Abstract: A constructive existence proof is given for equilibria of polymatrix games. The construction is based on a generalization of the almost-complementary paths used for the linear complementarity problem.

Abstract: Study of several related turbulence approximations with regard to dynamical properties and agreement of numerical predictions with laboratory and computer experiments. The approximations considered include the direct-interaction equations (Kraichnan, 1964), Herring's (1966) self-consistent-field theory, a generalization of Edwards' (1964) theory, the abridged Lagrangian-history, direct-interaction approximation (Kraichnan, 1966), the test-field model (Kraichnan, 1971), and an approximation, not previously described, in which one velocity field passively suffers convection by another. Most of the cited approximations are representable by stochastic model equations for the velocity amplitude. Explicit constructions are given for the stochastic models, in a form that can be approximated on a digital computer. These constructions are used to discuss the physical and mathematical differences between the model dynamics and actual Navier-Stokes dynamics.-

Abstract: One of the most fundamental and characteristic features of recursion theory is the fact that the recursive sets are not uniformly recursive. In this paper we consider the degrees a such that the recursive sets are uniformly of degree ≦a and characterize them by the condition a’ ≦ 0". A number of related results will be proved, and one of these will be combined with a theorem of Yates to show that there is no r.e. degree a < 0’ such that the r.e. sets of degree ≦a are uniformly of degree ≦a. This result and a generalization will be used to study the relationship between Turing and many-one reducibility on the r.e. sets.

Abstract: It is shown that an obvious generalization of the subjective metrics model by Bloxom, Horan, Carroll and Chang has a very simple algebraic solution which was previously considered by Meredith in a different context. This solution is readily adapted to the special case treated by Bloxom, Horan, Carroll and Chang. In addition to being very simple, this algebraic solution also permits testing the constraints of these models explicitly. A numerical example is given.

Abstract: Let a 1's and b 0's be randomly arranged in a row; k is defined to be the largest number of 1's appearing within any m consecutive points in the arrangement. We find Pk*(k) = Pr{k a/2, and give upper 5% points on k for some choices of a, b, and m. This generalization of the classical birthday problem corresponds to a test for nonrandom clustering that is a discrete analogue to a scan test for clustering of points on the line discussed by Nous [4].

Abstract: : A class of bilinear estimation problems involving single-degree-of- freedom rotation is formulated and resolved. Both continuous and discrete time estimation problems are considered. Error criteria, probability distributions, and optimal estimates on the circle are studies. An effective synthesis procedure for continuous time estimation is provided, and a generalization to estimation on arbitrary abelian Lie groups is included. An intrinsic difference between the discrete and continuous problems is discussed, and the complexity of the equations in the discrete time case is analyzed in this setting. Applications of these results to a number of practical problems including FM demodulation and frequency stability are examined.

Abstract: The problem of estimating the slope of a polynomial regression at a fixed point of the experimental region such that (a) the variance of the least-square estimate of the slope at the fixed point is a minimum and (b) the average variance of the least-square estimate of the slope is a minimum is discussed in this paper. In general these designs can be obtained using Kiefer-Wolfowitz [5] characterization of c-optimal designs, Federov [2] characterization of L-optimal designs, and Studden's [10] generalization of the Elfving Theorem [1]. After presenting a brief review of these characterization theorems, specific illustrations for the quadratic and cubic regressions are presented in detail.

Abstract: Outline.- A priori estimates for solutions of linear elliptic functional equations with constant coefficients.- A representation for continuous linear functionals on W o m,p (G) (1 ?) and its applications: A generalization of Garding's inequality and existence theorems.- Regularity and existence theorems for uniformly elliptic functional equations.

Abstract: In a category of modules the notions of />injectivity (with respect to a torsion radical p) and quasi-injectivity can be generalized to a notion of injectivity with respect to two preradicals simultaneously. Using this general definition an analog of Baer's condition for injectivity is obtained, as well as other generalizations of results for injective and quasi-injective modules. An alternate approach (not requiring the existence of injective envelopes) is given for abelian categories, with the results stated in dual form for projectivity.

Abstract: We prove and extend the results of [7], thereby obtaining a generalization of [4], [5] to the case of noncommutative formal (co)groups.

Abstract: The concept of α, β-elementary equivalence of two relational structures is defined. Sufficient and necessary conditions are given for this notion by generalizing Ehrenfeucht’s game in algebraic terms. Some results, in a first order language with generalized quantifier, are obtained.

Abstract: For normal positive linear functional /JL and v of a W* algebra 31, the following extension of a noncommutative Radon-Nikodym theorem by Sakai is given. There exist decompositions ju = jUi-}-jU2, v = Vi + V2 such that i>2 is the smallest normal positive linear functional on 91 satisfying v^^2 and 5(^2)-! s(/0, where s(a) denotes the support projection of a, and #2 is the smallest normal positive linear functional on ?H satisfying A22A2 and 5(^2) JLs(v). Further, there exists a non-negative self -adjoint operator AI = AI(V/{JL) (in general unbounded) such that Ai=\^dE\ with its spectral projections EJ in 9t, lim _£"}:= 1 — s£ and MO for all ^63^, where 5^ = 5(^1) — 5(^1) A(l — s(v)). There also exists another non-negative self -ad joint operator A2 = Ai(y / '/JL) such that its spectral projections E? are in 91, limEx=l — s^ and, for all They are related by Ai(v/ju)A2(jU/p)==A2(ju/v)Ai(v//i) = The Bures distance function d(ju9 v) is given by Received June 12, 1972. * On leave from Research Institute for Mathematical Sciences, Kyoto Univ., Kyoto, Japan.

Abstract: A generalization of the hyperstability criterion given in the literature is discussed. The new hyperstability conditions were obtained by utilizing the properties of the positive definite kernels. The results presented here are directly applicable for the synthesis of hyperstable adaptation algorithms for various applications.

Abstract: In this paper the linear transformation which is used in the unified analysis of electrical machines without & commutator is considered. It is shown that, a very straightforward derivation of the so-called commutator transformation is obtained if the underlying mathematical problem is well defined and linear system theory techniques are used to reduce a set of time-varying linear differential equations to a set of time-invariant ones. This leads to a generalization of Park's transformation to an arbitrary polyphase machine, and also yields a better insight into why the usual assumptions of unified machine theory are necessary

Abstract: The problem of computing the eigensystem of Ax = $\lambda$Bx when A and B are symmetric and B is positive definite is considered. A generalization of the Lanczos algorithm for reducing the problem to a symmetric tridiagonal eigenproblem is given. A numerically stable variant of the algorithm is described. The new algorithm depends heavily upon the computation of elementary Hermitian matrices. An ALGOL W procedure and a numerical example are also given.

Abstract: This paper offers the following generalization of the well known variational theorem of Leighton. Considering the non-linear equation , we show that following an arbitrary choice of function u(x)∊C2 and of G(u(x)), G(u(α)) = G(u(β)) = 0; G(u(x)) > 0 for all x in [α, β] the negative sign of the functional implies that all solutions of our equation will be bounded in absolute value by the number (m/K)1/(K-1), where on some sub-interval of [α, β]. In the linear case, K= 1, we show that the negative sign of the above functional and the condition imply the vanishing of all solutions on [α, β]

Abstract: ZIMMERMAN, BARRY J., and ROSENTHAL, TED L. Observation, Repetition, and Ethnic Background in Concept Attainment and Generalization. CHILD DEVELOPMENT, 1972, 43, 605-613. Attaining and generalizing a new concept were studied in Mexicanand Anglo-American fifth graders. The design factorially compared ethnicity x modeling or nonmodeling training x repetition or nonrepetition of a rule summary. All children received feedback on correct responses during performance-phase trials. Both modeling and repetition improved performance. Prior-modeling groups reduced errors faster than nonmodeling groups, whose errors decreased in the last block of trials. Concept generalization was aided by modeling and, especially, by repetition which mainly determined later verbalization of the rule. Anglooutperformed Mexican-American children, but the major results held for both ethnic groups.