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We establish an analog of the Cauchy-Poincare interlacing theorem for normal matrices in terms of majorization, and we provide a solution to the corresponding inverse spectral problem. Using this solution we generalize and extend the Gauss-Lucas theorem and prove the old conjecture of de Bruijn-Springer on the location of the roots of a complex polynomial and its derivative and an analog of Rolle's theorem, conjectured by Schoenberg.

The structure of a sofic shift space is investigated, and Krieger’s embedding theorem and Boyle’s factor theorem are generalized to a large class of sofic shifts.

Let A A be a separable simple A F AF -algebra with finitely many extreme traces. We give a necessary and sufficient condition for an essentially normal element x ∈ M ( A ) x\in M(A) , i.e., π ( x ) \pi (x) is normal ( π : M ( A ) → M ( A ) / A \pi : M(A)\to M(A)/A is the quotient map), having the form y + a y+a for some normal element y ∈ M ( A ) y\in M(A) and a ∈ A . a\in A. We also show that a normal element x ∈ M ( A ) x\in M(A) can be quasi-diagonalized if and only if the Fredholm index i n d ( λ − x ) = 0 ind(\lambda -x)=0 for all λ ∉ s p ( π ( x ) ) . \lambda \not \in sp(\pi (x)). In the case that A A is a simple C ∗ C^* -algebra of real rank zero, with stable rank one and with continuous scale, K 1 ( A ) = 0 , K_1(A)=0, and K 0 ( A ) K_0(A) has countable rank, we show that a normal element x ∈ M ( A ) x\in M(A) with zero Fredholm index can be written as x = ∑ n = 1 ∞ λ n ( e n − e n − 1 ) + a , \begin{equation*} x=\sum _{n=1}^{\infty }\lambda _n(e_n-e_{n-1})+a, \end{equation*} where { e n } \{e_n\} is an (increasing) approximate identity for A A consisting of projections, { λ n } \{\lambda _n\} is a bounded sequence of numbers and a ∈ A a\in A with ‖ a ‖ > ϵ \|a\|>\epsilon for any given ϵ > 0. \epsilon >0.

The Whitney-Mahowald theorem gave normal Euler number ( mod 4 ) (\bmod \,4) for embeddings of a closed 2 n 2n -manifold in Euclidean 4 n 4n -space. We generalize this theorem to embeddings of closed 2 n 2n -manifolds in an oriented 4 n 4n -manifold with an approach in the framework of unoriented bordism groups of maps.

The Stone representation theorem for Boolean algebras gives us a finite set of equations axiomatizing the class of Boolean set algebras. Boolean set algebras can be considered to be algebras of unary relations. As a contrast here we investigate algebras of n-ary relations (originating with Tarski). The new algebras have more operations since there are more natural set theoretic operations on n-ary relations than on unary ones. E.g. the identity relation appears as a new constant. The Resek-Thompson theorem we prove here gives a finite set of equations axiomatizing the class of algebras of n-ary relations (for every ordinal n). The (Resek-Thompson) theorem we are going to prove here is a "geometric" representation theorem for cylindric algebras. It provides an apparently satisfactory positive solution to the representation problem of cylindric algebras (summed up, e.g., in the introduction of [HMTI] and in, e.g., Henkin-Monk [74]). The theorem represents every "abstract" algebra satisfying the cylindric axioms (eight schemes of equations; cf. the remarks on the choice of the axioms at the end of the paper) by a "concrete" algebra of sets of sequences. The representing algebra is concrete in the sense that we do not have to know the operations of the algebra, it is enough to know its elements. I.e. if we know the elements of the algebra, we can "compute" the operations on them by using their concrete set theoretic structure. (This is similar to the Boolean case where if x, y are elements of a concrete algebra B then their meet must be the set theoretic x n y independently of the choice of B. Already in the Boolean case we have to know the greatest element of B in order to be able to compute the complement-x of x in B.) The first version of the theorem was obtained by Diane Resek and is proved as Theorem 5.27 on p. 285 of Resek [75]. Resek's result is also announced in [HMTII, p. vi, p. 101 (item 3.2.88)] and Henkin-Resek [75, Theorem 4.3], and is mentioned, e.g., in Maddux [82] preceding Problem 5.21; but no proof has appeared in print for this important theorem so far (for reasons indicated below). Using the techniques of Thompson [79], Richard J. Thompson generalized Resek's theorem to the form in which it appears below. Thompson's result is (partially) quoted in [HMTII, 3.2.88] without proof, and otherwise is unpublished. Thompson's proof is of a proof theoretic nature and proves more than the theorem stated below. Further discussion of that proof is found at the end of this paper. In the introduction of Received by the editors February 18, 1987 and, in revised form, August 5, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 03G15, 03C95; Secondary 03G25, 03C75. Research supported by Hungarian National Foundation for Scientific Research grant No. 1810. The second author is visiting the Mathematics Institute Budapest on an IREX fellowship. (r)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Using the semigroup product formula of P \text {P} . Chernoff, a central limit theorem is derived for products of random matrices. Applications are presented for representations of solutions to linear systems of stochastic differential equations, and to the corresponding partial differential evolution equations. Included is a discussion of stochastic semigroups, and a stochastic version of the Lie-Trotter product formula.

Let f be an entire function, and let β \beta and λ \lambda be positive numbers with β ≤ π \beta \leq \pi and β λ > π \beta \lambda > \pi . Let E ( r ) = { θ : log ⁡ | f ( r e i θ ) | > cos ⁡ β λ log ⁡ M ( r ) } E(r) = \{ \theta :\log |f(r{e^{i\theta }})| > \cos \beta \lambda \log M(r)\} . It is proved that either there exist arbitrarily large values of r for which E ( r ) E(r) contains an interval of length at least 2 β 2\beta , or else lim r → ∞ r − λ log ⁡ M ( r , f ) {\lim _{r \to \infty }}{r^{ - \lambda }}\log M(r,f) exists and is positive or infinite. For β = π \beta = \pi this is Kjellberg’s refinement of the cos π ρ \pi \rho theorem.

A version of the trace Paley-Wiener theorem for a reductive Padic group in the context of twisted harmonic analysis with respect to an outer automorphism is proved. Let G be a connected reductive algebraic group over a p-adic field F (of characteristic 0) and let G = G(F). Let Po be a minimal parablic subgroup of G. Let E be an outer automorphism of G of finite order r of G and suppose that E fixes PO. Elements x,y E G are called E-conjugate if y = g-1xe(g) for some g E G. A distribution on G which is invariant under twisted conjugacy is called e-invariant. The objects of study in twisted harmonic analysis are E-invariant distributions. The main examples are provided by twisted orbital integrals (invariant integrals over twisted conjugacy classes) and by twisted characters, which are obtained as follows. Let ir be an irreducible admissible representation of G and let E(7r) denote the representation of G through which g E G acts by r(E-1(g)). If 7r is equivalent to E(ir), then there exists an operator 7r(E), unique up to rth roots of unity, such that 7r(E)7r(g)7r(E)-1 = 7r(E(g)) and ir(E)r = 1. In other words, 7r extends to the semidirect product G* = C X (E). The twisted character of ir (which depends on the choice of ir(E)) is the distribution f -Tr(ir(f)ir(E)) on the space 1(G) of compactly supported locally constant functions on G. In applications of the trace formula to global problems in the theory of automorphic representations, it is often necessary to use results on the representation theory and harmonic analysis of reductive groups over p-adic fields. Because of the role played by twisted local harmonic analysis in applications of the twisted trace formula (e.g., [1, 10]), it is important to have available extensions to the twisted case of results known for ordinary harmonic analysis. Many results in harmonic analysis extend to the twisted case. For example, by [7], Harish-Chandra's theorem on the local integrability of characters holds for twisted characters. The purpose of this paper is to prove a twisted version (Proposition 11.2) of the trace Paley-Wiener theorem for p-adic groups of Bernstein, Deligne, and Kazhdan, given in [3]. The twisted Paley-Wiener theorem gives a characterization of functions of the form ir Tr(ir(f)i(e)) for some f E )(G). In ??1-6, the appropriate setting for the twisted theorem is developed. In ??710, results needed for the proof of Proposition 11.2 are proved. Propositioii 7.4 is a twisted version of Casselman's theorem on characters. The proof of Proposition 11.2 follows the plan of the argument given in [3], with modifications for the twisted Received by the editors December 12, 1986 and, in revised form, June 11, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 22E50. The author was partially supported by grants from the National Science Foundation and the Sloan Foundation. (E)1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

In a paper published in 1906t, Professor G. D. Birkhoff treated the meanvalue and remainder theorems belonging to polynomial interpolation, in which the linear differelntial operator lt(n) played a particular role. It is natural to expect that a generalization of many of the ideas of that paper may apply to the general linear differential operator of order n, and the author is attempting such a program. This generalization throws fundamentally new light on the theory of trigonometric interpolation. A very elegalnt paper by G. Polyae has just appeared treating mean-valute theorems for the general operator in a restricted interval. It is the special aim of the present paper to develop a general mean-value theorem, and to show how it can be specialized to obtain Polya's results. We consider a linear differential expression of order n

Sufficient conditions for the uniform convergence of polynomials Pj(z) of best qth power approximation to a given function f(z) of a complex variable on a smooth curve P were presented in an early paper by Jackson [1]. Although his theorem has found frequent application in approximation theory (see [2]), only slight extensions of the result are to be found in the literature. The present paper gives sufficient conditions for the convergence of the P,(z) to f(z) in the mean of order p > q on r when convergence in the mean on F of order p is known for some auxiliary sequence of polynomials, and so includes Jackson's theorem as the special case p = oo. Further extensions of that theorem are given for best approximation by trigonometric polynomials, rational functions with prescribed and free poles, and bounded analytic and meromorphic functions. The method of proof uses standard best approximation arguments together with a recent result of Walsh [3] which we state as

As originally proved by Hilbert, this theorem applied to polynomials whose coefficients were either elements of a field, or rational integers. In keeping with the modern tendency toward abstraction, however, one now finds the theorem proved for polynomials whose coefficients are elements of a commutative ring with unit element in which every set has a finite basis. When one turns to differential polynomials and differential ideals one finds that the exact analogue of the Hilbert theorem is lacking('). It is not true that every system of differential polynomials z contains a finite subset F1, * *, F8 such that

We prove a partition theorem (in the sense of the theorems of Ramsey [3], Erdos-Rado [1], and Rado [2]) which together with a forthcoming paper by Halpern and A. Levy will constitute a proof of the independence of the axiom of choice from the Boolean prime ideal theorem in Zermelo-Fraenkel set theory with the axiom of regularity. Although the theorem arises in logic, it is of a purely combinatorial character and, we believe, interesting in its own right. One application is as follows. Let P be a partition of R x R (R being the rational numbers) into two parts, i.e. P = {P0, P1}, P0 n P1 = 0, P0 u P1 = R x R. Thus P determines a matrix of 0's and l's; P(x, y) = 0 if e P0, P(x, y) = 1 if e-P1. What kind of solid submatrices are there? (a solid submatrix is a subset of R x R of the form A x B, whose entries are either all 0's or all l's). Results of Rado [2] tell us that for any positive integers m, n there are AI B CR, A| = n, IBI = m and A x B is solid. Our theorem implies that A, B can be found satisfying additional properties of separation or scattering in R. The finite version gives similar results for the case where R is replaced by any large finite set. The theorem is applicable to all finite dimensions (not just d = 2 as in the example). In fact much of the difficulty in the proof was generalizing from dimension 2 to higher dimensions. The proof is novel in the sense that we accomplish it by means of metamathematical techniques. Thus the proof we give is in some sense dissatisfying. We have tried to eliminate the use of metamathematics without success and would welcome a simplification in this direction(l).

It is shown that a condition on the size of the exceptional set in the second fundamental theorem of Nevanlinna cannot be improved. The method is based on a construction of Hayman and also makes use of a quantitative version of a result of F. Nevanlinna about the growth of the characteristic function of a meromorophic function omitting a finite number of points