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1# Introduction. Consider a compact Hausdorff space X and the set C(X) of all continuous complex-valued functions on X, Consider also a subset 21 of C(X) which is an algebra, which is closed in the uniform topology of C(X), which contains the constant functions, and which contains sufficiently many functions to distinguish points of X. Such an algebra 21 is called self-adjoint if the complex conjugate of each function in 2t is in 21. The classical Stone-Weierstrass Theorem states that if 21 is self-ad joint then 21 = C(X). If 21 has the property that the only functions in 21 which are real at every point of X are the constant functions then 21 is called anti-symmetric. Clearly anti­ symmetry and self-adjointness are opposite properties, in the sense that if 21 has both properties then X must consist of a single point. Hoffman and Singer [2] have studied these two properties and given several interesting examples. The present paper was inspired by their work but it more directly relates to a previous paper of Silov [3]. The purpose of the present paper is to prove the following decomposition theorem for a general algebra 21 of the type defined above.

For a domain R, an it-module A is called a Baer module if Ext £ 04, T) = 0 for every torsion /^-module T. Dual to Baer modules, a torsion it-module B is called a UT-moάule if Ext \(X, B) = 0 for every torsion free it-module X. In this paper properties of these two types of modules will be derived and characterizations of Priίfer domains, Dedekind domains and fields will be obtained in terms of Baer and l/T-module properties. One characterization will show the Baer modules are analogous to projective modules in the sense that a domain R is Dedekind if and only if, over R, submodules of Baer modules are Baer. In addition, just as a semisimple ring S can be characterized by the property that all 5-modules are injective, or, equivalently, aU 5-modules are projective, a domain R is a field exactly if every torsion it-module is UT or, equivalently, every torsion free it-module is a Baer module. Further properties of these two kinds of modules will provide sufficient conditions to bound the global dimension of a domain R. 0. Historical Note. The concept of a Baer module goes back to 1936 when R. Baer, in [1], proposed the problem asking for a complete characterization of all abelian groups G such that Extz (G, T) = 0, for all torsion abelian groups T. At that time he showed that any such abelian group must be torsion free, and free if it had countable rank. Then in 1959 R. Nunke, in [10], extended these results to modules over a Dedekind domain, proving that such a module was again torsion free, and projective if it had countable rank. Finally in 1969 P. Griffith, in [5], completely solved Baer's problem for abelian groups, and showed that any such abelian group, now called a Baer group or 2?-grouρ, must be free. In [6], the author extended Griffith's techniques to modules over a Dedekind domain, showing that if A is a Baer module over a Dedekind domain then A is projective. The major adjustments needed in this transition from abelian groups to modules over a Dedekind domain were accomplished by means of Corollary 2 on p. 279 of [12], Theorems 3 and 5 in [8], Lemma 8.3 and Theorem 8.4 in [10], as well as the exposition on ideals and valuations in § 18, 19 of 1. Preliminaries. Unless additional restrictions are stated, in this paper R denotes an arbitrary integral domain: that is, a commutative ring with 1 having no zero divisors. The quotient field oίR will be denoted by Q.

In this paper smooth maps /: Mn —> Nn with a zerodimensional critical set are considered. The singularities of these maps in the case n — 2 are known to be points where / is locally topologically equivalent to z -»zd {d~ 2,3, •••)• Originally these singularities were studied in connection with the regularity of Douglas' solution to Plateau's problem. In § 1, an Euler characteristic formula is developed which generalizes both the Riemann-Hurwitz equation from complex analysis and the usual Euler characteristic formula for covering maps. Section 2 is devoted to several technical lemmas while § 3 applies these lemmas to the case where M is the disc (with holes) and N is a compact orientable 2-manifold. It is shown that for the existence of such a map there is a lower limit depending upon the genus of N and on the number of holes of M. The singularities of these maps have been characterized by Church and Timourian [2]. In the case n — 2 these maps are locally topologically equivalent to z —-> zd (d = 2, 3, ) and for n > 2, these maps are covering projections. For n — 2, the singularities and maps are special cases of branch points and branched immersions. In this case, the Euler characteristic formula represents a generalization of the classical Riemann-Hurwitz equation for light interior transformations on 2-manifolds. When f'^dN) = dM, the formula produced here reduces to the Riemann-Hurwitz equation. For n > 2, the maps are covering projections and the Euler characteristic formula reduces to the usual equation. The mappings considered in this paper are not, in general, interior transformations on the boundary of M. These considered here, however, appear to have more applications with regard to questions which have arisen from Plateau's problem. Heinz [6] and Gulliver, Osserman, and Royden [5] have shown that these maps are of much more interest than solely in the context of minimal surfaces. A very readable account of the Riemann-Hurwitz formula may be found in Whyburn [10]. The authors wish to express their gratitude to the referee for his many helpful suggestions.

0. Introduction. In the present paper, we use the counting techniques of the author's earlier work [5] to prove the converse of a result of R.C. Bose and S.S. Shrikhande [3] on geometric and pseudo-geometric graphs (q + l,q+H). Section 1 is devoted to preliminaries on strongly regular graphs and group divisible designs. We also give a brief description of the problem under consideration and a statement of our main result Theorem 1.1. Section 2 contains the proof of Theorem 1.1. We refer to [3] for the necessary background. Throughout this paper / will denote an identity matrix and / a square matrix of all ones. Alsoy and O will denote row vectors of all ones and zeros respectively. Finally, \S\ denotes the cardinality of the set S.

The paper is concerned with a generalized type of convexity, which is called /-simplicial convexity. The name is derived from the simplex with I vertices, an Z-simplicial convex set being the union of all (ΐ — l)-simplexes with vertices in another set, i varying between 1 and I. The basic space is a linear one. For convex sets the Z-order (which is a natural number associated to an ί-simplicial convex set) is a decreasing function of I. Several inequalities between Iand ά-orders are established. In doing this the case of a convex set and that of a non convex set are distinguished. Besides these inequalities, the main result of the paper is the proof of non monotonicity of the border, given by an example in a 34-dimensional linear space.

This paper treats extension and retraction properties in the category *$/9 of compact metric spaces with periodic maps of a prime period p; the subspaces and maps in J^p are called equivariant subspaces and maps, respectively. The motivation of the paper is the following question: Let E be a Euclidean space and α: E X E-> E X E be the involution (x, y) -> (y, x), i.e., the symmetry with respect to the diagonal. Suppose that Z is a symmetric (i.e., equivariant) closed subset of ExE which is an absolute retract; that is, Z is a retract of E X E. When does there exist a symmetric (i.e., equivariant) retraction Ex E-+ZΊ This is an extension problem in the category J2/'p. If X and Y are spaces in J£fp, A is a closed equivariant subspace of X and /: A -> Y is an equivariant map, then the existence of an extension of / does not, in general, imply the existence of an equivariant extension. It is shown, however, that if A contains all the fixed points of the periodic map and dim(X— A) < oo, then a condition for the existence of an extension is also sufficient for the existence of an equivariant extension. In particular, it follows that a finite dimensional space X in Sf 'p is an equivariant ANR (i.e., an absolute neighborhood retract in the category Sf v) if and only if it is an ANR and the fixed point set of the periodic map on X is an ANR. Generally speaking, the paper deals with the question of symmetry in extension and retraction problems.

Serre's theorem on projective modules says roughly that if a projective R module is big enough it can map onto R. Forster and Swan discuss how big a free module is needed to map onto a given finitely generated module. This note examines a common generalization of these results and extends Swan's technique. This paper follows Swan [5]. The reader is urged to refer to Swan for a more complete exposition of some of the ideas. The author is also in debt to Professor Kaplansky whose unpublished exposition of Swan's result [2] isolated one of the ideas for this paper. Throughout the paper R will be a commutative ring with identity whose maximal ideal spectrum is a noetherian space and A is an R algebra which is a finitely generated R module. Following Swan we define J-Spec(iϋ) to be the set of all prime ideals of R which are intersection of maximal ideas with topology the subspace topology inherited from Spec(i2). If M is a finitely generated R module, then for each p e J-Spec(jβ) we define b(p, M) = 0 if Mp = 0 and b(p, M) = r + d where r = ά\m{RιP)v (M/pM)p and d = dim J-Spec(R/p) otherwise. We also call an element x e Mp basic if it will serve as part of a set of generators with the minimal number of elements, i.e., if Mp/Rpx requires fewer generators than Mp. THEOREM 1. R a commutative ring with JSpec(R) a noetherian

This paper is about maps of compact 3-manifolds which map the boundary of the domain (possibly nonhomeomorphically) into the boundary of the range. F. Waldhausen has shown that such a map between compact, orientable, irreducible 3-manifolds with nonempty, incompressible boundary is homotopic to a homeomorphism if and only if the map induces an isomorphism at the fundamental group level. The main theorem of this paper states that the above theorem remains valid if the assumption of incompressibl e boundary is dropped. A study of disk sums of bounded 3-manifolds will be required in order to prove the above-mentioned theorem. This investigation involves theorems about disk sums of bounded 3-manifolds analogous to the classical Eneser theorem for closed 3-manifolds. The reader may wish to consult [11] for a proof of Waldhausen's theorem mentioned above and [4] and [7] for variations of Waldhausen's theorem related to the theorems proved in this paper. All spaces and maps in this paper are assumed to belong to the precise linear category, and each subspace that we shall discuss is taken to be piecewise linearly embedded. If A is a subcomplex of the simplicial complex X, we use the notation U(A, X) to denote a regular neighborhood of A in a second derived subdivision of X. If X is a manifold, we use the notation bdX and intX to denote the boundary of X and the interior of X respectively. A 3-manifold M is said to be irreducible if each 2-sphere in M is the boundary of some 3-cell in M.

In this paper necessary and sufficient conditions are developed for certain classes of continuous functions f(X) = Y, where X and Y are arbitrary metric spaces, to have the property that dim K— dim f(K) for all closed K c X In particular it is shown that if /is closed and dim f(K) > dim K for some closed K c Xf then there exists a closed K' c X so that dim K' = 0 and dim f{Kf) > 0. These results are then used to show that if / is closed and finite to one so that the multiplicity function of / takes on at most k + 1 distinct values, then dim K ^ dim/(iΓ) S dim K + k for all closed KaX. The purpose of this paper is to investigate the relation of the dimension of a closed subset of the domain with the dimension of its image in the range under various classes of continuous functions. In the first part of the paper we investigate this relation for closed subsets of the domain which have dimension zero. Using these results we characterize the property of being dimension preserving on closed subsets for a large class of mappings. In the second part of the paper we then show several important types of mappings to be dimension preserving on closed subsets. In the last section we generalize a result of Hurewicz [3]. The results of this paper are related to those of a number of investigators among whom are: Alexandroff [1], R. Hodel [2], K. Nagami [7], J. Nagata [8, pp. 68-73], J. H. Roberts [9], J. Suzuki [10], and R. F. Williams [11], As indicated in the title, the setting for our study is the class of metric spaces. Notation-. Throughout the paper X and Y denote metric spaces and / a continuous function from X onto Y. By dim X is meant the

This paper analyzes a multimove infinite game with linear payoff function. The game had its origin in the consideration of a military problem, but is presented here solely for its mathematical interest. It is symmetric in every respect except that the initial conditions of the two players are different. On each move, each player allocates his resources to tasks that might be described roughly as attacking, defending, and scoring. His resources for the next move are diminished by the amount that his opponent's attack exceeds his own defense, while his score cumulates from move to move. The value of the game and the optimal strategies for the players are rigorously derived in the present paper. It is shown that one player has a pure optimal strategy and the other player must randomize.

The inequality by Polya and Schiffer considered in this paper is concerned with the sums of the n first reciprocal eigenvalues of the problem Au + Tui = 0 in G, u = 0 on SG. First we extend this inequality to the problem of an inhomogeneous membrane Au + Apu = 0 in G, u = 0 on SG. Then we prove a sharper form of it for a class of homogeneous membranes with partially free boundary. The proofs are based on a variational characterization for the eigenvalues and use conformal mapping and transplantation arguments. 'The work was supported by NSF Grant GU-2056. EXTENSIONS OF AN INEQUALITY BY POLYA AND SCHIFFER FOR VIBRATING MEMBRANES Catherine Bandle INTRODUCTION. The inequality by Polya and Schiffer considered in this paper is concerned with the eigenvalue problem Atp + A . The Gaussian curvature has the form K =(~A In p)/2p r &2 S2 £_ = — 5 — + — 7 . We shall assume that the in equality L Z Bx ay J " K 0. The a a conformal radius of the point a with respect to G is then defined as r (G) = 1/f T (a) [9, p. 16] . We set a a (2) R (G) = < a. (G) if Ko ^ 0 \/p(a) 'ra(G) if K Q = 0 Example; If G is a circle with the radius r 3 the center in the origin and p(z) = g(z), then R (G) = r . w (z) = R (G)f (z) maps G onto the circle {w; Iwl < R (G)}. a a a. a and z (w) denotes its inverse. We shall denote the circle a {w; |wI < e} by C . R a( ) h a s bean chosen in such a way that (3) Jj g(w)dudv = JJ p(z)dxdy -f o(c). c Since JJ g(w)dudv = < ^47rcc + o(€ ) if K Q £ O € W e 2 if

It is shown in this paper that total biorthogonal systems have the same cone if and only if they differ at most by rearrangement and by positive scalar multiplication. A connection is demonstrated between this result and work done by R. E. Fullerton in which he characterized the existence of an unconditional basis in terms of the existence of certain type cones. The paper is concluded by generalizing the first result to the situation in which two biorthogonal systems have cones which induce order isomorphic orderings. l Definitions and notations* In this paper we will assume that

Subdirectly irreducible idempotent semigroups were characterized in [3], and in that paper, their connection with the various equational classes of idempotent semigroups was discussed. All these results are in terms of identities, so that examples of subdirectly irreducibles in the equational classes are explicitly known only for small classes. It is easy to show from general considerations (see the last section of the present paper) that every proper equational subclass of the class of idempotent semigroups is generated (as an equational class) by one or two subdirectly irreducibles. In this paper we give an example of a subdirectly irreducible for each join irreducible equational class of idempotent semigroups, which generates the class. This list, together with known results, gives explicit examples of one or two finite subdirectly irreducibles which generate the various equational classes.

o ^ b, satisfying the conditions of Definition 1.2. This paper is concerned primarily with those linear operators, the P-operators, which are abstractions from that class of linear physical systems whose output signals at a given time do not depend on their input signals at a later time; and with a sub-family of the P-operators, the Pi-operators which include all stationary linear_ operators. The Poperators are the Volterra operators on QL. Necessary conditions and sufficient conditions for a P-operator to have an inverse which is a P-operator are found; and a necessary and sufficient condition for a Pi-operator to have an inverse which is a P-operator is given in Theorem 3.1. In addition it is shown that if Sf is a Pi-operator and JS^"1 is a P-operator then JS^"1 may be written as the product of two operators whose generating functions may be found by successive approximation techniques. An analogue of Lane's inversion theorem for stationary operators on QCOL is found as a special case of these results. In [1] subspaces of the space of functions which are quasicontinuous on an interval [α, b] for which every linear operator £f may be written as a σ-mean Stieltjes integral of the form J*ff(s) = f(t)dL(t, s) are investigated. In this paper we will be concerned with one such subspace, Q£, and with a class of linear operators on QLi the P-operators, which are essentially the abstractions from that class of linear physical systems whose output signals at a given time do not depend on their input signals at a later time. In particular we shall be concerned with determining conditions which will guarantee that a P-operator has an inverse which is a P-operator. In §2 some of the basic properties of P-operators are developed and in § 3 a subfamily of these operators, the Px-operators, are introduced. The Pj-operators have the property that if a PΓoperator, 3ίΓ, has an inverse which is a P-operator then the generating function for may be determined by successive approximation techniques. In

1. Introduction^ We consider here a facet of the problem of justifying the methods of the operational calculus and in particular the use of the "Dirac Delta Function". L. Schwartz's "Theorie des Distributions* ' [6] is the most complete exposition to date on generalized functions but the operational calculus as such is largely omitted. B. Van Der Pol [8] discusses the latter but not in the context of distributions . Ketchum and Aboudi [4] suggested using unilateral Laplace Transforms to construct a link between Schwartz's theory and the operational calculus. This paper will enlarge on the latter suggestion. Two principal results are obtained. An imbedding space is constructed and a comparison between the topologies is made. Let S denote the strip σx < R(z) < σ2, in the complex plane. Consider the one parameter family of functions {ezt}, where the parameter z ranges over S and — oo < t < oo. This family is not a linear space but each member possesses derivatives of all orders. In a manner analogous to Schwartz we define an L5-Distribution to be an analytic complexvalued functional on the above family of functions, where by analytic we mean with respect to the parameter z. If a is any complex scalar and F, σ are two such functionals then we require that F ezt + σ ezt — (F+σ)-σzt, and (aF) ezt = F-{aezt). The latter property then allows us to define the derivative in a manner similar to that of Schwartz, that is Ff -ezt = F-(e zt)' = F - zezt = zF-e zt. It also follows that the Laplace Transform supplies an integral representation of some of the functionals. The other L?-Distributions define generalized functions for similar integral representations. That is, each function analytic for z e S has for its values, the values of an L5-Distribution acting on a function ezt and the L9-Distribution has an integral representation utilizing the symbolic inverse Laplace Transform of the analytic function. In most of this paper we deal only with analytic functions whose inverse transforms exist but the definitions and theorems will be stated without this restriction where possible. Following a practice used by other authors, we will call the inverse Laplace Transform, symbolic or not, an L 5Distribution rather than the functional. Because of the relation between the functional and an analytic function we concentrate on the latter and utilize the already known properties of such functions. By emphasizing the integral representations rather than the functionals we utilize the

In this paper we canonically represent the isomorphism classes of all rings whose additive group is a direct sum of two infinite cyclic groups by a system of 4 by 2 matrices whose elements are rational integers. It is then shown how the canonical forms can be used to solve other problems relating to these rings. The results obtained are (1) that any integral domain in this class of rings is isomorphic to a quadratic extension of a subring of the integers, (2) the complete survey of rings in the class under study which are decomposable as a direct sum, and (3) the complete survey of rings in this class which are decomposable as an ordered product which is not a direct sum. The paper concludes with a description of other problems which can be solved by means of the canonical matrices using routine calculations.

{y'x)dy. Then if A: JEk is an even integer or if k = 3, the following result is established: limβ_>oo IB(x) = /ί(x) almost everywhere. In [5] V. L. Shapiro proved that the conjugate Pourier-Stieltjes integral of a finite Borel measure μ in the plane E2, taken with respect to a Calderόn-Zygmund kernel K(x) in Lip a, 1/2 < a < 1, is almost everywhere Abel summable to the principal-valued convolution K*μ. The purpose of this paper is to extend this result to E3 and to even-dimensio nal Ek for K{x) in Lip a, 0 < a < 1. The first author will obtain the corresponding result for the odd-dimensional cases Jc = 2s + 1, s ^ 2, in a paper to appear, by the use of special functions. Also, the results of the present paper should be compared with Theorem 2 of [6, p. 44].

In an earlier paper (B. J. Gardner, Pacific J. Math., 33 (1970), 109-116) the torsion classes of abelian groups which are closed under pure subgroups were characterized, and §§ 3-6 of the present paper are devoted to generalizations of results appearing there. If & is a homomorphically closed class of objects in an abelian category, a subobject A of an object B is called ^-pure if it is a direct summand of every intermediate subobject X for which XIA e . (This terminology is due to C. P. Walker). In particular, ^ may be a torsion class. The following question is investigated: If and Ήf are torsion classes of abelian groups, when is closed under ^f-pure subgroups? Although ordinary purity is not ^/-purity for any torsion class %f, a torsion class ^7~~ is closed under pure subgroups if and only if it is closed under ^Hrpure subgroups, where ^l is the class of all torsion groups. In §5, for an arbitrary torsion theory (*g/, ^ ) a rank function (^/-rank) is defined for nonzero groups in ^ . It is shown that every torsion class closed under ^/-pure subgroups is determined by its intersection with ^ and the groups of ^/-rank 1 it contains. When ^ = , the groups with ^ rank 1 are the rational groups, so the earlier results for ordinary purity suggest that in general some refinement of the representation should he possible. A further special case of the general problem is also solved: Let X and Y be rational groups, T(X), T(Y) the smallest torsion classes containing them. If X is a subring of the rationale then T(X) is always closed under T(Γ)-pure subgroups; if not, the condition is satisfied if and only if X has a greater type than Y. § 7 is devoted to proving the following result: A torsion class is closed under countable direct products, i.e. direct products of countable sets of groups, if and only if it is determined by torsion-free groups.

The purpose of this paper is to classify centers according to isomorphisms. We define three types of isomorphism, and for two of these types give necessary and sufficient conditions for two centers to be isomorphic. We also give necessary and sufficient conditions for the third type of isomorphism to be equivalent to one of the other two. These isomorphisms are discussed in a more general situation by Taro Ura [7]. This paper was motivated by discussions with Taro Ura and Otomar Hajek. In our investigation we construct a section which generates a neighborhood of the center by using a theorem from the theory of fibre bundles. This section may be constructed directly, using the existence of a transversal through each noncritical point of the dynamical system. Much insight, which is otherwise lost, into the structure of a center is obtained from the fibre bundle approach. The concept of a transversal is essential in our investigation. The basic material on transversal theory in planar dynamical systems is found in [3].

The purpose of this paper is to construct a nonlinear semi-group determined by a given (multi-valued) nonlinear operator A in a Banach space X, and to investigate the differentiability of this semi-group. The semi-group treated in this paper is the semigroup {T(t); t ^ 0} of nonlinear operators in X such that for each τ > 0, {T(t); O^t^τ} is equi-Lipschitz continuous on bounded sets. In order that an operator A in X determine such a semi-group {T(t); £ Ξ> 0} on D(A) with (d/dt)T(t)xeAT(t)x for almost all t^O and xeD(A), it is required that X have a uniformly convex dual, A be dissipative in a local sense, I-λA, λ positive and small, satisfy a range condition and an injectiveness condition, and finally the family of operators (J—XA)~~n9 n = 1, 2, 3, be locally equi-bounded. Let X be a Banach space and S a subset of X, and let {T(t); t ^[0} be a family of nonlinear operators from S into itself satisfying the following conditions: (i) Γ(0) = /(the identity) and T(t + s) = T(t)T(s) on S for ί

In 1944 von Neumann and Morgenstern introduced a theory of solutions (stable sets) for ^-person games in characteristic function form. This paper describes an eight-person game in their model which has a unique solution that is nonconvex. Former results in solution theory had not indicated that the set of all solutions for a game should be of this nature. First, the essential definitions for an ^-person game will be stated. Then, a particular eight-person game is described. Finally, there is a brief discussion on how to construct additional games with unique and nonconvex solutions. The author [2] has subsequently used some variations of the techniques described in this paper to find a ten-person game which has no solution; thus providing a counterexample to the conjecture that every ^-person game has a solution in the sense of von Neumann and Morgenstern.

φ(s>> *} = 0, 0 Φ(s); x] = Ux\ a η, δ > 0, 0, are real parameters, /2O&) is a known function, Φ(s) is to be determined and (3) m{h(x); s} = H(s), Tl^His); x) = h(x) , denote the Mellin transform of h(x) and its inversion formula respectively. The above equations are an extension of the dual integral equations solved in a recent paper by Erdelyi [2] by means of a systematic application of the Erdelyi-Kober operators of fractional integration [4]. Using the properties of some slightly extended forms of the Erdelyi-Kober operators we show, in a purely formal manner, that the solution of the triple integral equations can be expressed in terms of the solution of a Fredholm integral equation of the second kind. Srivastav and Parihar [5] have solved a very special case of the equations by a completely different method from that used in this paper. The method of solution employed here will be seen to follow closely that used by Cooke [1] to obtain the solution to some triple integral equations involving Bessel functions; indeed Cooke's equations may be regarded as a special case of equations (1) and (2) and it is shown that a solution of his equations can be readily obtained from that presented in this paper.

In an earlier paper [2]; the author used equivalence classes of analytic functions to construct an imbedding space for Schwartz Distributions. The mechanism for constructing the mapping was the bilateral Laplace Transform, in this way the traditional approach to operational calculus was preserved. In that paper a topology was imposed on the imbedding space from the space of analytic functions. We now obtain some additional results about the possible topologies defined on the space of analytic functions.

In this paper the global dimension of any complete, wellpowered abelian category with injective envelopes in calculated relative to the torsion theory of A. W. Goldie and is found to be always one or zero. The rings R such that the left module category n^^ has global dimension zero are precisely those such that every module having zero singular submodule is injective. These rings are characterized as being of the form !Γ©S (ring direct sum) where T is a ring having essential singular ideal and S is semi-simple with minimum condition. The rings with essential singular ideal are precisely those which are torsion as left modules over themselves. In a recent paper [3] the right derived functors of a torsion subfunctor of the identity were calculated for an abelian category £T sufficiently like the category R^f

Let k, A and B be commutative rings such that A and B are ^-algebras. In this paper it is shown that Ω(kq\A(g)kB), the module of high order differentials of A (x)A B can be expressed by making use of 42? }(A) and ΩkJ)(B). On the other hand let K/k be a finite purely inseparable field extension. Sandra Z. Keith has given a criterion for a /b-linear mapping of K into itself to be a high order derivation of K\k. The representation of Ω(kq)(A®kB) is used to show that Keith's result is valid for larger class of algebras.

This paper is concerned with defining Lipschitz spaces on Σn-U the surface of the unit sphere in R . The importance of this example is that Σn-± is not a group but a symmetric space. One begins with functions in Lp(Σn-i)9 1 p ^oo. Σn^ is a symmetric space and is related in a natural way to the rotation group SO(w). One can then use the group SO(n) to define first and second differences for functions in Lp(Σn-^. Such a function is the boundary value of its Poisson integral. This enables one to work with functions which are harmonic. Differences can then be replaced by derivatives.

In this paper it is shown that, with suitable hypotheses on the base field, any element of generic trace zero in an octonion algebra is a commutator and an associator, and any element of generic trace zero in a simple Jordan algebra is an associator.

Let X be a complex linear space endowed with a semi-inner product [ , ]. An operator A on X will be calledHermitian if [Ax, x] is real for all x 6 X; A is said to beadjoint abelian if [Ax, y] = [x, Ay] for all x and yeX. Sinceevery Banach space may be given a semi-inner product (notnecessarily unique) which is compatible with the norm, it ispossible to study such operators on general Banach spaces.This paper characterizes Hermitian and adjoint abelian opera-tors on certain Banach spaces which decompose as a directsum of Hubert spaces. In particular, the Hermitian operatorsare shown to have operator matrix representations which arediagonal, with the operators on the diagonal being Hermitianoperators on the appropriate Hubert space. The class of spacesstudied includes those Banach spaces with hyperorthogonalSchauder bases.

This paper contains three theorems on linear stochastic differential equations, where the differential equations are given in terms of McShane's first and second order related integrals. The first, which is modelled on the classical Picard Theorem, concerns the existence of solutions, the second gives boundedness of their moments, and the third provides them with adjoints. The Adjoint Theorem has the interesting property that its formulation requires the second order integral even when the original differential equation involves only the more standard first order integral.