Riemann Equations Research Articles

Liouville theorem for the Yang–Mills self-duality equations

It is shown that under certain conditions one may associatively matrix-multiply Lie-algebra-valued matrices with a componentwise Lie bracket. Using this, a simple algebraic constraint on a Lie-algebra-valued antisymmetric n×n matrix F, which in n=4 is essentially self-duality or anti-self-duality, is described. Somewhat in analogy with Liouville’s theorem for the Cauchy–Riemann equations in n=2, it is shown that, for n>4, the constraint implies that the Lie subalgebra generated by the matrix elements {Fμν } decomposes into copies of S_O_(n) plus a few degenerate cases. The result may be relevant to the structure of the quantum chromodynamic vacuum.

Generalization of Cauchy–Riemann equations

An extension of the analyticity notion is proposed including Cayley numbers. Relations, originating from the Cayley–Dickson procedure, between subsequent hypercomplex analyticities are revealed. Some related topics are discussed and a specific technique is invented.

A Kernel Approach to the Local Solvability of the Tangential Cauchy Riemann Equations

An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the $Y(q)$ condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in ${{\mathbf {C}}^n}$. In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when $Y(q)$ is not satisfied.

A kernel approach to the local solvability of the tangential Cauchy Riemann equations

An integral kernel approach is given for the proof of the theorem of Andreotti and Hill which states that the Y ( q ) Y(q) condition of Kohn is a sufficient condition for local solvability of the tangential Cauchy Riemann equations on a real hypersurface in C n {{\mathbf {C}}^n} . In addition, we provide an integral kernel approach to nonsolvability for a certain class of real hypersurfaces in the case when Y ( q ) Y(q) is not satisfied.

Lageabschätzung für einen Kondensator minimaler Kapazität

With the help of the method of interior variation we get informations about the geometrical form and locus from that continuum, which contains n ≥ 2 various points and has minimal generalized transfinite diameter connected with a fundamental solution for generalized Cauchy–Riemann equations.

Properties of isolated singularities of some functions taking values in real Clifford algebras

The study of solutions to special examples of elliptic differential equations, using Clifford algebras, has been developed by a number of authors [4–12, 14–16, 18, 19, 21–30, 32]. This study has also been applied [3, 13, 17, 28] within a number of areas of theoretical physics, including Yang–Mills field theory, and the Kähler equation. Most of the function theories associated to the solutions of these elliptic differential equations, referred to as generalized Cauchy–Riemann equations [21,32], generalize in a natural manner many aspects of classical one-variable complex analysis [1]. For instance, each of these function theories contain analogues of the Cauchy theorem, Cauchy integral formula and Laurent expansion theorem. However, no information has yet been obtained on the local topological behaviour of a general solution to any of these equations.

Complexified clifford analysis

The foundations of a function theory, in several complex variables, over complex Clifford algebras are developed The influence within this theory of complex analysis, in one variable, is demonstrated. A generalized Cauchy integral formula is provided, and complex harmonic functions are used to construct holomorphic functions which satisfy a generalized Cauchy Riemann equation introduced here.

Electromagnetism and the holomorphic properties of spacetime

The Cauchy–Riemann equations of holomorphy are extended to fields in higher-dimensional spaces in a framework of Clifford algebras. The equations of holomorphy in Minkowski spacetime turn out to be the Maxwell equations in vacuum. The Lorentz gauge condition is a result of the holomorphy. Sources can be included in an extension of the residue theorem, where charges correspond to the residues.

A ‘Unified’ Numerical Treatment of the Wave Equation and the Cauchy–Riemann Equations

A ‘unified’ second-order accurate finite difference approach to treating the wave equation and the Cauchy’Riemann equations is developed. The procedure is motivated by an analysis of properties of weak solutions.

Higher Order Analogues to the Tangential Cauchy Riemann Equations for Real Submanifolds of C n with C.R. Singularity

Higher Order Analogues to the Tangential Cauchy Riemann Equations for Real Submanifolds of C n with C.R. Singularity

Estimates for integral kernels of mixed type, fractional integration operators, and optimal estimates for the $$\bar \partial $$ operator

Sharp tangential Lipschitz estimates for the inhomogeneous Cauchy Riemann equations with Lp data on strongly pseudoconvex doma ins in complex manifolds are proved. Estimates in both isotropic and non-isotropic spaces of functions of bounded mean oscillation are proved.

B�cklund transformations as nonlinear Dirac equations

It is pointed out that the Backlund transformations for a physically interesting class of nonlinear partial differential equations can be interpreted as generalisations of the Cauchy Riemann equations or as nonlinear Dirac equations. The generalisations are inhomogenisations of the Cauchy Riemann equations (or their hyperbolic analogue), whose condensed form makes the transformations easy to remember, which suggests ways to generalise to more than 2 dimensions, and which suggest that complex analysis techniques may be helpful in understanding the transformations.

A FORMULA AND ESTIMATES FOR THE SOLUTIONS OF THE TANGENTIAL CAUCHY–RIEMANN EQUATION

In this paper an explicit integral formula is derived for solutions of the homogeneous tangential Cauchy–Riemann equation on the boundary of a strongly pseudoconvex domain in , and best possible estimates are obtained. Using this formula, some questions on the approximation of holomorphic functions on the boundary of a strongly pseudoconvex domain are studied. Bibliography: 27 titles.

The support of Mikusiński operators

A class of Mikusiriski operators, called regular operators, is studied. The class of regular operators is strictly smaller than the class of all operators, and strictly larger than the class of all distributions with left bounded support. Regular operators have local properties. Lions' theorem of supports holds for regular operators with compact support. The fundamental solution to the Cauchy- Riemann equations is not regular, but the fundamental solution to the heat equation in two dimensions is regular and has support on a half-ray.

On Holomorphic Extension from the Boundary

Let D be a bounded domain of the complex n-space Cn(n≥2), or more generally a pair (M,D) a finite manifold (cf. Definition 2.1), and we assume the boundary ∂D is a smooth and connected submanifold. It is well known by Hartogs-Osgood’s theorem that every holomorphic function on a neighbourhood of ∂D can be continued holomorphically to D. Generalizing the above theorem we shall prove that if a differentiable function on ∂D satisfies certain conditions which are satisfied for the trace of a holomorphic function on a neighbourhood of ∂D, then it can be continued holomorphically to D (Theorem 2-5). The above conditions will be called the tangential Cauchy Riemann equations.

Method of variable directions for the Cauchy — Riemann equation system

Methods of variable directions bave become widely known in recent years for the numerical integration of parabolic and elliptic differential equations with several space variables (see, e.g. [1]–[4]). An attempt is made in this paper to transfer these methods to elliptic equation systems by the simplest example of the Hilbert problem, which involves the determination of two functions u( z, y) and v( x, y) satisfying the Cauchy — Rlemann system du dx − dv dy = 0, du du + dv dx = 0 (1) and the limiting condition α ( s) ( s) u + β ( s) v = ( s) ( α 2 + β 2 ≠ 0). (2)

The Theory of the Propagation in Liquid Helium II of `Temperature-Waves' of Finite Amplitude

The two-fluid model of liquid helium II is applied to the problem of predicting the behaviour of waves and pulses of second sound of finite amplitude. It is found that equations quite analogous to the Riemann and Rankine-Hugoniot equations for ordinary sound waves and shock fronts can be established. These equations predict the `overtaking effect' with the establishment of a `temperature-shock' for pulses of second sound of finite amplitude The temperature-shocks should be of `front-edge' type (propagation of an abrupt rise in temperature) if the liquid helium is at a temperature below 1.96° K., but should become `back-edge', corresponding to an abrupt fall in temperature, between 1.96° K and the λ-point. There is experimental evidence in support of both these predictions. It is shown that, just as for ordinary shock fronts, the irreversible effects due to the passage of a temperature shock should be proportional to the cube of the height of the shock.