Let X denote a compact Riemann surface of genus g. Let K denote the canonical line bundle on X and denote by K n the n-fold tensor power of K. Put dn=dimcH~ Kn)=(2n-1 ) (g -1 )+b ln . At each point P~X, there is a sequence of d, integers 1 = 71 (P) < 72(P) < " " < 7a~(P) < 2 n ( g 1) + 1, called the sequence of n-gaps at P. An integer 7 is an n-gap at P if and only if there exists O~H~ K ~) wit...

Zusammenfassung Gegenstand dieses Beitrags ist die asymptotische Störkompensation für lineare Deskriptorsysteme mittels Ausgangsregler, wobei die Regelgrößen nicht als messbar vorausgesetzt werden. Durch Lösung dieser Problemstellung mittels des dualen Zugangs zur Zustandsregelung werden direkt Regler bestimmt, die eine klassische Zustandsdarstellung besitzen. Der Entwurf erfolgt ohne eine Tran...

1. Holomorphic functions in one and several variables. Definition of a complex manifold. Canonical splitting of the complexified tangent and cotangent space. T R ⊂ T ⊗ C = T ′ ⊕ T ′′ , where T R is the real tangent space at a point, T ′′ = ∂/∂z i annihilates holomorphic functions and T ′ = ∂/∂z i annihilates antiholomorphic functions. On C, Df (w) = ∂f ∂z w + ∂f ∂z w. Quasiconformal maps. Stone...

The group generated by the Weierstrass points of a smooth curve in its Jacobian is an intrinsic invariant of the curve. We determine this group for all smooth quartics with eight hyperflexes or more. Since Weierstrass points are closely related to moduli spaces of curves, as an application, we get bounds on both the rank and the torsion part of this group for a generic quartic having a fixed nu...

Néron showed that an elliptic surface with rank 8, and with base B = P1Q, and geometric genus =0, may be obtained by blowing up 9 points in the plane. In this paper, we obtain parameterizations of the coefficients of the Weierstrass equations of such elliptic surfaces, in terms of the 9 points. Manin also describes bases of the Mordell-Weil groups of these elliptic surfaces, in terms of the 9 p...

is the geometric multiplicity of λk which is also the number of Jordan blocks corresponding to λk . • The orders of the Jordan Blocks of λk must sum to the algebraic multiplicity of λk . • The number of Jordan blocks corresponding to an eigenvalue λk is its geometric multiplicity. • The matrix A is diagonalizable if and only if, for any eigenvalue λ of A , its geometric and algebraic multiplici...

In this paper we study certain aspects of the complete integrability of the Generalized Weierstrass system in the context of the Sinh-Gordon type equation. Using the conditional symmetry approach, we construct the Bäcklund transformation for the Generalized Weierstrass system which is determined by coupled Riccati equations. Next a linear spectral problem is found which is determined by nonsing...

Let β1, . . . , βn be linearly independent vectors in a vector space. For all j with 0 ≤ j ≤ n and all vectors α1, . . . , αk, if β1, . . . , βn are in the span of β1, . . . , βj, α1, . . . , αk, then j + k ≥ n. The proof of the claim is by induction on k. For k = 0, the claim is obvious since β1, . . . , βn are linearly independent. Suppose the claim is true for k−1, and suppose that β1, . . ....

The structure group and the involutive differential system that characterize the pseudo-group of contact transformations on a jet space are determined.

In mathematics, complete classification of structures, such as groups and rings, is often a primary goal. Linear transformations are no exception to this. Certain canonical forms exist to classify linear transformations, therefore creating a unique representative of linear transformations in the same similarity class. Diagonal representation is of course one of the simplest examples of a canoni...