Partial Primitives for Clifford Algebra-Valued Functions

In this paper we generalize the concept of primitivation of monogenic functions taking values in a Clifford algebra, which is on its own a generalization to higher dimension of the primitivation problem for holomorphic functions in the complex plane. This problem can be stated as follows: given a monogenic function on , i.e. a solution for the generalized Cauchy-Riemann operator D on , construct a monogenic function such that . In view of the fact that, for monogenic functions g, this can be written as g = f, a straightforward generalization consists in replacing the scalar generator of translations in the x ₀-direction by a generator of another transformation group. In this paper we consider translations in more dimensions.     

Clifford Algebra-Valued Wavelet Transform on Multivector Fields

This paper presents a construction of the n = 2 (mod 4) Clifford algebra Cl _n,0-valued admissible wavelet transform using the admissible similitude group SIM(n), a subgroup of the affine group of \mathbbRⁿ{\mathbb{R}^{n}} . We express the admissibility condition in terms of the Cl _n,0 Clifford Fourier transform (CFT). We show that its fundamental properties such as inner product, norm relation, and inversion formula can be established whenever the Clifford admissible wavelet satisfies a particular admissibility condition. As an application we derive a Heisenberg type uncertainty principle for the Clifford algebra Cl _n,0-valued admissible wavelet transform. Finally, we provide some basic examples of these extended wavelets such as Clifford Morlet wavelets and Clifford Hermite wavelets.     

A Matrix Recurrence for Systems of Clifford Algebra-Valued Orthogonal Polynomials

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory (Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Gürlebeck, Bock, Lávi?ka, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.     

On Fundamental Solutions in Clifford Analysis

Euclidean Clifford analysis is a higher dimensional function theory offering a refinement of classical harmonic analysis. The theory is centred around the concept of monogenic functions, which constitute the kernel of a first order vector valued, rotation invariant, differential operator ?{\underline{\partial}} called the Dirac operator, which factorizes the Laplacian. More recently, Hermitean Clifford analysis has emerged as a new branch of Clifford analysis, offering yet a refinement of the Euclidean case; it focusses on a subclass of monogenic functions, i.e. the simultaneous null solutions, called Hermitean (or h−) monogenic functions, of two Hermitean Dirac operators ?_z{\partial_{\underline{z}}} and ?_z^f{\partial_{\underline{z}^\dagger}} which are invariant under the action of the unitary group, and constitute a splitting of the original Euclidean Dirac operator. In Euclidean Clifford analysis, the Clifford–Cauchy integral formula has proven to be a corner stone of the function theory, as is the case for the traditional Cauchy formula for holomorphic functions in the complex plane. Also a Hermitean Clifford–Cauchy integral formula has been established by means of a matrix approach. Naturally Cauchy integral formulae rely upon the existence of fundamental solutions of the Dirac operators under consideration. The aim of this paper is twofold. We want to reveal the underlying structure of these fundamental solutions and to show the particular results hidden behind a formula such as, e.g. ?E = d{\underline{\partial}E = \delta}. Moreover we will refine these relations by constructing fundamental solutions for the differential operators issuing from the Euclidean and Hermitean Dirac operators by splitting the Clifford algebra product into its dot and wedge parts.     

Oscillation Theorems for Quasilinear Elliptic Differential Equations

By using vector Riccati transformation and averaging technique, some oscillation criteria for the quasilinear elliptic differential equation of second order, ΣNi,j=1Di[Ψ（y）Aij（x）｜Dy｜^p-2Djy]＋p（x）f（y）=0, are obtained. These theorems extend and include earlier results for the semilinear elliptic equation and PDE with p-Laplacian.     

Regularity Theorems and Heat Kernel for Elliptic Operators

We give a new proof of Aronson's upper gaussian bound on theheat kernel for parabolic equations with time-independent realmeasurable coefficients. This approach also gives gaussian boundsin the case of a complex perturbation of real coefficients andin the case of uniformly continuous and complex-valued coefficients.     

Oscillation Theorems for Nonlinear Second Order Elliptic Equations

Some oscillation theorems are given for the nonlinear second order elliptic equationsum from i,j=1 to N D_i[a_(ij)(x)Ψ(y)||▽y||~(p-2)D_(jy)] c(x)f(y)=0.The results are extensions of modified Riccati techniques and include recent results of Usami.     

Proving Theorems in Elementary Geometry with Clifford Algebraic Method

本文结合吴方法及平面几何的Ｃｌｉｆｏｒｄ代数表示，提出了几何定理机器证明的一种完备的方法．用这种方法证明定理时，三角化的过程及证明的过程通常较以前的方法更简短而且它们是可以几何解释的．     

Multi-vector Spherical Monogenics, Spherical Means and Distributions in Clifford Analysis

New higher-dimensional distributions have been introduced in the framework of Clifford analysis in previous papers by Brackx, Delanghe and Sommen. Those distributions were defined using spherical co-ordinates, the ＂finite part＂ distribution Fp x＋^μ on the real line and the generalized spherical means involving vector-valued spherical monogenics. In this paper, we make a second generalization, leading to new families of distributions, based on the generalized spherical means involving a multivector-valued spherical monogenic. At the same time, as a result of our attempt at keeping the paper self-contained, it offers an overview of the results found so far.     

Existence Theorems for Certain Elliptic and Parabolic Semilinear Equations

Existence theorems for the nonlinear parabolic differential equation −∂u/∂t + Δu + |u|^p + f(x, t) = 0 in ⁿ × [0, ∞) with zero initial value are established given explicit conditions on the nonhomogeneous termf(x, t). An existence theorem is also demonstrated for the corresponding elliptic equation.     

Liouville Type Theorems for Elliptic Equations with Gradient Terms

In this paper we obtain Liouville type theorems for nonnegative supersolutions of the elliptic problem ${-\Delta u + b(x)|\nabla u| = c(x)u}$ in exterior domains of ${\mathbb{R}^N}$ . We show that if lim ${{\rm inf}_{x \longrightarrow \infty} 4c(x) - b(x)^2 > 0}$ then no positive supersolutions can exist, provided the coefficients b and c verify a further restriction related to the fundamental solutions of the homogeneous problem. The weights b and c are allowed to be unbounded. As an application, we also consider supersolutions to the problems ${-\Delta u + b|x|^{\lambda}|{\nabla} u| = c|x|^{\mu} u^p}$ and ${-\Delta u + be^{\lambda |x|}|\nabla u| = ce^{\mu |x|}u^p}$ , where p > 0 and λ, μ ≥ 0, and obtain nonexistence results which are shown to be optimal.     

关于稳定性的几个基本定理

本文给出了非自治常微分方程稳定性，一致稳定性，渐近稳定性与一致渐近稳定性的几个定理。这些定理允许Lyapunov函数的导数为变号函数，放宽了Marachkov定理对系统状态变量变化率有界的要求，推广了文[4-7]的相应结果。     

Regularity Theorems for Elliptic and Hypoelliptic Operators via the Global Relation

In this paper we give a new proof regarding the regularity of solutions to hypoelliptic partial differential equations with constant coefficients. On the assumption of existence, we provide a spectral representation for the solution and use this spectral representation to deduce regularity results. By exploiting analyticity properties of the terms within the spectral representation, we are able to give simple estimates for the size of the derivatives of the solutions and interpret them in terms of Gevrey classes.