The purpose of this paper is twofold. The first is to give some forms of the second main theorem in Nevanlinna theory for meromorphic mappings from parabolic manifolds intersecting moving targets in general position with truncated counting functions, which are improvements of some recent results. The second is to apply the above forms to the proof of an algebraic dependence theorem for meromorphic mappings on parabolic manifolds sharing moving targets regardless of multiplicity.

In this work, using elementary tools we determine those Lattès maps which are at the same time Belyi maps by explicitly determining their ramification data. It turns out that in the generic case, i.e. when the automorphism group is Z/2Z, the corresponding family of Lattès maps are Belyi maps if the isogeny is multiplication by two or four. Elliptic curves with extra automorphisms also determine families of Belyi maps. We provide examples of some Belyi Lattès maps together with a formula for such maps which may be used to write Belyi maps of arbitrarily high degree. We conclude the paper with a discussion of the field of definition of such Belyi pairs.

Chowla's (inverse) problem is a deduction of linear independence over the rationals of circular functions at rational arguments from L.(1, x) ≠ 0, while determinant expressions for the (relative) class number of (subfields of) a cyclotomic field are referred to as the Maillet-Demyanenko determinants. In Wang, Chakraborty and Kanemitsu (to appear), Chowla's problem and Maillet-Demyanenko determinants (CPMD) in the case of Bernoulli polynomial entries (odd part) are unified as different-looking expressions of the (relative) class number on the grounds of the base change formula for periodic Dirichlet series, Dedekind determinant and the Euler product. Our aim here is to show that the genesis of the new theory of discrete Fourier transform as well as the Dedekind determinant is the characters of a finite Abelian group and its convolution map, thus revealing that CPMD boils down to analysis of the class number by group characters. We settle the case of Clausen function (log sine function) entries (even part) as an example. Other cases are similar.

Weakly holomorphic modular forms for modular groups are holomorphic away from the cusp. We study a certain family of weakly holomorphic modular forms and the locations of their zeros. We prove that all of the zeros in the standard fundamental domain for the modular group lie on a lower boundary arc, providing conditions.

The aims of this paper are expansions of the measure theory in Fesenko (Amer. Math. Soc, Transl. Ser. 2 219 (2006)) and Waller (New York J. Math. 25 (2019), 396-442). We define countably additive translation-invariant measures and integrals on product spaces of F and of GL_d(F), and on a parabolic subgroup of GL_d(F). Moreover, we will prove that measurable functions on these product spaces are repeatedly integrable.

We introduce a class of discrete dynamical systems that we call virtually expanding. This is an open subset of self-covering maps on a closed manifold which contains all expanding maps and some partially hyperbolic volume-expanding maps. We show that the Perron-Frobenius operator is quasi-compact on a Sobolev space of positive order for such a class of dynamical systems.

Trisection maps are certain stable maps from closed 4-manifolds to R². A simpler but reasonable class of trisection maps was introduced by Baykur and Saeki, called a simplified (g, k)-trisection. We focus on the right-left equivalence classes of simplified (2, 0)-trisections. Simplified trisections are determined by their simplified trisection diagrams, which are diagrams on a genus-two surface consisting of simple closed curves of vanishing cycles with labels. The aim of this paper is to study how the replacement of reference paths changes simplified trisection diagrams up to upper-triangular handle-slides. We show that, for a simplified trisection f satisfying a certain condition, there exist at least two simplified (2, 0)-trisections f' and f" such that f , f' and f" are right-left equivalent to each other but their simplified trisection diagrams are not related by automorphisms of a genus-two surface and upper-triangular handle-slides.

The purpose of this article is to describe the singularities of one-dimensional oscillatory integrals, whose phases have a certain singularity, in the form of an asymptotic expansion. In the case of the Laplace integral, an analogous result is also given.

By using the property of generalized pseudo-anti-commuting, Ric Φ + Φ Ric = f Φ, for real hypersurfaces in the complex quadric Q^m = SO_m+2/SO₂SO_m, we give a complete classification of Ricci-Bourguignon soliton Hopf real hypersurfaces in the complex quadric Q^m. Then, as an application, we show a complete classification of Hopf Ricci-Bourguignon gradient solitons in the complex quadric Q^m.

A probabilistic proof by means of the adelic formulation is given to the classical mean-value theorem for [0, 1]-valued multiplicative arithmetical functions f. Then a more general mean-value theorem is derived for composite functions φ(f) of f with (semi-)continuous functions φ.

On the basis of fractional calculus, the integral of controlled paths along Hölder rough paths is given explicitly as Lebesgue integrals for fractional derivative operators, without using any arguments from a discrete approximation. In this paper, we introduce a backward version of the integral and provide fundamental relations between both integrals from the perspective of the backward representation of the rough integral.

The purpose of this paper is to study the basic properties of a degenerate operator with critical gradient and critical Hardy term, controlled by two free parameters. This operator arises from the critical Caffarelli-Kohn-Nirenberg inequality. We analyze the fundamental solutions in a weighted distributional identity and obtain the Liouville theorem for the Lane-Emden equation with that operator, by using the classification of singular solutions with isolated singularities of the related Poisson problem in a bounded domain containing the origin.

We determine when the modified Rankin-Cohen bracket of the Eisenstein series of weight two and a Hecke eigenform is again an eigenform.