The bilinear field of values

A complete characterization is given of a bilinear field of values involving orthonormal sets of vectors.     

The ratio field of values

The ratio field of values, a generalization of the classical field of values to a pair of n-by-n matrices, is defined and studied, primarily from a geometric point of view. Basic functional properties of the ratio field are developed and used. A decomposition of the ratio field into line segments and ellipses along a master curve is given and this allows computation. Primary theoretical results include the following. It is shown (1) for which denominator matrices the ratio field is always convex, (2) certain other cases of convex pairs are given, and (3) that, at least for n=2, the ratio field obeys a near convexity property that the intersection with any line segment has at most n components. Generalizations of the ratio field of values involving more than one matrix in both the numerator and denominator are also investigated. It is shown that generally such extensions need not be convex or even simply connected.     

Quasisymmetric and quasiantisymmetric bilinear forms over a field

Translated from Matematicheskie Zametki, Vol. 52, No. 6, pp. 140–148, December, 1992.     

Systems of bilinear and quadratic equations in a finite field

Summary An explicit formula is found for the number of solutions in a finite field of the system of polynomial equations(11), when the coefficients satisfy certain conditions.     

The number of equivalence classes of nondegenerate bilinear and sesquilinear forms over a finite field

Generating functions in the form of infinite products are given for the number of equivalence classes of nondegenerate sesquilinear forms of rank n over GF(q²) and for the number of equivalence (or congruence) classes of nondegenerate bilinear forms of rank n over GF(q).     

Brauer characters with cyclotomic field of values

It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675-686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)).     

The theory of models of bilinear mappings

Omsk City. Translated from sibirskii Matematicheskii Zhurnal, Vol. 31, No. 3, pp. 94–108, May–June, 1990.     

On the field of values of oblique projections

We highlight some properties of the field of values (or numerical range) W(P) of an oblique projector P on a Hilbert space, i.e., of an operator satisfying P²=P. If P is neither null nor the identity, we present a direct proof showing that W(P)=W(I-P), i.e., the field of values of an oblique projection coincides with that of its complementary projection. We also show that W(P) is an elliptical disk (i.e., the set of points circumscribed by an ellipse) with foci at 0 and 1 and eccentricity 1/‖P‖. These two results combined provide a new proof of the identity ‖P‖=‖I-P‖. We discuss the influence of the minimal canonical angle between the range and the null space of P, on the shape of W(P). In the finite dimensional case, we show a relation between the eigenvalues of matrices related to these complementary projections and present a second proof to the fact that W(P) is an elliptical disk.     

The synthesis of multi-programme controls in bilinear systems

The problem of synthesizing multi-programme controls is considered for a bilinear time-dependent control system with mutli-dimensional input. An existence and representation theorem is proved for a control under which the initial system is capable of performing a given set of programmed motions, each of which is asymptotically stable in Lyapunov's sense.     

Domains containing the field of values of a matrix

Two groups of estimates are given for the field of values of a general complex matrix. The first group gives a domain containing both the field of values and the convex hull of the diagonal elements, while the second group gives a domain containing both the field of values and the convex hull of the eigenvalues. The domain given by the second group reduces to the smallest domain possible if the matrix is normal. The results have useful applications in control theory and in stability theory for differential equations.     

The apolar bilinear form in geometric modeling

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives.