Related papers: Automorphism of solutions to Ramanujan's different…
In this article, we study the set of all solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of…
We define a family of polynomial ring homomorphisms generalizing the well-known Nagata automorphism. We establish necessary and sufficient conditions under which these homomorphisms are automorphisms, and verify that they satisfy the…
The paper consists of two parts. The aim of the first, and main, part is to explain, in an elementary way, Hasse's proof of Ramanujan-Nagell's Theorem. In the second part, we formulate some natural extensions of Ramanujan-Nagell's equation.
It is known that automorphisms of quasi-circular domains fixing the origin are polynomial mappings. By introducing the so-called resonance order and quasi-resonance order, we provide a uniform upper bound for the degree of such polynomial…
In this paper we study the homeomorphic properties of the solutions to one dimensional backward doubly stochastic differential equations under suitable assumptions, where the terminal values depend on a real parameter. Then, we apply them…
We prove the subconvexity conjecture for sup-norms of automorphic forms for congruence subgroups of SL(n, Z) that satisfy the Ramanujan conjecture at infinity.
We show that the Nakayama automorphism of a Frobenius algebra $R$ over a field $k$ is independent of the field (Theorem 4). Consequently, the $k$-dual functor on left $R$-modules and the bimodule isomorphism type of the $k$-dual of $R$, and…
In his notebooks, Ramanujan presented without proof many remarkable formulae for the solutions to generalized modular equations. Much later, proofs of the formulae were provided by making use of highly nontrivial identities for theta series…
Let K<x,y> be the free associative algebra of rank 2 over an algebraically closed constructive field of any characteristic. We present an algorithm which decides whether or not two elements in K<x,y> are equivalent under an automorphism of…
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and…
Let X be a flexible variety of F be an isomorphism of closed one-dimensional subschemes of $X$. We develop criteria which guarantee that F extends to au automorphism of X.
Recently, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I--IV for automorphisms of a polynomial ring. An automorphism…
We analyze the extendability of the solutions to a certain second order differential equation on a Riemannian manifold $(M,g)$, which is defined by a general class of forces (both prescribed on $M$ or depending on the velocity). The results…
We study the plane automorphisms given by polynomials with certain degree decompositions.
Let $I$ be the ideal of relations between the leading terms of the polynomials defining an automorphism of $K^n$. In this paper, we prove the existence of a locally nilpotent derivation which preserves $I$. Moreover, if $I$ is principal,…
The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous…
When studying boundary value problems for some partial differential equations arising in applied mathematics, we often have to study the solution of a system of partial differential equations satisfied by hypergeometric functions and find…
The classical transformation of Jacobi's theta function admits a simple proof by producing an integral representation that yields this invariance apparent. This idea seems to have first appeared in the work of S. Ramanujan. Several examples…
We present here the explicit parametric solutions of second order differential equations invariant under time translation and rescaling and third order differential equations invariant under time translation and the two homogeneity…
Results providing conditions on a family of integro-differential operators to determine a formal automorphism are established. Equivalently, the problem can be read in terms of existence and uniqueness of formal solutions of Cauchy problems…