Related papers: Non-commutative linear algebra and plurisubharmoni…
In this paper, we study the convergence in the capacity of sequence of plurisubharmonic functions. As an application, we prove stability results for solutions of the complex Monge-Amp\`ere equations.
The Monge-Ampere equation, plays a central role in the theory of fully non linear equations. In fact we will like to show how the Monge-Ampere equation, links in some way the ideas comming from the calculus of variations and those of the…
The review of modern study of algebraic, geometric and differential properties of quaternionic (Q) numbers with their applications. Traditional and "tensor" formulation of Q-units with their possible representations are discussed and groups…
Properties of the Cauchy-Riemann-Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3-surface to the cotangent bundle of a complex projective space are computed. A relationship…
In this paper, we introduce the first-order differential operators $d_0$ and $d_1$ acting on the quaternionic version of differential forms on the flat quaternionic space $\mathbb{H}^n$. The behavior of $d_0,d_1$ and $\triangle=d_0d_1$ is…
We consider a new class of quaternionic mappings, associated with the spatial partial differential equations. We describe all mappings from this class using four analytic functions of the complex variable.
In this paper, we consider degenerate quaternionic Monge-Amp\`ere equations in weighted energy class $\mathcal{E}_{\chi}(\Omega)$ where $\Omega$ is a quarternionic domain in $\mathbb{H}^n$ and $\chi$ is a weight function which satisfies…
The richly developed theory of complex manifolds plays important roles in our understanding of holomorphic functions in several complex variables. It is natural to consider manifolds that will play similar roles in the theory of holomorphic…
In this paper we presents an algorithm for finding a solution of the linear nonhomogeneous quaternionic-valued differential equations. Moveover, several examples shows the feasibility of our algorithm.
The theory of quaternionic modular forms has been studied for decades as an example of the modular forms of many variables. The purpose of this study is to provide some congruence relations satisfied by such quaternionic modular forms.
Most of theoretical physics is based on the mathematics of functions of a real or a complex variable; yet we frequently are drawn to try extending our reach to include quaternions. The non-commutativity of the quaternion algebra poses…
In [15] Labourie develops a theory of immersed surfaces of prescribed extrinsic curvature which has since found widespread applications in hyperbolic geometry, general relativity, Teichm\"uller theory, and so on. In this chapter, we present…
This paper studies the complex Monge-Amp\`ere equations for $\mathcal F$-plurisubharmonic functions in bounded $\mathcal F$-hyperconvex domains. We give sufficient conditions for this equation to solve for measures with a singular part.
A systematic study of non-trivial cubic extensions of the four-dimensional Poincar\'e algebra is undertaken. Explicit examples are given with various techniques (Young tableau, characters etc).
We extend some definitions and give new results about the theory of slice analysis in several quaternionic variables. The sets of slice functions which are respectively slice, slice regular and circular w.r.t. given variables are…
We argue that there should exist a "noncommutative Fourier transform" which should identify functions of noncommutative variables (say, of matrices of indeterminate size) and ordinary functions or measures on the space of paths. Some…
We introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy their important properties. Moreover, they exist in…
Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…
We define and study multivariate exponential functions, symmetric with respect to the alternating group A_n, which is a subgroup of the permutation (symmetric) group S_n. These functions are connected with multivariate exponential…
A physically more adequate definition of a quaternionic holomorphic (H-holomorphic) function of one quaternionic variable compared to known ones and a quaternionic generalization of Cauchy-Riemann's equations are presented. At that a class…