Generator functions and their applications

We had introduced so called generators functions to precisely follow the regularity of analytic solutions of Navier-Stokes equations earlier (see Grenier and Nguyen [Ann. PDE 5 (2019)]. In this short note, we give a short presentation of these generator functions and use them to construct analytic solutions to classical evolution equations, which provides an alternative way to the use of the classical abstract Cauchy-Kovalevskaya theorem (see Asano [Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), pp. 102–105], Baouendi and Goulaouic [Comm. Partial Differential Equations 2 (1977), pp. 1151–1162], Caflisch [Bull. Amer. Math. Soc. (N.S.) 23 (1990), pp. 495–500], Nirenberg [J. Differential Geom. 6 (1972), pp. 561–576], Safonov [Comm. Pure Appl. Math. 48 (1995), pp. 629–637]).

Uniqueness of certain differential polynomial of $L$-functions and meromorphic functions sharing a polynomial

The purpose of this paper is to obtain some sufficient conditions to determine the relation between a meromorphic function and an $L$-function when certain differential polynomial generated by them sharing a one degree polynomial. The main theorem of the paper extends and improves all the results due to W.J. Hao, J.F. Chen [Discrete Dyn. Nat. Soc. 2018, 2018, article ID 4673165], F. Liu, X.M. Li, H.X. Yi [Proc. Japan Acad. Ser. A Math. Sci. 2017, 93 (5), 41-46], P. Sahoo, S. Halder [Tbilisi Math. J. 2018, 11 (4), 67-78].

On the order of strong starlikeness and the radii of starlikeness for of some close-to-convex functions

In this paper we show several sufficient conditions for close-to-convex functions to be strongly starlike of some order. The results continue the line of study from the first author’s paper on the order of strong starlikeness of strongly convex functions, (Nunokawa in Proc Japan Acad Ser A 69(7):234–237, 1993). Also it appears an small improvement of a certain classical results of Ch. Pommerenke. As an application, we also derive estimates for the radii of star-likeness for close-to-convex functions.

On the Existence of Solutions of Ordinary Differential Equations in Banach Spaces

In this paper we prove an existence theorem for ordinary differential equations in Banach spaces. The main assumptions in our results, formulated in terms of the Kuratowski measure of noncompactness, are motivated by the paper [CONSTANTIN, A.: On Nagumo’s theorem, Proc. Japan Acad. Ser. A Math. Sci. 86 (2010), 41-44].

ON MURATA DENSITIES

In this paper, we set up an abstract theory of Murata densities, well tailored to general arithmetical semigroups. In [On certain densities of sets of primes, Proc. Japan Acad. Ser. A Math. Sci.56(7) (1980) 351–353; On some fundamental relations among certain asymptotic densities, Math. Rep. Toyama Univ.4(2) (1981) 47–61], Murata classified certain prime density functions in the case of the arithmetical semigroup of natural numbers. Here, it is shown that the same density functions will obey a very similar classification in any arithmetical semigroup whose sequence of norms satisfies certain general growth conditions. In particular, this classification holds for the set of monic polynomials in one indeterminate over a finite field, or for the set of ideals of the ring of S-integers of a global function field (S finite).

Addendum

A Diophantine Equationby J. W. S. CasselsDr A. Mąkowski of the University of Warsaw has informed me that the result of my paper published in the Glasgow Mathematical Journal 27 (1985), 11–18 was anticipated by S. Uchiyama in his paper “On a diophantine equation”, Proc. Japan Acad. Ser. A, 55 (1979), 367–369.

Images Of Class- Spaces

In this paper the author characterizes images of class- spaces (as defined by Ishii, Tsuda and Kunugi, Proc. Japan Acad. 44, (1968), 897-903) under almost-open maps, bi-quotient maps, pseudoopen maps and quotient maps.