A Factorization Result for Classical and Similitude Groups

For most classical and similitude groups, we show that each element can be written as a product of two transformations that preserve or almost preserve the underlying form and whose squares are certain scalar maps. This generalizes work of Wonenburger and Vinroot. As an application, we re-prove and slightly extend a well-known result of Mœglin, Vignéras, and Waldspurger on the existence of automorphisms of p-adic classical groups that take each irreducible smooth representation to its dual.

The factorization of the linear differential operator

In this paper, necessary and sufficient conditions for factorization of a linear differential operator are presented. As a consequence of the factorization result some criterion of polynomial factorization is obtained. As a special case of the main result we have got Polynomial Remainder Theorem.

Bidiagonal Factorization

This chapter introduces and methodically develops the important and useful topic of bidiagonal factorization. Factorization of matrices is one of the most important topics in matrix theory, and plays a central role in many related applied areas such as numerical analysis and statistics. Investigating when a class of matrices admits a particular type of factorization is an important study, which historically has been fruitful. Often many intrinsic properties of a particular class of matrices can be deduced via certain factorization results. For example, it is a well-known fact that any (invertible) M-matrix can be factored into a product of a lower triangular (invertible) M-matrix and an upper triangular (invertible) M-matrix. This LU factorization result leads to the conclusion that the class of M-matrices is closed under Schur complementation, because of the connection between LU factorizations and Schur complements. This chapter focuses on triangular factorization extended beyond just LU factorization, however.

Estimation of the intensity of stationary flat processes

The intensity of a stationary process of k-dimensional affine subspaces (k-flats) of ℝ d with directional distribution from a given family R is estimated by observing the process in a compact window. To this end we introduce a type of unbiased estimator (the R-estimator) using the available information about the directional distribution. Special cases are estimators for the intensity of stationary k-flat processes (1) with known directional distribution, (2) with directional distribution invariant with respect to a subgroup of the group of rotations in ℝ d and (3) with unknown directional distribution. We give sufficient conditions for the R-estimator to be the uniformly best unbiased estimator for the intensity of stationary Poisson k-flat processes with directional distribution in R. Equivalent statements for certain types of stationary Cox flat processes can be deduced directly from the results in the Poisson case. Moreover, we consider stationary ergodic flat processes with directional distribution in R and general stationary flat processes with unknown directional distribution, all with a non-degeneracy property. In both cases our estimator turns out to be the uniformly best unbiased estimator from a restricted set of estimators. The result for general stationary flat processes is proved with the help of a factorization result for the second factorial moment measure.

Estimation of the intensity of stationary flat processes

The intensity of a stationary process of k-dimensional affine subspaces (k-flats) of ℝd with directional distribution from a given family R is estimated by observing the process in a compact window. To this end we introduce a type of unbiased estimator (the R-estimator) using the available information about the directional distribution.Special cases are estimators for the intensity of stationary k-flat processes (1) with known directional distribution, (2) with directional distribution invariant with respect to a subgroup of the group of rotations in ℝd and (3) with unknown directional distribution.We give sufficient conditions for the R-estimator to be the uniformly best unbiased estimator for the intensity of stationary Poisson k-flat processes with directional distribution in R. Equivalent statements for certain types of stationary Cox flat processes can be deduced directly from the results in the Poisson case.Moreover, we consider stationary ergodic flat processes with directional distribution in R and general stationary flat processes with unknown directional distribution, all with a non-degeneracy property. In both cases our estimator turns out to be the uniformly best unbiased estimator from a restricted set of estimators. The result for general stationary flat processes is proved with the help of a factorization result for the second factorial moment measure.

Mutually conjugate solutions of formally self-adjoint differential equations

SynopsisLet L be a formally self-adjoint linear differential operator of order m with strictly positive leading coefficient and let m = 2n + 1 if m is odd, m = 2n if m is even. Let y1, y2,…, yn be n given mutually conjugate solutions of Ly = 0 on I, where I is some interval, whose Wronskian is non-zero on I. Then L = (−1)nQ*Q or L = (−1)nQ*DQ where Q is a differential operator of order n, Q* is the adjoint operator and D denotes differentiation. This fact is used to construct further solutions yn+1,−, ym of Ly = 0 so that y1,…, ym is a basis for the solutions of Ly = 0 and for which yi and yn+j are mutually conjugate if i ≠ j. If y1 ≠ 0 on I the degree of L may be lowered by 2 to obtain a formally self-adjoint operator L1 for which mutually conjugate solutions are constructed. If this process is continued a factorization result is obtained which is related to a result of Pólya.