Solvability of symmetric word equations in positive definite letters

Let S(X, B) be a symmetric (‘palindromic’) wordin two letters X and B. A theorem due to Hillar and Johnsonstates that for each pair of positive definite matrices B andP, there is a positive definite solution X to the word equationS(X, B)=P. They also conjectured that these solutions are finiteand unique. In this paper, we resolve a modified version ofthis conjecture by showing that the Brouwer degree of such anequation is equal to 1 (in the case of real matrices). It followsthat, generically, the number of solutions is odd (and thusfinite) in the real case. Our approach allows us to addressthe more subtle question of uniqueness by exhibiting equationswith multiple real solutions, as well as providing a secondproof of the result of Hillar and Johnson in the real case.     

Hyperkahler Structures and Group Actions

A 4n-dimensional Riemannian manifold (M, g) is hyperkählerif it possesses three anti-commuting complex structures I, J,K such that the metric g is Kähler with respect to eachof them. The reduced holonomy group of such a manifold is necessarilya subgroup of Sp(n) so the Ricci tensor of g vanishes and (M,g) can be regarded as a positive definite solution to Einstein'sequations in vacuum.     

On the representation of integers by quadratic forms

Let n 4 and let Q [X₁, ..., X_n] be a non-singular quadraticform. When Q is indefinite we provide new upper bounds for theleast non-trivial integral solution to the equation Q = 0, andwhen Q is positive definite we provide improved upper boundsfor the greatest positive integer k for which the equation Q= k is insoluble in integers, despite being soluble modulo everyprime power.     

On weakly eutactic forms

We make precise some properties of the Hermite function in relationwith the Morse theory introduced by Avner Ash in his papers‘On eutactic forms’,Canad. J. Math. 29 (1977) 1040–1054and ‘On the existence of eutactic forms’,Bull. LondonMath. Soc. 12 (1980) 192–196, and with the cellular decompositionof the space of positive definite quadratic forms. We also establisha link between Ash's and Bavard's mass formulae.     

A Factorization-Related Method for Elliptic Partial Differential Equations with Iterative Improvement

We consider a method for solving elliptic boundary-value problems.The method arises from a finite-difference discretization whichhas one form in the interior region, but is modified near theboundary. This permits the problem to be solved in terms ofsparse upper and lower triangular matrices. The result of thisdirect method is then improved by an iterative technique, whichis further enhanced by a multigrid-type process. For the type of problems we consider here, the total combinedmethod requires only O(N²) time and O(N²) space to compute thesolution of a system of N x N mesh points to good accuracy.The method is applied to a case where normal discretizationleads to a matrix that is not positive definite.     

Phantom holomorphic projections arising from Sturm’s formula

A (positive definite integral) quadratic form is called diagonally 2-universal if it represents all positive definite integral binary diagonal quadratic forms. In this article, we show that, up to equivalence, there are exactly 18 (positive definite integral) quinary diagonal quadratic forms that are diagonally 2-universal. Furthermore, we provide a “diagonally 2-universal criterion” for diagonal quadratic forms, which is similar to “15-Theorem” proved by Conway and Schneeberger.     

A Bohr Phenomenon For Elliptic Equations

In 1914 Bohr proved that there is an r (0,1) such that if apower series converges in the unit disk and its sum has modulusless than 1 then, for |z| < r, the sum of absolute valuesof its terms is again less than 1. Recently, analogous resultshave been obtained for functions of several variables. The aimof this paper is to place the theorem of Bohr in the contextof solutions to second-order elliptic equations satisfying themaximum principle. 2000 Mathematics Subject Classification: 35J15, 32A05, 46A35.     

On the Convergence of the Variable Metric Algorithm

The variable metric algorithm is a frequently used method forcalculating the least value of a function of several variables.However it has been proved only that the method is successfulif the objective function is quadratic, although in practiceit treats many types of objective functions successfully. Thispaper extends the theory, for it proves that successful convergenceis obtained provided that the objective function has a strictlypositive definite second derivative matrix for all values ofits variables. Moreover it is shown that the rate of convergenceis super-linear.     

Quadratic Termination Properties of Minimization Algorithms I. Statement and Discussion of Results

There is a family of gradient algorithms (Broyden, 1970) thatincludes many useful methods for calculating the least valueof a function F(x), and some of these algorithms have been extendedto solve linearly constrained problems (Fletcher, 1971). Somenew and fundamental properties of these algorithms are given,in the case that F(x) is a positive definite quadratic function.In particular these properties are relevant to the case whenonly some of the iterations of an algorithm make a completelinear search. They suggest that Goldfarb's (1969) algorithmfor linearly constrained problems has excellent quadratic terminationproperties, and it is proved that these properties are betterthan has been stated in previously published papers. Also anew technique is identified for unconstrained minimization withoutlinear searches.     

Uniform Families of Polynomial Equations Over a Finite Field and Structures Admitting an Euler Characteristic of Definable Sets

Over a fixed finite field F_p, families of polynomial equations for i = 1,..., k_N, that areuniformly determined by a parameter N, are considered. The notionof a uniform family is defined in terms of first-order logic.A notion of an abstract Euler characteristic is used to givesense to a statement that the system has a solution for infiniteN, and a statement linking the solvability of a linear systemfor infinite N with its solvability for finite N is proved.This characterisation is used to formulate a criterion yieldingdegree lower bounds for various ideal membership proof systems(for example, Nullstellensatz and the polynomial calculus).Further, several results about Euler structures (structureswith an abstract Euler characteristic) are proved, and the caseof fields, in particular, is investigated more closely. 1991Mathematics Subject Classification: primary 03F20, 12L12, 15A06;secondary 03C99, 12E12, 68Q15, 13L05.     

The Effects of Rounding Error on an Algorithm for Downdating a Cholesky Factorization

Let the positive definite matrix A have a Cholesky factorizationA= R^TR. For a given vector xsuppose that ? =A - xx^T has a Choleskyfactorization ? = ^T.This paper considers an algorithm for computing from R and x and an extension for removing a row from the QR factorizationof a regression problem. It is shown that the algorithm is stablein the presence of rounding errors. However, it is also shownthat the matrix can be a very ill-conditioned function of R and x.     

Effective Isometric Embeddings for Certain Hermitian Holomorphic Line Bundles

We consider bihomogeneous polynomials on complex Euclidean spacesthat are positive outside the origin and obtain effective estimateson certain modifications needed to turn them into squares ofnorms of vector-valued polynomials on complex Euclidean space.The corresponding results for hypersurfaces in complex Euclideanspaces are also proved. The results can be considered as Hermitiananalogues of Hilbert's seventeenth problem on representing apositive definite quadratic form on Rⁿ as a sum of squares ofrational functions. They can also be regarded as effective estimateson the power of a Hermitian line bundle required for isometricprojective embedding. Further applications are discussed.     

矩阵方程X+A~*X~(-n)A=P的Hermite正定解及其扰动分析

考虑非线性矩阵方程X A~*X~(-n)A=P,其中A是m阶非奇异复矩阵,P是m阶Hermite正定矩阵.本文利用不动点理论讨论了该方程Hermite正定解的存在性及包含区间,给出了极大解的性质及求极大,极小解的迭代算法.研究了极大解的扰动问题,利用微分等方法获得了两个新的一阶扰动界,并给出数值例子对所得结果进行了比较说明.     

Bounds on Norms of Compound Matrices and on Products of Eigenvalues

An upper bound on operator norms of compound matrices is presented,and special cases that involve the l₁, l₂ and l norms are investigated.The results are then used to obtain bounds on products of thelargest or smallest eigenvalues of a matrix. 1991 MathematicsSubject Classification 15A15, 15A18, 15A42.     

The Finite Subsets of Z2 Having the Extension Property

The paper considers finite subsets Z^d which possess the extensionproperty, namely that every collection {c_k}_k– of complexnumbers which is positive definite with respect to is the restrictionof the Fourier coefficients of some positive measure on T^d.All finite subsets of Z² which possess the extension propertyare described.     

A Proof of a Generalized Nakayama Conjecture

In a recent paper, Külshammer, Olsson, and Robinson proveda deep generalization of the Nakayama conjecture for symmetricgroups. We provide a similar but a shorter and relatively elementaryproof of their result. Our method enables us to obtain a moregeneral H-analogue of the Nakayama conjecture where H is a setof positive integers. 2000 Mathematics Subject Classification20C30 (primary), 20C15, 20C20 (secondary).     

Some Integrals of Gamma, Polygamma and Volterra Functions

Infinite integrals of the gamma and polygamma functions areexpressed in terms of the Volterra and related functions usingthe Laplace transformation method. A number of infinite, convolution,indefinite and definite integrals of v(x), v(x, ), µ(x,ß) and µ(x, ß, ) has been evaluated.A short table of the Volterra function v(x) in the range 0 x 10–0 is presented.     

Stability of Methods for Matrix Inversion

Inversion of a triangular matrix can be accomplished in severalways. The standard methods are characterized by the loop ordering,whether matrix-vector multiplication, solution of a triangularsystem, or a rank-1 update is done inside the outer loop, andwhether the method is blocked or unblocked. The numerical stabilityproperties of these methods are investigated. It is shown thatunblocked methods satisfy pleasing bounds on the left or rightresidual. However, for one of the block methods it is necessaryto convert a matrix multiplication into the solution of a multipleright-hand side triangular system in order to have an acceptableresidual bound. The inversion of a full matrix given a factorizationPA=LU is also considered, including the special cases of symmetricindefinite and symmetric positive definite matrices. Three popularmethods are shown to possess satisfactory residual bounds, subjectto a certain requirement on the implementation, and an attractivenew method is described. This work was motivated by the questionof what inversion methods should be used in LAPACK.     

How and Why to Solve the Operator Equation AX-XB = Y

The entities A, B, X, Y in the title are operators, by whichwe mean either linear transformations on a finite-dimensionalvector space (matrices) or bounded (= continuous) linear transformationson a Banach space. (All scalars will be complex numbers.) Thedefinitions and statements below are valid in both the finite-dimensionaland the infinite-dimensional cases, unless the contrary is stated.1991 Mathematics Subject Classification 15A24, 47A10, 47A62,47B47, 47B49, 65F15, 65F30.     

Dirac Operators on a Compact Lie Group

Let K be a compact connected Lie group with a bi-invariant metric( , ) which is k-spin. A Dirac operator D on ‘pure’definite spinors is associated to a left k-invariant, metricconnection on the tangent bundle; D is left k-invariant andelliptic. It is shown that the dimension of the kernel of D (the spaceof harmonic spinors) depends on the choice of connection. Infact for a real family of connections _a(obtained via the adjointrepresentation of K) the subrepresentation of the left regularrepresentation of K on the space of harmonic spinors is determined,and a dimension formula is given.