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．     

Riemann boundary value problem for harmonic functions in Clifford analysis

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.     

On Quadratic Forms Between Spheres

For a quadratic form f: S^m Sⁿ, we first estimate the dimensions of its preimages and determine the cases when the upper bound is achieved. Combining results of Wood and Yiu, we then derive a complete classification of f when m 2n-2. By using orthogonal multiplication, we also introduce a new construction of quadratic forms between spheres, generalizing the classical Hopf construction.     

Clifford分析中广义超正则函数的一个非线性边值问题

讨论了Cliffrd分析中广义超正则函数的一个非线性边值问题.首先将广义超正则函数分解为两个奇异积分算子,然后给出了广义超正则函数的Plemelj公式及相关奇异积分算子的性质,最后利用Schauder不动点原理证明了广义超正则函数的一个非线性边值问题的解的存在性及积分表达式.     

Geometric Algebra for Subspace Operations

The set theory relations , \,,, and have corollaries in subspace relations. Geometric algebra is introduced as a useful framework to explore these subspace operations. The relations , \, and are easily subsumed by geometric algebra for Euclidean metrics. A short computation shows that the meet () and join () are resolved in a projection operator representation with the aid of one additional product beyond the standard geometric algebra products. The result is that the join can be computed even when the subspaces have a common factor, and the meet can be computed without knowing the join. All of the operations can be defined and computed in any signature (including degenerate signatures) by transforming the problem to an analogous problem in a different algebra through a transformation induced by a linear invertible function (a LIFT to a different algebra). The new results, as well as the techniques by which we reach them, add to the tools available for subspace computations.     

Zeroes of slice monogenic functions

In this article, we study the structure of zeroes of power series with Clifford algebra‐valued coefficients. Especially, if it has paravector‐valued coefficients, we obtain some sufficient and necessary conditions of power series that have zeroes, as well as a method to compute the zeroes if exist. Copyright 2011 John Wiley & Sons, Ltd.     

A Group of Involutions Characterizing a Clifford Algebra of Type {\mathcal{C}}\,{\mathcal{L}}_{0,3}

In this note, a group characterization for Clifford algebras of type is given.     

On a backward parabolic problem with local Lipschitz source

We consider the regularization of the backward in time problem for a nonlinear parabolic equation in the form _ut+Au(t)=f(u(t),t)$u_t+Au(t)=f(u(t),t)$, u(1)=φ$u(1)=\phi$, where A is a positive self-adjoint unbounded operator and f is a local Lipschitz function. As known, it is ill-posed and occurs in applied mathematics, e.g. in neurophysiological modeling of large nerve cell systems with action potential f   in mathematical biology. A new version of quasi-reversibility method is described. We show that the regularized problem (with a regularization parameter β>0$\beta >0$) is well-posed and that its solution _Uβ(t)$U_{\beta }(t)$ converges on [0,1]$[0,1]$ to the exact solution u(t)$u(t)$ as β→0⁺$\beta \to 0^{+}$. These results extend some earlier works on the nonlinear backward problem.     

Factorisations des extensions galoisiennes

ABSTRACT

In general, Clifford algebras of quadratic forms are finite dimensional; therefore, their representations are easy to describe. However, for homogenous polynomial forms of degree dbm > 2, the situation is different because their Clifford algebras are infinite dimensional. In this article, we get a finite set of pairwise orthogonal idempotents of sum 1 in these algebras. This permits us to obtain interesting properties for d-dimensional representations of polynomial forms of degree d; for example, we show that the image C of the Clifford algebra by such representation is an endomorphism algebra of finitely generated projective Z(C)-module of d-rank, direct sum of finitely generated projective Z(C)-module of 1-rank. Before establishing this, we give a new proof of the Poincaré-Birkhoff-Witt theorem for these algebras with the help of a general composition lemma. At the end of this work, we give a linearization of diagonal binary and ternary forms of degree dbm > 3.     

ON CAUCHY－POMPEIU FORMULA FOR FUNCTIONS WITH VALUES IN A UNIVERSAL CLIFFORD ALGEBRA

This paper obtains the Cauchy-Pompeiu formaula on certain distinguished boundary for functions with values in a unviersal Clifford algebra ,This formulas is Just an extension of the Cauchy‘s integral formula obtained in [11].     

ON SETS OF ZEROES OF CLIFFORD ALGEBRA-VALUED POLYNOMIALS

In this note, we study zeroes of Clifford algebra-valued polynomials. We prove that if such a polynomial has only real coefficients, then it has two types of zeroes： the real isolated zeroes and the spherical conjugate ones. The total number of zeroes does not exceed the degree of the polynomial. We also present a technique for computing the zeroes.     

A boundary value problem for hypermonogenic functions in Clifford analysis

This paper deals with a boundary value problem for hypermonogenic functions in Clifford analysis. Firstly we discuss integrals of quasi-Cauchy's type and get the Plemelj formula for hypermonogenic functions in Clifford analysis, and then we address Riemman boundary value problem for hypermonogenic functions.     

ORDERING IN AUTOMATED THEOREMPROVING OF DIFFERENTIAL GEOMETRY

1.BackgroundMom1979,WuWentsunbeganthestudyofautomatedtheoremprovingindifferentialgeometry[1].Inhismethod,thehypotheses,representedbydifferentialpolynomials,aretriangulatedaccordingtoanorderofthevariablesinthepolynomials;theconclusionisprovedbythetriangulatedresultcalledcharacteristicset.Wuusedelement-basedorderingmethodwhichcanbeillustratedbythisexample'LettingxKybetwosmoothfunctionsofthesamevariable,thenxKX'KX"K'Ky'y,Ky"'.ThisorderisthefoundationofWu'seliminationprinciple.However,a…     

An initial-boundary value problem for the Maxwell equations

In this paper, by using methods from complex analysis and quaternionic analysis, we investigate an initial-boundary value problem for the Maxwell equations and obtain the general solutions and solvable conditions of the problem respectively in different cases. In addition, by using a similar method, we also discuss an initial-boundary value problem for a hyperbolic complex system of first order equations in R³.     

Half Dirichlet problem for matrix functions on the unit ball in Hermitian Clifford analysis

The simultaneous null solutions of the two complex Hermitian Dirac operators are focused on in Hermitian Clifford analysis, where the Hermitian Cauchy integral was constructed and will play an important role in the framework of circulant (2×2) matrix functions. Under this setting we will present the half Dirichlet problem for circulant (2×2) matrix functions on the unit ball of even dimensional Euclidean space. We will give the unique solution to it merely by using the Hermitian Cauchy transformation, get the solution to the Dirichlet problem on the unit ball for circulant (2×2) matrix functions and the solution to the classical Dirichlet problem as the special case, derive a decomposition of the Poisson kernel for matrix Laplace operator, and further obtain the decomposition theorems of solution space to the Dirichlet problem for circulant (2×2) matrix functions.     

A computer-assisted proof of Saari's conjecture for the planar three-body problem

The five relative equilibria of the three-body problem give rise to solutions where the bodies rotate rigidly around their center of mass. For these solutions, the moment of inertia of the bodies with respect to the center of mass is clearly constant. Saari conjectured that these rigid motions are the only solutions with constant moment of inertia. This result will be proved here for the planar problem with three nonzero masses with the help of some computational algebra and geometry.