用于余标架附加麦克斯韦方程组的分数阶德拉姆复形
A fractional de Rham complex for coframe-attached Maxwell equations
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中文总结 AI 辅助
研究在有界矩形上构建分数阶切泛函理论，通过定义锚定分数阶算子等方法构造外代数及相关链复形，证明坐标变换刚性定理并建立麦克斯韦型系统，推导电荷守恒和分数阶波动方程。
中文摘要 AI 辅助
我们在有界矩形上发展了一种坐标锚定的逐点分数阶切泛函理论。通过研究坐标方向上黎曼 - 刘维尔积分的取值范围，我们通过精确求逆定义了锚定分数阶算子，并证明了一个表示定理：每个坐标分数阶切泛函是逆算子在基点处求值的标量倍数，且归一化后的是唯一的。在内部点，所得分数阶切空间忠实地作用于公共锚定空间。然后我们在由分数阶坐标原函数生成的多项式代数上构造一个外代数。其分数阶外微分定义了一个幂零代数链复形，有明确的多项式庞加莱同伦，并且通过正对角缩放与普通多项式德拉姆复形相关。由于这个分数阶微分对于普通楔积不是分次导数，全局对象是一个德拉姆型链复形。我们还证明了关于保持分数阶余标架的正锥保持线性坐标变换的刚性定理，并建立了一个洛伦兹余标架附加麦克斯韦型系统，在多项式系数类中推导电荷守恒和分数阶波动方程。
英文摘要
We develop a coordinate-anchored pointwise theory of fractional tangent functionals on bounded rectangles. By working on the range of the coordinatewise Riemann-Liouville integral, we define anchored fractional operators by exact inversion and prove a representation theorem: every coordinate fractional tangent functional is a scalar multiple of evaluation of the inverse operator at the base point, and the normalized one is unique. At interior points, the resulting fractional tangent space acts faithfully on the common anchored space. We then construct an exterior algebra over the polynomial algebra generated by the fractional coordinate primitives. Its fractional exterior differential defines a nilpotent algebraic chain complex, admits an explicit polynomial Poincare homotopy, and is related to the ordinary polynomial de Rham complex by a positive diagonal rescaling. Since this fractional differential is not a graded derivation for the ordinary wedge product, the global object is a de Rham-type chain complex. We also prove a rigidity theorem for positive-cone-preserving linear coordinate changes preserving the fractional coframe and formulate a Lorentzian coframe-attached Maxwell-type system, deriving charge conservation and fractional wave equations within the polynomial coefficient class.
