Extrema detection of bivariate spline functions

In this paper, we develop a systematic method for detecting the extrema of bivariate spline functions and of their derivatives. It is assumed that the splines are constituted by employing normalized, uniform B-splines as the basis functions, and the detection procedure fully utilizes the spline properties. All the extrema can be found except those with singular Hessian matrix. By numerical examples, we demonstrate the effectiveness of the method.     

Extrema of elementary symmetric polynomials of the eigenvalues of the matrix P1KP+L

The paper describes extrema of elementary symmetric polynomials of mth order of the eigenvalues of the matrix P¹KP+L where P is an n by k matrix (n?k) and P¹ its conjugate transpose, such that P¹P=I; and K,L are Hermitian, positive- definite matrices of dimensions n by n, k by k, respectively.     

Principal eigenvalues with indefinite weight functions

Both existence and non-existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved. The existence of a continuous family of principal eigenvalues is demonstrated.

     

Normal and seminormal eigenvalues of matrix functions

LetP() be ann×n analytic matrix function andW(P) be its numerical range. In this paper classical results on the normality of matrix eigenvalues on W(P) are generalized to the context of such matrix functions. Special attention is paid to corners of W(P) and to the special case of matrix polynomialsP().     

Extrema Constrained by a Family of Curves and Local Extrema

This paper considers the connections between the local extrema of a function f:DR and the local extrema of the restrictions of f to specific subsets of D. In particular, such subsets may be parametrized curves, integral manifolds of a Pfaff system, Pfaff inequations. The paper shows the existence of C ¹ or C ²-curves containing a given sequence of points. Such curves are then exploited to establish the connections between the local extrema of f and the local extrema of f constrained by the family of C ¹ or C ²-curves. Surprisingly, what is true for C ¹-curves fails to be true in part for C ²-curves. Sufficient conditions are given for a point to be a global minimum point of a convex function with respect to a family of curves.     

Monodromy eigenvalues and zeta functions with differential forms

For a complex polynomial or analytic function f, there is a strong correspondence between poles of the so-called local zeta functions or complex powers ∫|f|^2sω, where the ω are C^∞ differential forms with compact support, and eigenvalues of the local monodromy of f. In particular Barlet showed that each monodromy eigenvalue of f is of the form , where s₀ is such a pole. We prove an analogous result for similar p-adic complex powers, called Igusa (local) zeta functions, but mainly for the related algebro-geometric topological and motivic zeta functions.     

Extrema of matrix functionals

The problem of determining conditional extrema of functionals with matrix arguments is considered. We derive the necessary and sufficient mathematical conditions for the existence of extrema of functionals satisfying constraints of the form of matrix equalities on the arguments. The construction of extrema is based on functions and matrices of indeterminate Lagrange multipliers. As applications we consider an example of determining the optimal strength coefficient matrix in a dynamical system with an adaptive Carleman filter and an example, from the theory of statistical decisions, of minimizing the volume of the dispersion error ellipsoid. Our approach has wide applications not only in optimization problems from automatic control theory but also in mathematical statistics and the theory of material strength and plasticity.Translated from Dinamicheskie Sistemy, No. 5, pp. 103–106, 1986.     

Leading coefficients of the eigenvalues of perturbed analytic matrix functions

This note contains some supplements to our earlier notes [LN II], [LN III], where the Newton diagram was used in order to obtain in a straightforward way information about the perturbed eigenvalues of an analytic and analytically perturbed matrix function.     

Bounds for the stieltjes transform of spectral functions of singular eigenvalues

The main problem consists in obtaining G-estimates for singular eigenvalues of real matrices A=(a_ij),i = ,i = if the only things known are X_i, i = , independent observations over the matrix A+, where is a stochastic matrix. Under certain conditions on , A, n, m, and s results are obtained for the Stieltjes transform of spectral functions of the singular eigenvalues of the matrix A.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 464–469, April, 1990.     

On eigenvalues of meet and join matrices associated with incidence functions

Let (P,?,∧) be a locally finite meet semilattice. Let

     

Extrema of a Real Polynomial

In this paper, we investigate critical point and extrema structure of a multivariate real polynomial. We classify critical surfaces of a real polynomial f into three classes: repeated, intersected and primal critical surfaces. These different critical surfaces are defined by some essential factors of f, where an essential factor of f means a polynomial factor of f–c ₀, for some constant c ₀. We show that the degree sum of repeated critical surfaces is at most d–1, where d is the degree of f. When a real polynomial f has only two variables, we give the minimum upper bound for the number of other isolated critical points even when there are nondegenerate critical curves, and the minimum upper bound of isolated local extrema even when there are saddle curves. We show that a normal polynomial has no odd degree essential factors, and all of its even degree essential factors are normal polynomials, up to a sign change. We show that if a normal quartic polynomial f has a normal quadratic essential factor, a global minimum of f can be either easily found, or located within the interior(s) of one or two ellipsoids. We also show that a normal quartic polynomial can have at most one local maximum.     

Convergence of eigenvalues and Green functions under surgery type degeneration of Riemannian manifolds

We study the asymptotics for the eigenvalues and eigenfunctions and the Green function for sequences of Riemannian metrics that converge to a smooth compact limit of different topology in a controlled manner, as encountered in surgery constructions. A model case is the Bergman metric on a family of degenerating Riemann surfaces with reducible limit. Received February 26, 1996 / Accepted March 15, 1996     

Relations among eigenvalues of Sturm-Liouville problems with different types of leading coefficient functions

For any Sturm-Liouville problem with a separable boundary condition and whose leading coefficient function changes sign (exactly once), we first give a geometric characterization of its eigenvalues λn using the eigenvalues of some corresponding problems with a definite leading coefficient function. Consequences of this characterization include simple proofs of the existence of the λn's, their Prüfer angle characterization, and a way for determining their indices from the zeros of their eigenfunctions. Then, interlacing relations among the λn's and the eigenvalues of the corresponding problems are obtained. Using these relations, a simple proof of asymptotic formulas for the λn's is given.     

Asymptotic distributions of functions of the eigenvalues of some random matrices for nonnormal populations

The authors investigated the asymptotic joint distributions of certain functions of the eigenvalues of the sample covariance matrix, correlation matrix, and canonical correlation matrix in nonnull situations when the population eigenvalues have multiplicities. These results are derived without assuming that the underlying distribution is multivariate normal. In obtaining these expressions, Edgeworth type expansions were used.     

一元隐函数取极值的一般充分条件

在本文中,我们给出了判定一元隐函数取极值的一般充分条件,为判定一元隐函数取极值提供了一般的判定方法.