Volume 17, Issue 5

General formulas giving numerical constraints for projective invariants of embedded, complex, projective manifolds are given. Complex, projective, linearly normal manifolds of degree eleven are classified in the framework of adjunction theory.

Motivated by Manin’s expression relating modular symbols to critical values of L -functions, Goldfeld and the author have constructed group cocycles which are related in an analogous way to values of derivatives of L -functions. In this note, we propose a geometric interpretation of these cocycles that is based on the cohomology of the non-compactified modular curve.

The paper is concerned with the applicability of the method of Kantorovich majorants to nonlinear singular integro-differential equations with shift. Also the generalized Kantorovich majorization principle is applied to a class of nonlinear singular integral equations with shift. The results are illustrated in the generalized Hölder spaces.

A linear dynamical system resulting from the interconnection of subsystems is considered. Assuming that this interconnection is ‘‘temporal’’, i.e. starting at a given initial time in the continuous-time case and ending at a given final time in the discrete-time case, such a system is also said to be ‘‘temporal’’. Temporal interconnections generate ‘‘uncontrollable impulsive behaviors’’ which are not found in the classical theory, though they have been studied for more than 20 years in the case of systems with constant coefficients. Determining the structure of the impulsive behavior of a temporal system is a key problem in the theory of linear dynamical systems. It is addressed here, using module theory, for systems with time-varying coefficients, in both the continuous- and discrete-time cases. These two cases are merged into a general framework. The impulsive behavior of a temporal system satisfying a suitable regularity condition has a structure which is fully elucidated. It turns out that the determination of this structure in practice is an algebraic—not an analytic—problem, which makes the calculations simpler and easier to computerize. The theory is illustrated through several examples.

We compute the Bergman reproducing kernel for monogenic functions for half-ball, more general orthogonal ball sectors, and for fractional wedge domains. In the results we obtain the terms to be expected from analogy with complex analysis, viz in the first case the Bergman kernels for the half-space and the entire ball, and in the second the sum of rotated half-space Bergman kernels, but in both cases there also occur supplementary, purely hypercomplex correction terms. Finally, applying a periodisation argument we obtain closed and explicit formulas for the Bergman kernels of wedge shaped domains that are rectangularly bounded.

We compute the Bonnet Plancherel formula associated to a monomial representation induced from a normal subgroup of an exponential solvable Lie group. We give also a variant of the Penney-Fujiwara formula associated to the same representation.

Let S be a linear algebraic semigroup defined over a field. The kernel of S , denoted ker( S  ), as the minimal two-sided semigroup-theoretical ideal of S , exists and is a Zariski closed subset of M . In many situations, certain properties of a linear algebraic monoid are controlled to a large degree by the structure of its kernel. In this paper, the structure of the kernel of S is described in terms of algebraic group, algebraic variety and the Rees construction. Some properties of an irreducible algebraic monoid related to ker( M  ) are studied.