*) such that Hc*,a* has a semi-parabolic fixed point and |a*|is sufficiently small, then the corresponding invariant sets converge to those of Hc*,a*. This in particular generalizes the previous result of Radu and Tanase (Trans. Amer. Math. Soc. 370(6) (2018), 3949-3996) to the case of horocyclic convergence.
This paper studies the harmonic means of positive functions on weighted Herz-Morrey spaces. We establish the norm inequalities for the harmonic means of positive functions on weighted Herz-Morrey spaces. This result is an extension of the harmonic means of positive functions on weighted Lebesgue spaces. As a special case of the main result, we have the norm inequalities of the harmonic means of positive functions on weighted Herz spaces.
