In this paper we describe uniied formulas for unitary and hyperbolic reeections and rotations, and show how these uniied transformations can be used to compute a Hermitian triangular decomposition ^ R H D ^ R of a strongly nonsingular indeenite matrix ^ A given in the form ^ The uniication is achieved by the introduction of signature matrices which determine whether the applicable transformatio...

Fulton’s algorithm is used to calculate the intersection multiplicity of two plane curves about a rational point. This work extends Fulton’s algorithm first to algebraic points (encoded by regular chains) and then, with some generic assumptions, to l many hypersurfaces. Out of necessity, we give a standard-basis free method (i.e. practically efficient method) for calculating tangent cones at po...

for X,Y ∈ ḡ, m,n ∈ Z, where X(m) = X⊗t with X ∈ ḡ and m ∈ Z and (·|·) is the normalized invariant inner product of ḡ. We identify ḡ with ḡ⊗C ⊂ g. Fix the triangular decomposition ḡ = n̄− ⊕ h̄⊕ n̄+, and the Cartan subalgebra of g as h = h̄⊕CK ⊕CD. We have h = h̄ ⊕CΛ0 ⊕Cδ, where Λ0 and δ are elements dual to K and D, respectively. Let L(λ) be the irreducible highest weight representation of g of highe...

Let g be a Lie algebra over a field k of characteristic zero, and a fix positive integer N. The Lie algebra ĝ = g ⊗k k[t]/t N+1 k[t] is called a truncated current Lie algebra. In this paper a highest-weight theory for ĝ is developed when the underlying Lie algebra g possesses a triangular decomposition. The principal result is the reducibility criterion for the Verma modules of ĝ for a wide cla...

The aim of this note is the study of an integer matrix whose determinant is related to the Möbius function. We derive a number-theoretic inequality involving sums of a certain class of Möbius functions and obtain a sufficient condition for the Riemann hypothesis depending on an integer triangular matrix. We also provide an alternative proof of Redheffer’s theorem based upon a LU decomposition o...

This paper gives componentwise perturbation analyses for Q and R in the QR factorization A = QR, QTQ = I , R upper triangular, for a given realm×nmatrixA of rank n. Such specific analyses are important for examplewhen the columns ofA are badly scaled. First order perturbation bounds are given for both Q and R. The analyses more accurately reflect the sensitivity of the problem than previous suc...

where d×x denotes a Haar measure on G. Up to constants, for (additive) Haar measure dx on A, d×x = dx/|detx|. For brevity, write |x| for |detx| when possible. [0.1] Convergence The integral defining us converges absolutely in Re(s) > n− 1: Recall the Iwasawa decomposition G = P ·K with P the parabolic subgroup of upper-triangular matrices. Since K is open in G, Haar measure on G restricted to K...

In this paper we prove that Neville elimination can be matricially described by elementary matrices. A PLU-factorization is obtained for any n×m matrix, where P is a permutation matrix, L is a lower triangular matrix (product of bidiagonal factors) and U is an upper triangular matrix. This result generalizes the Neville factorization usually applied to characterize the totally positive matrices...

Classical orthogonal polynomials and matrix polynomials being orthogonal with respect to some Hermitean positive deenite matrix of measures share several properties , e.g., three term recurrencies, Christooel{Darboux formulas; there are connections to the triangular decomposition of the (inverse) moment matrix and to eigenvalue{problems for the banded matrix of recurrence coeecients. Also, a co...