Abstract: For two-fold integrals, a lower bound is derived for the number of nodes in a cubature formula of degree 2s-1. There is a formula of degree 2s-1 for which the number of nodes attains this lower bound, iff a certain condition is fullfilled. By this condition, all formulas of degree 2s-1 with that minimal number of nodes can be constructed. Examples and a generalization to then-dimensional case are given.

Abstract: 0. Introduction 203 A. History and explanation of the problem 203 B. Background from logic and elimination theory 208 1. Generalizing the quantifier elimination problem . 210 A. Notations and terminology 210 B. The Frobenius symbol 212 C. Galois stratification and generalization of the diophantine problem .213 2. The intersection-union process 215 A. The intersection-union process over a finite field 215 B. The intersection-union process over a perfect field 217 3. A generalization of the theorems of Bertini and Noether . 219 4. Diophantine problems over all residue class fields of a number field . . .225 5. Diophantine problems over finite fields . 230 A. Over all extensions of a fixed finite field 230 B. Over all finite fields 231

Abstract: The most convenient numerical method for defining surfaces is by biparametric vectors. The paper describes the conditions for ensuring second order continuity. A second definition of continuity is given which provides a mathematical generalization to order n between two surface patches.

Abstract: The worst-case, minimum number of comparisons complexity Vi(n) of the i-th selection problem is considered. A new upper bound for Vi(n) improves the bound given by the standard Hadian-Sobel algorithm by a generalization of the Kirkpatrick-Hadian-Sobel algorithm, and extends Kirkpatrick's method to a much wider range of application. This generalization compares favorably with a recent algorithm by Hyafil.

Abstract: The investigation of the properties of small fluctuations serves as a basis for the generalization of equilibrium and irreversible thermodynamics, to encompass all spatially homogeneous situations of stable chemical steady states. In terms of a Gibbs ensemble picture, the reactants are coupled to particle reservoirs, often in a more intriguing way than in usual grand canonical statistics. It is shown how these couplings can be represented as generalized forces, by postulating an equipartition theorem. Response functions connected with the action of external forces are introduced, and the fluctuation–dissipation theorem is derived as a consequence of the proper definition of forces. The symmetry properties of the systems, with respect to time reversal, provide a classification scheme for the large body of possible steady states.

Abstract: where SS, is the residual sum of squares (RSS) from a regression based on the original n observations and SS2 is the RSS from a regression based on all n + m observations. Under the null hypothesis that both sets of observations are generated by the same regression model, in which the disturbances are normally distributed and obey the classical assumptions, the test statistic (1) has an F distribution with (m, n k) degrees of freedom. If m > k the test is still valid although in such circumstances the analysis of covariance test is usually to be preferred'; see Chow (1960, p. 598). The proof that the statistic (1) follows an F distribution under the null hypothesis may be found in Chow (1960) and also in Fisher (1970). The present note sets out an alternative proof, which follows almost immediately given certain results on "recursive residuals." As well as providing fresh insight into the Chow test this approach leads to a useful generalization, namely that a sequence of such tests is statistically independent when the model remains unchanged. This result has important implications for the construction of procedures designed to detect structural change. It may also be useful in connection with testing for outliers.

Abstract: Let $X$ and $Y$ be two unbounded random variables. Then two necessary conditions are proved concerning the structure of the bivariate distribution function of $X$ and $Y$ when it is expanded in the orthonormal polynomials of its marginal distributions. The first condition concerns the shrinking of the polynomial representation into a diagonal form, and the second is a generalization of the Sarmanov-Bratoeva theorem.

Abstract: New criteria as a generalization of the resultant of two polynomials are given for relative primeness of polynomial matrices together with system theoretic interpretations and unification of the existing and new criteria.

Abstract: A multidimensional Tauberian theorem is established which generalizes the one-dimensional Tauberian theorem of Hardy and Littlewood.Bibliography: 6 titles.

Abstract: This paper studies iterative methods for the global flow analsis of computer programs. We define a hierarchy of global flow problem classes, each solvable by an appropriate generalization of the "node listing" method of Kennedy. We show that each of these generalized methods is optimum, among all iterative algorithms, for solving problems within its class. We give lower bounds on the time required by iterative algorithms for each of the problem classes.

Abstract: The problem under consideration is a maximization problem over a constraint set defined by a finite number of inequality and equality constraints over an arbitrary set in a reflexive Banach space. A generalization of the Kuhn-Tucker necessary conditions is developed where neither the objective function nor the constraint functions are required to be differentiable. A new constraint qualification is imposed in order to validate the optimality criteria. It is shown that this qualification is the weakest possible in the sense that it is necessary for the optimality criteria to hold at the point under investigation for all families of objective functions having a constrained local maximum at this point

Abstract: The aim of the present note is to prove a nonlinear generalization of the well-known integral inequality due to Bihari. This generalization is useful in obtaining pointwise estimates of solutions of nonlinear Volterra integral equations.

Abstract: Recently, Shea and Wainger obtained a variant of the WienerLévy theorem for nonintegrable functions of the form a(t) = b(t) + ß(t), where b(t) is nonnegative, nonincreasing, convex and locally integrable, and ß(t), tß(t) e L1 (0, oo). It is shown here that the moment condition tß(t) e Ü may be omitted from the hypotheses of this theorem. These results are useful in the study of stability problems for some Volterra integral and integrodifferential equations. Introduction. It is well known that under rather weak hypotheses (see [9, Chapter 4] and [3]) the solutions of the linear Volterra equations (1) uii) = fit) /J ait s)uis) ds (0 < t < oo), (2) u'it) = fit) /J ait s)uis) ds (M(0) = u0 ; 0 < t < oo)

Abstract: Several weak base (in the sense of A V Arhangel'skit) metrization theorems are established, including a weak base generalization of the Nagata-Smirnov Metrization Theorem

Abstract: H. Weyl's \"unitary trick\" is generalized to the context of semisimple symmetric Lie algebras with Cartan subspaces, over fields of characteristic zero. As an illustration of its usefulness, the result is used to transfer to characteristic zero an important theorem in the representation theory of real semisimple Lie algebras.