1. Introductory remarks 1 2. Energy and area 3 3. Exact Lagrangian submanifolds and the non-squeezing theorem 7 4. Sobolev spaces 9 5. The linear Cauchy–Riemann operator 20 5.1. Proof of the Calderón–Zygmund Theorem for p , 2 26 6. Local properties of J-holomorphic curves 30 6.1. Smoothness 31 6.2. The Carleman Similarity Principle and its applications 35 6.3. Isolation of intersection points a...

A homogeneous boundary condition is constructed for the parabolic equation (∂t + I − Δ)u = f in an arbitrary cylindrical domain Ω×R (Ω ⊂ Rn being a bounded domain, I and Δ being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator ∂t + I −Δ, but also f...

This is (raw) lecture notes of the course read on 6th European intensive course on Complex Analysis (Coimbra, Portugal) in 2000. Our purpose is to describe a general framework for generalizations of the complex analysis. As a consequence a classification scheme for different generalizations is obtained. The framework is based on wavelets (coherent states) in Banach spaces generated by “admissib...

This is (raw) lecture notes of the course read on 6th European intensive course on Complex Analysis (Coimbra, Portugal) in 2000. Our purpose is to describe a general framework for generalizations of the complex analysis. As a consequence a classification scheme for different generalizations is obtained. The framework is based on wavelets (coherent states) in Banach spaces generated by “admissib...

We develop quaternionic analysis using as a guiding principle representation theory of various real forms of the conformal group. We first review the Cauchy-Fueter and Poisson formulas and explain their representation theoretic meaning. The requirement of unitarity of representations leads us to the extensions of these formulas in the Minkowski space, which can be viewed as another real form of...

We give a new proof of Szekeres’ formula for P (n, k), the number of partitions of the integer n having k or fewer positive parts. Our proof is based on the recursion satisfied by P (n, k) and Taylor’s formula. We make no use of the Cauchy integral formula or any complex variables. The derivation is presented as a step-by-step procedure, to facilitate its application in other situations. As cor...

Abstract In this paper, with the aid of Faà di Bruno formula and by virtue properties Bell polynomials second kind, authors define a kind notion degenerate Narumi numbers polynomials, establish explicit formulas for derive (degenerate) Cauchy numbers.

In this paper we provide a Weierstrass representation formula for translating solitons and singular minimal surfaces in ${\mathbb{R}^3}$. As application study when the euclidean Gauss map has harmonic argument solve general Cauchy problem class of surfaces.

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward discretisation of the Cauchy-Riemann equation. A lot of classical results in Riemann theory have a discrete counterpart, Hodge star, harmonicity, Hodge theor...