Linear forms with algebraic coefficients. I

Linear Differential Algebraic Equations with Constant Coefficients

Differential-algebraic equations (DAEs) arise in a variety of applications. Their analysis and numerical treatment, therefore, plays an important role in modern mathematics. The paper gives an introduction to the topics of DAEs. Examples of DAEs are considered showing their importance for practical problems. Some essential concepts that are really essential for understanding the DAE systems are...

Forms and Linear Algebraic Groups

Linear Stochastic Differential-algebraic Equations with Constant Coefficients

We consider linear stochastic differential-algebraic equations with constant coefficients and additive white noise. Due to the nature of this class of equations, the solution must be defined as a generalised process (in the sense of Dawson and Fernique). We provide sufficient conditions for the law of the variables of the solution process to be absolutely continuous with respect to Lebesgue mea...

Mean Values of Algebraic Linear Forms

Suppose K/Q is a totally real extension of degree d = [K : Q]. Let Qp denote the completion of with respect to the p-adic valuation when p is a rational prime number and let Q,, denote IR when the valuation is the usual absolute value. This latter case is thought of as corresponding to the case where the prime p is 'infinite'. Suppose : K->QP is a (Q-)linear form. We say that is p-ad...

Logarithmic forms with twisted coefficients

Given a compact Kähler manifold, we consider the complement U of a divisor with normal crossings and a unitary local system g̃ on it. We consider a differential graded Lie algebra of forms with holomorphic logarithmic singularities and vanishing residues. We construct a spectral sequence converging to the L2 cohomology H •(U, g̃)(2) corresponding to the anti-holomorphic filtration of this algebra...