Studies of optimization problems and certain kinds of differential equations have led in recent years to the development of a generalized theory of differentiation quite distinct in spirit and range of application from the one based on L. Schwartz's “distributions.” This theory associates with an extended-real-valued function ƒ on a linear topological space E and a point x ∈ E certain elements of the dual space E* called subgradients or generalized gradients of ƒ at x. These form a set ∂ƒ(x) that is always convex and weak*-closed (possibly empty). The multifunction ∂ƒ: x →∂ƒ(x) is the sub differential of ƒ.

Rules that relate ∂ƒ to generalized directional derivatives of ƒ, or allow ∂ƒ to be expressed or estimated in terms of the subdifferentials of other functions (whenƒ = ƒ1 + ƒ2,ƒ = g o A, etc.), comprise the sub differential calculus.

A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [20] (and sketched below) that if k is a number field such that SL(2, 5) and one other nonsolvable group Ŝ5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [20] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q, hence we Obtain

THEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbersQ.

Let G be a closed subgroup of one of the classical compact groups 0(n), U(n), Sp(n). By a reflection we mean a matrix in one of these groups which is conjugate to the diagonal matrix diag (–1, 1, …, 1). We say that G is a topological reflection group (t.r.g.) if the subgroup of G generated by all reflections in G is dense in G.

It was shown recently by Eaton and Perlman [5] that, in case of 0(n), the whole group 0(n) is the unique infinite irreducible t.r.g. In this paper we solve the analogous problem for U(n) and Spin). Our method of proof is quite different from the one used in [5]. We treat simultaneously all the three cases.

Introduction. In 1858 Chebyshev showed that xn+l can be approximated uniformly on [–1, 1] by polynomials of degree at most n with an error 2–n. Let 0 ≦ σ ≦ (n + l)tan2(π/2n + 2). In 1868 Zolotarev established that x n + 1σxn can be approximated uniformly on [ –1, 1] by polynomials of degree at most (n – 1) with an error 2–n(l + σ/n + l)n+1. It is interesting to note that for the case σ = 0, Zolotarev's result includes Chebyshev's result. Achieser ([1], p. 279) proved the following analogue for rational approximation. Let a0 ≠ 0, a1, a2, a3, …, an be any given real numbers. Then for every N > n,

where λ is numerically the smallest root of the polynomial

with

Recently invariant theory of linear groups has been used to determine the structure of several weight enumerators of codes. Under certain conditions on the code, the weight enumerator is invariant under a finite group of matrices. Once all the polynomial invariants of this group are known, the form of the weight enumerator is restricted and often useful results about the existence and structure of codes can be found. (See [5], [8], [14], and [15].) Many of the groups in these applications are of degree 2; in this paper all the invariants of finite 2 X 2 matrix groups over C are determined.

All spaces considered here are metrizable. k will always denote an infinite cardinal. The successor of k will be denoted by k+.

Of particular interest will be the Baire spaces where each Tn is a discrete space of cardinal k. The product topology on B(k) is the same as the topology given by the (complete) “first-difference” metric, p : p(s, t) = 1/n if Si = ti for 1 ≦ i ≦ n —1 and sn = tn. A great deal of information about these spaces can be found in [4].

A subset A of X is called k-analytic (in X) if there exist, for each t ∈ B(k), closed subsets F(t1), …, F(t1, …, tn), … of X such that

In this paper we are going to use a result of H. Harada and Y. Sai concerning composition of nonisomorphisms between indecomposable modules and the theory of almost split sequences introduced in the representation theory of Artin algebras by M. Auslander and I. Reiten to obtain the inductive step in the second Brauer-Thrall conjecture.

Section 1 is devoted to giving the necessary background in the theory of almost split sequences.

As an application we get the first Brauer-Thrall conjecture for Artin algebras. This conjecture says that there is no bound on the length of the finitely generated indecomposable modules over an Artin algebra of infinite type, i.e., an Artin algebra such that there are infinitely many nonisomorphic indecomposable finitely generated modules. This result was first proved by A. V. Roiter [8] and later in general for Artin rings by M. Auslander [2] using categorical methods.

The main purpose of this note is to give a characterization of p-pure unital subrings of the p-adic completion of the ring of integers R of an algebraic number field K localized at a maximal ideal p. This yields a characterization of the valued subfields of the p-adic field. In this context there turn up valuations of rational function fields in many indeterminates which seem to be new. The proof that the underlying function is indeed a valuation is quite easy here, however direct computations would involve a large amount of combinatorics. Our approach seems to fit well with Kronecker's, apparently forgotten, approach to ideal theory in rings of algebraic integers [3]. The concept of p-pure unital subrings arose from a study by the first author and A. Laroche of quasi-p-pure-injective (q.p.p.i.) abelian groups ([1], p. 582).

The following lemma, part a) due to S. N. Bernstein and part b) due to A. A. Markov, is fundamental to the proofs of many inverse theorems in polynomial approximation theory.

LEMMA 1. [2, p. 62 and p. 67] Let ∏n denote the real polynomials of degree at most n. Let p ∈ ∏n, then

a)

b)

From the lemma one can deduce, for example:

THEOREM 1. Iƒ there is a sequence of polynomials pn ∈ ∏n and a δ > 0 so that ‖f – pn‖[a, b] ≧ A/nk + δ then f is k times continuously differentiate on (a, b).

We shall refer to inequalities, such as those of Lemma 1 that bound the derivative r′ of a rational function of degree n in terms of its supremum norm ‖r‖[a, b] and n, as Bernstein-type inequalities.

There are numerous instances in which the partners in an extension of commutative rings R ⊂ T have the same prime ideals, i.e., in which Spec(R) = Spec(T). Although this equality is intended to be taken set-theoretically, the identification easily extends to the corresponding spaces endowed with their Zariski topologies (see Proposition 3.5(a)), but of course need not extend to an identification of Spec(R) and Spec(T) as affine schemes. Perhaps the most striking recent illustration of this phenomenon arises from the work of Hedstrom and Houston [14] in which R is a pseudo-valuation domain and T is a suitable valuation overring. Other examples may be found by means of the D + M construction, either in its traditional form [12, p. 560] or in the generalized situation introduced by Brewer and Rutter [5].

Various aspects of the prime spectrum of a distributive continuous lattice have been discussed extensively in Hofmann-Lawson [7]. This note presents a perhaps optimally direct and self-contained proof of one of the central results in [7] (Theorem 9.6), the duality between distributive continuous lattices and locally compact sober spaces, and then shows how the familiar dualities of complete atomic Boolean algebras and bounded distributive lattices derive from it, as well as a new duality for all continuous lattices. As a biproduct, we also obtain a characterization of the topologies of compact Hausdorff spaces.

Our approach, somewhat differently from [7], takes the open prime filters rather than the prime elements as the points of the dual space. This appears to have conceptual advantages since filters enter the discussion naturally, besides being a well-established tool in many similar situations.

In generalized Riemann integration theory it is becoming increasingly clear that a particular collection of sets has some properties of a topology; it is a useful topology when general requirements hold, and the present paper examines the background. Thomson [23, 24] altered my original theory of the variation and Riemann-type integration that has Lebesgue properties, defining the variation of a function of interval-point pairs over the whole of a space T by using partial divisions of T instead of divisions covering T entirely, and also defining a Lebesgue-type integral. His reason might have been that a decomposable division space seems impossible in a general compact or locally compact space. McGill mentioned this to me, and in [15] connected Thomson's setting with topological measure and Topsøfe [25], giving an interesting theorem on the variation of the limit of a monotone increasing generalized sequence of open sets.

In this paper, we consider the class of groups G(l, m; k) which are defined by the presentation

where k, l, m are integers, and |l| > m > 0, k > 0. Groups in this class possess many properties which seem unusual, especially for one-relator groups. The basis for the results obtained below is the determination of endomorphisms.

For certain of the groups, we are able to calculate their automorphism groups. One consequence of this is to produce examples of one-relator groups with infinitely generated automorphism groups. This answers a question raised by G. Baumslag (in a colloquium lecture at the University of Waterloo). Our examples are, perhaps, the simplest possible; J. Lewin [10] has found an example of a finitely presented group with an infinitely generated automorphism group.

Let X be a Banach space and B a bounded subset of X. For each x ∈ X, define R(x) = sup{‖x – y‖ : y ∈ B}. If C is a nonempty subset of X, we call the number R = inƒ{R(x) : x ∈ C} the Chebyshev radius of B in C and the set the Chebyshev center of B in C. It is well known that if C is weakly compact and convex, then and if, in addition, X is uniformly convex, then the Chebyshev center is unique; see e.g., [9].

Let {Bα : α ∈ ∧} be a decreasing net of bounded subsets of X. For each x ∈ X and each α ∈ ∧, define

In this paper we wish to study fields which can be written as intersections of real closed fields. Several more restrictive classes of fields have received careful study (real closed fields by Artin and Schreier, hereditarily euclidean fields by Prestel and Ziegler [8], hereditarily Pythagorean fields by Becker [1]), with this more general class of fields sometimes mentioned in passing. We shall give several characterizations of this class in the next two sections. In § 2 we will be concerned with Gal , the Galois group of an algebraic closure F over F. We also relate the fields to the existence of multiplier sequences; these are infinite sequences of elements from the field which have nice properties with respect to certain sets of polynomials. For the real numbers, they are related to entire functions; generalizations can be found in [3].

If r is a mapping of a set X onto a set Y, then a selection for r is a mapping S of Y into X for which r o s is the identity. Analogously, if F is a mapping of a set Y into the power set 2Z of a set Z, then a selection for F is a mapping f of Y into Z such that ƒ(y) is an element of F(y) for all y in Y. These two notions are formally equivalent: Given r mapping X onto Y, define F(y) = r–l(y) and Z = X. Conversely, given F mapping Y into 2Z, define X to be the subset of Y × Z consisting of the pairs (y, z) for which z belongs to F(y), and define r on X by r(y, z) = y.

Let G be a metabelian group and R an integral domain of characteristic zero, such that no rational prime divisor of │G│ is invertible in R. By RG we denote the group ring of G over R. In this note we shall prove

THEOREM. If RG ≌ RH as R-algebras, then G ≌ H

The question whether this result holds was posed to me by S. K. Sehgal. The result for R = Z is contained in G. Higman's thesis, and he apparently also proved a more general result. At any rate, I think that the methods of the proof are interesting eo ipso, since they establish a “Noether-Deuring theorem” for extension categories.

In proving the above result, it is necessary to study closely the category of extensions (ℊs, S), where the objects are short exact sequences of SG-modules

This paper continues the nonstandard duality theory of locally convex, topological vector spaces begun in Section 5 of [3]. In Section 1, we isolate an external property, called the pseudo monad, that appears to be one of the central concepts of the theory (Definition 1.2). In Section 2, we relate the pseudo monad to the Fin operation. For example, it is shown that the pseudo monad of a µ-saturated subset A of *E, the nonstandard model of the vector space E, is the smallest subset of A that generates Fin (A) (Proposition 2.7).

The nonstandard model of a dual system of vector spaces is considered in Section 3. In this section, we use pseudo monads to establish relationships among infinitesimal polars, finite polars (see (3.1) and (3.2)) and the Fin operation (Theorem 3.7).

Let be an algebra over a field . For x, y, z in , write (x, y, z) = (xy)z – x(yz) and x-y = xy + yx. The attached algebra is the same vector space as , but the product of x and y is x · y. We aim to prove the following result.

THEOREM 1. Let be a finite-dimensional, power-associative, simple algebra of degree two over a field of prime characteristic greater than five. For all x, y, z in , suppose

1

Then is noncommutative Jordan.

The proof of Theorem 1 falls into three main sections. In § 3 we establish some multiplication properties for elements of the subspace in the Peirce decomposition . In §4 we construct an ideal of which we then use to show that the nilpotent elements of form a subalgebra of for i = 0, 1.

Consider the basic problem in the calculus of variations, that of minimizing

1.1

over a class of functions x satisfying certain boundary conditions at 0 and 1. One of the classical first order necessary conditions for optimality is the second Erdmann condition, which asserts, in the case in which L is independent mof t, that

1.2

along any local solution x. This formula is the customary basis for solving many of the classical problems, such as the brachistochrone. When it is possible to define via the Legendre transform a Hamiltonian H(t, x, p) corresponding to L, the second Erdmann condition, again in the autonomous case, is the assertion that

1.3

a relation which always evokes classical Hamiltonian mechanics and conservation laws.