Search

In a recent paperf entitled A note on the linear fractional group, Todd obtained an abstract definition for the group LF(2, 2) in terms of n+2 generators. Apparently he gave no consideration to the question of the independence of the defining relations, for they can be considerably simplified. First, in view of the condition RSi = Si+iR (which is the same as i£S0^~* = Si), the three generators U, R and S0 are sufficient to generate the entire group. If we give a definition in teims of these three generators alone, the relations

In [6] Higman and N eumann showed that the Frattini subgroup @(A * B) of the ordinary free product of nontrivial groups A and B is trivial and they asked (implicitly) whether or not the Frattini subgroup of a generalized free product G = A *H B of groups A, B with amalgamated subgroup H is always to be found inside H. Whittemore [16] took the first steps toward answering this problem. Subsequently her results have been fairly substantially generalized in a series of papers (e.g., [I, 2, 4, 151). This paper first offers further evidence to support the conjecture that Q(G) < H in all cases and then turns to the problem of the location of the Frattini subgroup of HNN groups. One notable corollary of this latter result is Corollary 6.2. As yet, however, the authors have been unable to see if the results they have obtained are of any value in trying to settle the problem of Neuwirth as to whether the Frattini subgroup of a knot group is trivial1 In Section 2 we first show that the two results which have to date been central in the theory are both corollaries of another rather artifical-looking theorem (Theorem 2.1) but the full hypothesis of the statement of Theorem 2.1 is used in proving, in Section 3, that if G = A *H B and if H has the maximum condition on subgroups then D(G) < H. This result itself depends on the result, found in Section 4 that if A, B are countable and His an arbitrary (countable) group

0. Introduction. Let M be a differentiable manifold of class C°°. By a Morse function / on M, we mean a differentiable function / on M having only non-degenerate critical points. A well-known topological result of Reeb states that if M* is compact and there is a Morse function / on M having exactly 2 critical points, then M is homeomorphic to an ?^-sphere, S (see, for example, [3], p. 25). In a recent paper, [4], Nomizu and Rodriguez found a geometric characterization of a Euclidean ^-sphere SaR in terms of the critical point behavior of a certain class of functions Lp, p e R , on M. In that case, if p e R, x e M, then Lp(x) = (d(x, p))\ where d is the Euclidean distance function. Nomizu and Rodriguez proved that if M {n ^ 2) is a connected, complete Riemannian manifold isometrically immersed in R such that every Morse function of the form Lp, p e R , has index 0 or n at any of its critical points, then M is embedded as a Euclidean subspace, R, or a Euclidean ^-sphere, S. This result includes the following: if M is compact such that every Morse function of the form Lp has exactly 2 critical points, then M = S. In this paper, we prove results analogous to those of Nomizu and Rodriguez for a submanifold M of hyperbolic space, H, the spaceform of constant sectional curvature —1. For p € H, x e M, we define the function Lp(x) to be the distance in H from p to x. We then define the concept of a focal point of (M, x) and prove an Index Theorem for Lp which states that the index of Lp at a non-degenerate critical point x is equal to the number of focal points of {M, x) on the geodesic in H from x to p. In section 2, we prove that a metric sphere S c H can be characterized by the condition that every Morse function of the form Lp, p e H, has exactly 2 critical points. In section 3, we give an example which shows that a result analo

In this paper it is shown that every strongly locally homogeneous Hausdorff continuum is locally connected. It is known that every connected representable space is homogeneous and that every locally connected representable space is strongly locally homogeneous; this paper investigates the problem of whether or not every connected representable space is locally connected. The discovery that homogeneity did not characterize the circle among planar continua motivated other classifications of continua in terms of the actions of their homeomorphism groups, and the pseudoarc remains the most tortuous proving grounds for testing such topological properties. Those homogeneity-like properties that are not possessed by the pseudo-arc appear to be closely related to local connectedness. Such properties include strong local homogeneity, representability, (strong) 2-homogeneity and countable dense homogeneity. In this paper it is shown that every strongly locally homogeneous Hausdorff continuum is locally connected. It is known that every connected representable space is homogeneous and that every locally connected representable space is strongly locally homogeneous; this paper investigates the problem of whether or not every connected representable space is locally connected. It follows from results of Ben Fitzpatrick, Jr. and Ralph Bennett that every connected locally compact separable metric representable space is locally connected. Arguments of Fitzpatrick and de Groot are used to show that if (X, τ) is a representable connected complete metric space such that τ does not have a base of totally disconnected sets, then (X, τ) is locally connected (and hence arcwise connected and strongly locally homogeneous). The authors do not know of even a homogeneous connected complete metric space with a base of totally disconnected sets; however, there exists a separable connected metric topology that has such a base. Let (X, τ) be a topological space. We let H(X) denote the group of all homeomorphisms from the space (X, τ) onto itself and let i denote the identity of H(X). If A c X, then A' = {he H(X): h\A = i I A) and if G is a subgroup of H(X), then G' = {x e X: g(x) = x for each geG}. If GaH{X) and AcX, then G(A) = {g(a):geG and aeA}. We write G(x) in room of G({x}). Throughout this paper all spaces are assumed to be Hausdorff.

The manifold in this paper is assumed to be connected differentiable of class C∞. Let Dr(M) and Ӿr(M) be the set of all diffeomorphisms and vector fields of class Cr on a manifold M with Whitney Cr topology, respectively. In [2], the concept of weak stability is defined. The definition is equivalent to the following ((2.1) of this paper); f∈Dr(M) or X ∈ Ӿr(M) is weakly (allowably) stable if and only if there is a neighborhood U of f or X in Dr(M) or Ӿr(M) such that for any (a suitable) g or Y ∈ U the set of all elements topologically equivalent to g or Y is dense in U, respectively. Here, f, g ∈ Dr(M) are said to be topologically equivalent if they are topologically conjugate and X, Y ∈ Ӿr(M) are said to be topologically equivalent if there is a homeomorphism mapping any trajectory of X onto a trajectory of Y preserving the orientations of the trajectories. Similarly, weak Ω-stability is defined for f and X.

A conjecture concerning characterization of blocks of representations of symmetric groups by means of Young diagrams, given formerly by one of the writers, was affirmed recently by R. Brauer and G. de B. Robinson jointly. Namely, it was proved in Brauer’s paper, relying on his general, and profound, theory of blocks in finite groups, that the number of p-blocks in a symmetric group of degree n, is exactly equal to the number of p-cores, or p-kernels in the terminology of NI, with n- Ip (l= 0, 1, 2,…) nodes, and that certain statements designated as (A), (B) and (C) would secure that two irreducible representations with a same p-core belong to a same block (whence the conjecture itself when combined with the estimate of the number of blocks). Then these statements (A), (B), (C) were proved actually in Robinson’s paper.

Suppose P(x, D) is a linear partial differential operator on an open set ~ contained in R ~ and that A is a closed subset of ~. Given a class ~(~) of distributions on ~, the set A is said to be removable for ~(~) if e ach /E ~(~), which satisfies P(x, D ) / = 0 in ~ A , also satisfies P(x, D)/= 0 in ~. The problem considered in this paper is the following. Given a class ~(~) of distributions on ~, what restriction on the size of A will ensure that A is removable for ~(~). We obtain results for Lroc (~) (p ~< ~ ) , C(~), and Lipa (~). The first result of this kind was the Riemann removable singularity theorem: if a function / is holomorphic in the punctured unit disk a n d / ( z ) = o ( H -1) as z approaches zero, then / is holomorphic in the whole disk. Bochner [1] generalized Riemann's result by considering the class ~(~) of functions f on ~ such that ](x)=o(d(x, A) -q) uniformly for x in compact subsets of ~, and giving a condition on the size of A which insures tha t A is removable for ~(~) (Theorem 2.5 below). Bochner's theorem is remarkable in that the condition on the size of A only depends on the order of the operator P(x, D). The theorem applies, therefore, to systems of differential operators, such as exterior differentiation in R n and ~ (the Cauchy-Riemann operator) in C n. The same can be said for the other results in this paper. The proof of Bochner's theorem provided the motivation for our results. I t is interesting to note tha t a very general result (Corollary 2.4) f o r / ~ (~) (due to Li t tman [7]) is an easy corollary of Bochner's work. Here the condition on the singular set A is expressed in terms of Minkowski content. In section 4 the case of Ll~oc(~) is studied again, and results in section 2 are improved by replacing Minkowski content with Hausdorff measure. In addition, the cases C(~)

In the first part of this paper the structure of w-aposyndetic continua is studied. In particular, those continua which are w-aposyndetic but fail to be in + l)-aposyndetic are investigated. Unicoherence is shown to be a sufficient condition for an %-aposyndetic continuum to be in + l)-aposyndetic. In the final portion of the paper a stronger form of unicoherence is defined. As a point-wise property, aposyndesis and connected im kleinen are shown to be equivalent in continua with this property. Throughout this paper a continuum is a compact connected metric space and M will denote a continuum. If N is a subcontinuum of M, the interior of N in M will be denoted by int N. Suppose pe M and F is a closed subset of M such that p ί F. M is aposyndetic at p with respect to F if there is a subcontinuum N of M such that p e int N c N c M — F. Let n be a positive integer. If M is aposyndetic at p with respect to each subset of M consisting of n points, then M is n-aposyndetic at p. M is n-aposyndetic if it is ^-aposyndetic at each point. By convention if M is 1-aposyndetic then M is said to be aposyndetic. For other terms not defined herein, see [3], [4] and [6]. LEMMA 1. Suppose M is n-aposyndetic, peM,F is a subset of M — {p} consisting of n + 1 points, and M is not aposyndetic at p with respect to F. If F 1 and F2 are disjoint nonempty subsets of F such that F = Fι U F21 there exist subcontinua H and K such that Fιa H — K, F2 c K- H,pemt(HΓ\K), and M= H\JK.

Let Z = {O, 1, 2} and define two binary operations /\ and V on Z as follows: 0/\ 1 = 0,0 V 1 = 1, 1 /\ 2 = 1,1 V 2 = 2,2 A 0 = 2, 2 V 0 = 2, both operations are idempotent and commutative. This paper deals with the equational class Z generated by the algebra . The class Z contains the class of all distributive lattices and Z is a subclass of the class of weakly associative lattices (trellis, T·lattice) in the sense of E. Fried and H. Skala. The purpose of this paper is to prove that Z shares the most important properties of the class of distributive lattices.

In this paper a proof of the Jordan curve theorem will be presented. Some familiarity with the basic notions of nonstandard analysis is assumed. The rest of the paper is selfcontained except for some standard theorems about polygons. The theorem will be proved in what ought to be a natural way: by approximation by polygons. This method is not usually found in the standard proofs since the approximating sequence of polygons is often unwieldly. But by using nonstandard analysis, one can approximate a Jordan curve by a single polygon that is infinitesimal ly close to the curve. This allows types of reasoning which are extremely difficult and unnatural on sequences of polygons. Preliminaries. The basic concepts of nonstandard analysis and some acquaintance with polygons are assumed. Some basic definitions and theorems of point set topology are also assumed. Throughout this paper the following notations and conventions will be used: (1) All discussion, unless otherwise stated, is assumed to be about a nonstandard model of the Euclidean plane. 'Otherwise stated" will often mean that the notion or concept will be prefaced by the word "standard". (2) A standard concept and its extension will be denoted by the same symbol. If it is necessary to distinguish between them, reference to the model in which they are to be interpreted will be made. (3) If A and B are sets of points and x is a point, then \x, A\ will denote the distance from x to A and \B, A\ = mίxeB\x, A\. (Thus if A Π B Φ 0 then \A, B\ = 0.) \x,y\ will denote the distance from the point x to the point y. (4) / will denote a fixed continuous function on [0,1] into the Euclidean plane with the property that x < y and f(x) — fyy) if and only if x — 0 and y = 1. C will denote the range of /. ( 5 ) x ~ y will mean that the distance from x to y is infinitesimal. If x is near-standard then °χ will denote the standard y such that x ~ y. (6) \i x and y are points then xy will denote the ordered, closed line segment that begins at x and ends at y. (7) If x and y are points then intv (x, y) is the set of all points z of xy such that z Φ x and z Φ y.

The structure of a stationary Gaussian process near a local maximum with a prescribed height u has been explored in several papers by the present author, see [5]–[7], which include results for moderate u as well as for u→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ Rn}, with mean zero and the covariance function r(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type. ¶ In Section 1 it is shown that if ξ has a local maximum with height u at 0 then ξ(t) can be expressed as $\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$ ¶ WhereA(t) and b(t) are certain functions, θu is a random vector, and Δ(t) is a non-homogeneous Gaussian field. Actually ξu(t) is the old process ξ(t) conditioned in the horizontal window sense to have a local maximum with height u for t=0; see [4] for terminology. ¶ In Section 2 we examine the process ξu(t) as u→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial in t1…, tn with random coefficients. This result is quite analogous with the one-dimensional case. ¶ In Section 3 we study the locations of the local minima of ξu(t) as u → ∞. In the non-isotropic case r(t) may have a local minimum at some point t0. Then it is shown in 3.2 that ξu(t) will have a local minimum at some point τu near t0, and that τu-t0 after a normalization is asymptotically n-variate normal as u→∞. This is in accordance with the one-dimensional case.

1. I n t r o d u c t i o n . Recently, E. MAKAI 1 determined the best-possible constant in a theorem of TURAN. The theorem is quoted as TU~AN'S first main theorem, since it appeared in his book 2 as the solution of his "first problem". It will be cited in Sec. 4 below. This best-possible constant was obtained independently (though somewhat later) by the present author, whose proof appeared to be considerably shorter that the one of Mr. MAKAI. The matter now seems to have been rounded off so nicely that it is worth while to present a self-contained account of it. The arrangement is as follows. In Sec. 3 we present a new proof of an elegant theorem, due to MAKAI. ~ Sec. 4 gives the proof of the main theorem, and Sec. 6 the new proof of the fact that the constant is best-possible. Finally, Sec. 7 gives a discussion of the cases where the equality sign occurs. It should be remarked that Sec. 4 was implicitly contained in the paper of I. DANCS, 4 which inspired both Mr. MAKAI and the present author.

In a recent paper, [6], Nomizu and Rodriguez found a geometric characterization of umbilical submanifolds Mn ⊂ Rn+p in terms of the critical point behavior of a certain class of functions Lp, p ⊂ Rn+p, on Mn. In that case, if p ⊂ Rn+p, x ⊂ Mn, then Lp(x) = (d(x,p))2, where d is the Euclidean distance function.

A group G is called an R-group if for every pair of elements x and y, and every natural number nf it follows from x n = y that x = y. In other words, G is an ϋϊ-group if G has not more than one solution for every equation x = a. If G is an iϋ-group, G is locally infinite. The converse, however, need not be true even if G is restricted to the group of a knot in S. For example, let G be the group of K(m, ri), the torus knot of type (m, n). G has a presentation G = (α, b: a = b). Then the equation x = a has infinitely many distinct solutions, x = a, (ba)a(ba)"ιf (ba) a(ba)~, This observation gives immediately a negative answer to Problem N in [3]. Neuwirth asks if a knot group can be ordered. In fact, the group of K{m, n)(\ m |, | n | ^ 2) cannot be ordered, since an ordered group is always an i?-group. Therefore, Problem N now leads slightly weaker problems: Can a knot group other than torus knot groups be ordered? Or, is a knot group other than torus knot groups an iZ-group? The purpose of this paper is to give a sufficient condition for the group of a fibred knot to be an iϋ-group. (See Theorem 2.) Using this condition, we can prove, for example, that the group of the figure eight knot is an i2-group. (See Proposition 3 or Proposition 5.)

1. This paper announces results concerning the cohomology groups H*(N, N) where A*" is in a certain class of finite-dimensional nilpotent Lie algebras over a field k and T is an abelian Lie algebra faithfully represented as a maximal diagonalizable algebra of derivations of N; we shall refer to such an iV as a T-algebra. The additional hypotheses to be placed on the pair N, T are inspired by the case when J is a Cartan subalgebra and T+N=B is a Borel subalgebra of a complex semisimple Lie algebra. In that case Kostant has shown [2] that H%N, N)=0 for i^.2 and the authors applied this result in [3] to conclude that H*(B, B) = 0. (A similar argument shows H*(P9 P ) = 0 for P parabolic.) Here we are concerned with the relations between the vanishing ofH%N, N), especially for i=2 , and the structure of the algebras N. Let W denote the set of weights of T in N. If dirn(r)=dim(7\T/A)=m then the subset of W arising from the induced representation of T on N/N has precisely m elements, say {a1? • • • , am}. Every a e Wthen has a unique representation a = 2 ^a* with each c? a nonnegative integer and ct 0. For such an a we call the sum (in Z) 2 i the height of a and denote it by |a|. For a in W, denote by Aa the weight space for a in N. DEFINITION. A T-algebra is called positive if (i) dim(70=dim(A7iV), (ii) N is graded by the heights of the weights, i.e., if N(j) = Q)\al=j Na then [N(j)9N(k)]cN(j+k). REMARK. Condition (ii) is superfluous in characteristic 0. However, in characteristic p>0 it has such consequences as N=0 for r > ( / > l ) d i m ( r ) .

commutes. In this paper, a necessary and sufficient condition for the existence of a stabilizing homomorphism (hence isomorphism) between any two crossed product extensions is obtained. The result is applied to obtain a necessary and sufficient condition for the existence of an automorphism φ: E -> E making the diagram 0 >G >E >Π >1

The two which have received the greatest attention are theclassical (Ore) quotient ring and the maximal (Utumi) quotient ring.The classical quotient ring has a relatively straightforward description,but it is only defined for rings which satisfy the so-called Ore con-dition. In contrast the maximal quotient ring is less easy to describebut is defined for all rings. In both cases there are distinct notionsof left and right quotient rings and we will always consider leftquotient rings.For group rings the classical quotient ring has been studied byHerstein and Small [2], Passman [5, 6], M. Smith [7], and P. F. Smith[81, and the maximal quotient ring has been studied by Burgess [1].This paper investigates the relationship of the maximal quotientrings of group rings, subgroup rings, and the centers of group rings.The object is to obtain for the maximal quotient ring analogues oftheorems of Passman and M. Smith on the classical quotient ring.Their techniques are used for the group ring arguments while thequotient ring arguments reflect the formalism of the maximal quotientring.

In his fundamental paper on a posteriori probability,* Bayes considered a certain event M having an unknown probability p of its occurring in a single trial. In deriving his a posteriori formula he assumed that all values of p are equally likely, and he recommended this assumption for similar problems in which nothing is known concerning p. In the corollary to proposition 8 he derives the value r/n\ 1

The generalized Schoenflies theorem of M. Brown [2], [3] can be restated in the following way: If S~ is the equator of S, then any locally flat embedding ƒ : S~—>S can be extended to a homeomorphism Ft S-»S . The purpose of this paper is to show that, if n^ 5, the extension F can be constructed in a controlled manner; in particular, if ƒ:S~^»5 is close to the inclusion embedding, then F: S-*+S can be chosen to be close to the identity homeomorphism. Consequently if, ƒ, g :S'-^S are locally flat embeddings, n^ 5, and ƒ is close to g, then there is a homeomorphism H: S—^S which is close to the identity such that

In this paper we examine the relation between i?-sets, which are a purely set-theoretic concept, and various concepts associated with planar maps, for instance, four-colorings, five-colorings, Ήamiltonian circuits, and Petersen's theorem. Moreover, the introduction of the notion of a i?-set into graph theory enables us to ask questions which may be more tractable than the four-color conjecture and shed light on it. 1* Introduction and definitions* Let F be a family of sets. A set that meets every member of F and yet contains none of the members of F is called a i?-set for F. Observe that if B is a Bset for F, then so is its complement, (\JEeFE) — B. In fact, F has a £-set if and only if \JEeF E can be partitioned into two sets, A and B, such that neither A nor B contains a member of F. Observe that if F has a J5-set, and if G S F, then G has a 5-set. Also, if G C F, and every member of F contains some member of G, and if G has a i?-set, then F also has a B-set. (The notion of j?-set goes back to Bernstein, who used it in 1908 to deal with a topological question.) We shall be concerned with maps covering the surface of the sphere, S2. For the most part, we will assume that these maps are 3-regular, that is, each vertex has degree three. Each region of the map will be a topological cell. Two regions are adjacent if they share at least one edge. A sequence of distinct regions Ru R2, , Rn+1, n^>l such that Ri is adjacent to Ri+1, 1 ^ i ^ n is a path of regions. If we have Rn+1 = Rι, and Ru R2, •••!?» are still distinct, we call the path a region-cycle of length n (or n-r eg ion-cycle). A region-cycle consisting simply of the regions around a vertex we call a basic cycle. In a 3-regular map the basic cycles have length three. If the union of any two regions in a map is simply connected, then the regions bordering any given region form a region-cycle, which we call a face cycle. Its length is just the number of edges of the surrounded region. We shall not be interested in region-cycles of length two, unless their union is not simply connected. A region-cycle is odd if its length is odd; otherwise, it is even.

1. When a permutation group G on a set Ω is given, a transitive extension G of G is defined to be a transitive permutation group on the set Γ which is a union of Ω and a new point ∞ such that the stabilizer of ∞ in G1 is isomorphic to G as a permutation group on Ω. The purpose of this paper is to prove that many known simple groups which can be represented as doubly transitive groups admit no transitive extension. Precise statement is found in Theorem 2. For example, the simple groups discovered by Ree [5] do not admit transitive extensions. Theorem 2 includes also a recent result of D. R. Hughes [3] which states that the unitary group U3(q) q>2 does not admit a transitive extension. As an application we prove a recent theorem of H. Nagao [4], which generalizes a theorem of Wielandt on the non-existence of 8-transitive permutation groups not containing the alternating groups under Schreier’s conjecture.

In this paper the combinatorial properties of rigid plane skeletal structures are investigated. Those properties are found to be adequately described by a class of graphs.

1. A number of papers discussing the spectral decomposition and eigenfunction expansion for partial differential operators appeared in the last few years. Browder [l], [2], [3], [4], Gârding [5] and Mautner [12 ] proved the existence of an abstract eigenfunction expansion for elliptic partial differential operators. In 1953 A. Ya. Povzner [13] considered the detailed spectral decomposition of — A-\-q(x). This was completed by T. Ikebe [6] who used the theory of wave operators as developed by Kato [8] and Kuroda [10], [ll]. In this note we investigate an eigenfunction expansion for the operator P(D)+q(x) where P(D) is a linear homogeneous elliptic partial differential operator with constant coefficients. Detailed proofs of the results will appear elsewhere. 2. The Euclidean w-space will be denoted by Rn or Mn with ele­ ments x=(xu • - • , xn) or k=(ki> • • • , kn) respectively. ff(x)dx de­ notes integration with respect to Lebesgue measure. We set d — Dj = for 1 ^ j S nidxj Let P(x) be a homogeneous elliptic polynomial, i.e. P(x)jâqx| 2p where 2p is the order of P(x). Then P(D) =P(Dh • • • , Dn) is a linear homogeneous elliptic partial differential operator. All through this note we assume that 4p>n. It is well known that P(D) can be ex­ tended to a selfadjoint operator P(D) in Lz(Rn). Let g(x)GC2[„/2] with q{x) — 0{\x\~n~h) for some h>0. Then by Theorem 1 of [ll], P(D)+q(x) is a selfadjoint operator in L

Let I’ be a maximal order over a complete discrete rank one valuation ring R in a central simple algebra over the quotient field of R. The purpose of this paper is to determine necessary and sufficient conditions for I’ to be equivalent to a crossed product over a tamely ramified extension of R.It is a classical result that every central simple algebra over a field k is equivalent to a crossed product over a Galois extension of k. Furthermore, it has been proved by Auslander and Goldman in [2] that every central separable algebra over a local ring is equivalent to a crossed product over an unramified extension.

In the first part of this paper are considered real polynomialsP(ζ), ζ∈R n, complete and nondegenerate in the sense that there is a set of (even) multi-indices α j ,j=1,...,N, such that, for |ζ|>K, ζ real, $$cP(\xi ) \leqslant \sum {\xi ^{\alpha j} } \leqslant CP(\xi ).$$ (See V. P. Mihailov,Soviet Math. Dokl. 164 (1965), MR 32: 6047). It is then proved by an explicit computation, for every given even multi-index γ, that there are a real number θ>0 and an integerr, 0≤r

The concept of a “set of ρ-discs” for complex-valued functions defined in the open unit disc U was introduced in [4; 5, p. 265] and various results concerning the existence of such sets associated with spirals and Stolz angles were derived. These sets are non-Euclidean analogues to the classical cercles de remplissage of Milloux [7; 8]. In the present paper additional theorems about Stolz angle phenomena and the existence of ρ-discs in more general subsets of U are established and new proofs are indicated for some results of Seidel [9, Theorem 2 and Corollary 1].

1. Introduction. C-matrices appear in the literature at various places; for a survey, see [11]. Important for the construction of Hadamard matrices are the symmetric C-matrices, of order v = 2 (mod 4), and the skew C-matrices, of order v = 0 (mod 4). In § 2 of the present paper it is shown that there are essentially no other C-matrices. A more general class of matrices with zero diagonal is investigated, which contains the C-matrices and the matrices of (v, k, X)-systems on k and k + 1 in the sense of Bridges and Ryser [6]. Skew C-matrices are interpreted in § 3 as the adjacency matrices of a special class of tournaments, which we call strong tournaments. They generalize the tourna­ ments introduced by Szekeres [24] and by Reid and Brown [21]. In § 4 we introduce the notion of negacyclic C-matrices, analogous to the similar notion introduced by Berlekamp in the setting of coding theory (cf. [4, p. 211]). Eigenvalues of negacyclic matrices are characterized and standard forms are obtained. Negacyclic C-matrices are interpreted in § 5 as the matrices of a special class of the relative difference sets introduced by Butson [7]. Exploiting some results of Elliott and Butson [10], we obtain a "multiplier theorem" for negacyclic C-matrices, and adapting a result of [2], we show that any negacyclic C-matrix has a nontrivial multiplier. Necessary conditions for the existence of a negacyclic C-matrix of order v are obtained in § 6. The nonexistence of nega­ cyclic C-matrices of all orders v ^ 226, v ^ 1 + Ph, with p prime, has been verified. This leads to the conjecture that they do not exist, unless v = 1 + PkPaley [19] constructed C-matrices of all orders v = 1 + pk, p prime. In § 7 it is shown that every Paley matrix is equivalent to a negacyclic C-matrix, a fact

This paper continues the study of Hn(x)=Hn(x, q)= $$\Sigma _0^n \left[ {\begin{array}{*{20}c} n \\ r \\ \end{array} } \right]x^r$$ and related polynomials discussed bySzego [7]. In particular, it is shown that Hn(x) has various formal properties analogous to properties of theHermite polynomials.

This paper contains in part the author’s Ph.D. dissertation written under the supervision of Professor F. Bagemihl at the University of Wisconsin-Milwaukee. This research was supported by grants from the National Science Foundation and the University of Wisconsin Graduate School.

In this paper we shall study relations among the class B of the Bayes solutions in the strict sense, the class W of the Bayes solutions in the wide sense and the closure c(B), in a certain sense, of the class B. For abbreviation, we shall use the word “Bayes class” for the class B and “Wald class” for the class W.

The question whether a system of n — 1 real algebraic numbers (n = 2, 3, •) chosen from an algebraic field of degree not higher than n9 yields periodicity by Jacobi's Algorithm is still as open and challenging as hundred years ago. The present paper gives an affirmative answer to this problem in the following case: let K(w) be an algebraic number field generated by w = (Dn — d: m)ί/n, where m, n, d, D are natural numbers satisfying the conditions m^ 1, n έ 3, d | D, 1 ^ d ^ D/2(n — 1). Then n — 1 numbers can be chosen from K(w), so that their Jacobi Algorithm becomes purely periodic. The length of the period equals n2 (or n, if d = m = 1). This is the longest period of a periodic Jacobi Algorithm ever known. In three corollaries the following special cases are investigated

SHORTER NOTES The purpose of this department is to publish very short papers of an unusually elegant and polished character, for which there is nor- mally no other outlet. A PROOF OF THE NONRETRACTIBILITY OF A CELL ONTO ITS BOUNDARY MORRIS W. HIRSCH By appealing to the simplicial approximation theorem [2, p. 64] it suffices to prove that there is no simplicial retraction of a subdivi- sion of a closed ra-simplex E onto its boundary dE. Suppose/: £—>d£ is a simplicial retraction. Let a be the barycenter of an (n— l)-simplex A EdE. The point is this:/_1(a) is a compact one- dimensional manifold whose boundary is contained in dE. The com- ponent of /_1(a) containing a is thus a broken line segment with one endpoint at a; but the other endpoint cannot exist. It would have to be a point of dE different from a which maps onto a under /, contra- dicting the assumption that/| dE is the identity. The proof that/_1(o) has the stated properties is simple and classi- cal (cf. [3]). Any ra-simplex B mapping onto A has precisely two faces mapping onto A, so that BC\f^1(a) is the line segment in B joining the barycenters of the two faces. These line segments fit together to form a manifold whose boundary is in dE because every (n— 1)- simplex C of E is incident to either one or two ra-simplices, according to whether CEdE or not. The same proof applies if £ is a compact triangulated manifold with boundary dE. More generally, the proof works if £ is a finite ra-dimensional complex such that each (ra— 1)-simplex is a face of not more than two ra-simplices, and dE is the union of those (ra—1)- simplices incident to at most one w-simplex. In the case where £ is a compact differentiable manifold, one may use the differentiable approximation theorem in place of the sim- plicial one, and take a to be a regular value. This of course requires a theorem such as Brown's [l, Theorem 3.Ill], Dubovitski's Theorem 4], or Sard's [5] on the existence of regular values. References 1. A. B. Brown, Functional dependence, Trans. Amer. Math. Received by the editors October 8, 1962. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Soc. 38 (1935)

In this paper, the zeta functions on the real general linear group, formed relative to arbitrary positive definite zonal spherical functions, are defined and studied. The main result asserts that such functions satisfy a functional equation and can be meromorphical ly continued to the whole complex plane. 1* Introduction* Let R denote the field of real numbers. Let M = M(n, R) denote the w2-dimensional vector space over R comprised of all n x n matrices with real coefficients, G — GL(n, R) the subset of M consisting of those elements with nonzero determinant. As is known, G is a locally compact unimodular group whose Haar measure dG(x) (xeG) is given, up to a constant factor, by dσ(x) = \x\~ ndM(x) (\x\ = |det(α?)|), dM(x) being normalized Lebesgue measure on M. Let K= {xeGix-'x = 1}

This paper treats a class of stochastic processes called biadditive processes and gives a proof of a decomposition of their sample functions. Informally, a biadditive proces X(s, t) is a process indexed by two time parameters whose " increments " over disjoint rectangles are independent. The increments of such a process are the second differences X(s2, U) - X(8U U) - X(s2, U) + X(8U ti) where s± < s2 and U < t2. The decomposition theorem states that every centered biadditive process is the sum of four independent biadditive processes: one with jumps in both variables, two with jumps in one variable and continuous in probability in the other, and a fourth process which is jointly continuous in probability. This decomposition is similar to one for processes with independent increments and in the proofs of both results a major role is played by the theory of centralized sums of independent random variables. More formally, let P1 = {slf s2, , sn} and P2 = {tl9 t2, , tm) be two partitions of [0, sn] and [0, tm] respectively. Define Px x P2 to be the corresponding partition of [0, sn] x [0, tm] into rectangles whose vertices are the (si9 ί, )'s. Let ΔiS denote the increment

In this paper we prove that under certain conditions the sum and the product of two commuting spectral operators on a locally convex space are again spectral operators. We also obtain expressions for the spectral measures of such sum and product. The technique used is, in principle, similar to one employed by Pedersen [9]. Also, if X is a locally compact and σ-compact space and μ is a positive Radon measure on X, we consider the space ΩP(X, μ)(l

0. Introduction. This note announces some new methods in algebraic topology, based on the results of [1], [2] and [3]. The relative lifting problem [5, p. 415] is fundamental to that subject. It includes the extension problem, the retraction problem, the lifting problem, the section problem, the relative section problem and the computation of [X, Y] problem, among its particular cases. These problems (excluding the last one) are usually taken to concern just the existence of extensions, retractions, etc. ; we use the terms in the wider sense, to include both the existence question and the homotopy classification question, for the respective extensions, retractions, etc. We will : (i) prove that many cases of the above problems (including the computation of the absolute and relative cohomology groups of the total space of a fibration, and the computation of certain homotopy groups) are equivalent to "parallel problems", involving the restricted fibered mapping projection (pq;a); (ii) give some examples of solutions of parallel problems. Our solutions include one lifting problem and two extension problems. The extension results have some immediate consequences; they give new derivations for the exact cohomology sequences of Serre and Wang. Gysin's sequence is derived separately. Future papers will discuss these and other applications in detail. Our argument is valid in several convenient categories, including the category of i-spaces [4], [2], [7] and the category of quasi-topological spaces [6], [1], [2]. This category of I-spaces, for a definition see [2, p. 276], contains the usual category of Hausdorff fc-spaces (= compactly generated spaces = Kelley spaces) as a subcategory.

(unbounded spectrum, domain Φ X) in finite-dimens ional spaces which linear operators exhibit only in infinite-dime nsional spaces. We present an outline of the paper. In § 2 we define p(T) where p is a polynomial with coefficients in the field Φ involved in X. We prove that (pq)(T) = p(T)q(T), (poq)(T) = p{q{T)), and point out that sometimes (p + q){T) Φ p(T) + q(T), etc. In § 3 we turn to relations in dual pairs. In this situation, adjoints can be defined. We build an automorphism λ —> λ of Φ into the theory of dual pairs, so as not to exclude the Hubert space situation, which dual pairs are intended to imitate. (Thus the transpose is a special kind of adjoint.) Closedness is defined algebraically, but in a way compatible with the topological concept. Closure of T 7* and other algebraic properties of * are established. Finally, it is shown that if T is closed and its resolvent is not void then p(T) is also closed. Section 4 considers the self-dual case. We give a simple condition (4.3) always true in Hubert space, that T*T be self-adjoint, T being closed. In § 5 we give the spectral analysis of self-ad joint linear relations in Hubert space. In a 1:1 manner these correspond to the unitary operators, via the Cay ley transform. However, it can be shown directly that X is the direct sum of orthogonal subspaces Y, Z which reduce T (= T7*) giving in Za self-ad joint operator and in Fthe inverse of the zero-operator. 2 Linear relations* A relation T between members of a set X and members of a set Y is merely a subset of X x Y. For x e X, T(x) = {y (x, y) e T}. The domain of T consists of those x such that T(x) is not void. T is called single-valued if T(x) never contains more than one element. The range of T is the union of all T(x).

A homeomorphism h of En or Sn onto itself is stable if 3 homeomorphisms &i, hi, • • • , hm and nonvoid open sets Z7i, Ui, • • • , Um such that h = hmhm-i • • • hi and hi\ Ui = I for i = l , 2, • • • , m. All orientation preserving homeomorphisms of En or Sn are stable provided n = 1, 2, or 3. There is no example known in any dimension of an orientation preserving homeomorphism which is not stable. In fact, the conjecture that all orientation preserving homeomorphisms of En or 5 n are stable is equivalent to the annulus conjecture (see [3 ]). I t is known that any homeomorphism of £3 onto itself can be approximated by a piecewise linear one (see [2] or [o]). The purpose of this paper is to announce that if n ^ 7 and h is a stable homeomorphism of En or Sn onto itself, then h can be approximated by a piecewise linear homeomorphism, and also, in the case of En, by a diffeomorphism. The set of all homeomorphisms on En or Sn forms a group under composition and the subset of stable homeomorphisms forms a normal subgroup. The stable group on Sn is simple while the stable group on En is not. Due to this fact, there is a shorter proof in the case of Sn than in the case of Eny and it is this proof which will be outlined here. The author thanks John Stallings for his assistance.

An axiomatic setting for the theory of convexity is provided by taking an arbitrary set X and constructing a family ^ of subsets of X which is closed under intersections. The pair consisting of any ordered vector space and its family of convex subsets thus become the prototype for all such pairs (X, ^). In this connection, Levi proved that a Radon number r for ^ implies a Helly number h ^ r — 1; it is shown in this paper that exactly one additional relationship among the Carathέodory, Helly, and Radon numbers is true, namely, that if ^ has Carathέodory number c and Helly number h then ^ has Radon number r ^ ch+1. Further, characterizations of (finite) Caratheodory, Helly, and Radon numbers are obtained in terms of separation properties, from which emerges a new proof of Levi's theorem, and finally, axiomatic foundations for convexity in euclidean space are discussed, resulting in a theorem of the type proved by Dvoretzky.

Generators and defining relations for the group A n of automorphisms of a free group of rank n were derived by J . Nielsen [11]. For n = 2, this is a fairly easy task, but for n ~> 3 it requires very difficult combinatorial arguments which have not been simplified since the appearance of Nielsen's paper. In order to obtain an easier approach to the investigation of An and a bet ter understanding of its structure, it seems natural to s tudy its subgroups. For all n, the elements of An which induce the identical automorphism in the commutator quotient group Fn/F'~ form a normal subgroup K of An. Bachmuth [1] calls this the group of IA automorphisms of F n. Magnus [8] showed tha t this subgroup is generated by the automorphisms Ktj : ai ~ a ja i a i 1

p o h = f \ L. A map g: K-> E such that p ° g — f and g I L = h we call a lifting of f rel h. In this paper single obstruction Γ(f) e Hf (K, L, f if) is defined, ί? is a so-called Z?-spectrum, and H* ( if) is cohomology in that spectrum. If a lifting of / rel h exists, Γ{f) = 0; this condition is also sufficient if the fiber of p is /^-connected and dim (K/L) ^ 2k + 1. If g0 and #i are liftings of / rel h, a single obstruction

In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open sets are resolutive [5], [1]. The aim of the present paper is to substitute in these results the continuous functions by Borel-measurable functions and to leave out the restriction that the open sets are relatively compact. H. Bauer has replaced the axiom 3 of Brelot’s axiomatic by two weaker axioms: the axiom of separation (Trennungsaxiom) and the axiom K1. Since the axiom of separation is not fulfilled in some important cases (e.g. the compact Riemann surfaces) we shall weaken this axiom too, substituting it by one of its consequences: the minimum principle for hyperharmonic functions.

Two point sets are mutually separated if they are mutually exclusive and neither of them contains a limit point of the other. A point set M is said to be not connected or connected according as M is or is not the sum of two non-vacuous mutually separated point sets. A subset X of a connected point set M is said to be a cutting of M, or is said to cut M, provided the set of points M—X is not connected; X is called an irreducible cutting of M provided X cuts M but no proper subset of X cuts M. If A and B are subsets of a connected point set Af, the subset X of M is said to separate A and B in My or to cut M between A and B, provided that M~X is the sum of two mutually separated sets Ma(X) and Mb(X) containing A and B respectively. In this paper the ordinary notation of point set theory will be employed, for example, X = X + X ' , whereX is the set of all limit points of the set X, K c H means that the set K is a subset of the set H, K H means the set of points common to K and Hy F(R) denotes the boundary of the set i?, etc. In addition, if X is a cutting of a connected point set M then, unless otherwise stated, the equation M—X — Ma(X) + Mb(X) is to be interpreted as meaning that M—X is the sum of two mutually separated sets Ma(X) and Mb(X) which contain the sets A and B respectively. 1. Cuttings of Connected Sets in General. The point sets considered in this section are assumed to lie in a separable metric space which will be denoted by 5. It is apparent from the proofs of some of the theorems, however, that they hold in even more general space.

In this paper is presented a partial list of formulas giving the orders of the ruled hypersurfaces and ruled surfaces whose rulings are incident with given curves and surfaces in 4-space. If the number of incidences is equivalent to six simple conditions, the number of lines having these incidences is finite. The formulas herein presented are analogous to those giving the orders of the surfaces whose rulings are lines satisfying three conditions and the number of lines satisfying four conditions in 3-space.* Some of these formulas are obvious, others require proof. Those that are not herein included are left out for future consideration. We shall let h be the number of apparent double points, p the deficiency, of a given curve C; H the number of apparent double points, P the deficiency, of a 3-space section c of a given surface F; and t the number of apparent triple points of F or the number of lines through a given point meeting F three times, f First consider the hypersurface F" whose lines satisfy four conditions. For a line to meet a given curve once is equivalent to two conditions. Hence the °o lines cutting across two given curves Cs C* form a hypersurface whose order is obviously

It is well known that the fundamental group π1(X) of an arcwise connected topological space X operates on the n-th homotopy group πn(X) of X as a group of automorphisms. In this paper I intend to construct geometrically a group 𝒰(X) of automorphisms of πn(X), for every integer n ≥ 1, which includes a normal subgroup isomorphic to π1(X) so that the factor group of 𝒰(X) by π1(X) is completely determined by some invariant Σ(X) of the space X. The complete analysis of the operation of the group on πn(X) is given in §3, §4, and §5

The purpose of the present paper is to give a new proof to the triad excision theorem of Blakers and Massey [1], in case m≦2 and n≦2, by the aid of path spaces and in connection with a recent work of J. P. Serre [2].

1. Definitions. A transformation group (G, X) consists of a group G acting on a topological space X to form a group of homeomorphisms of X onto itself. I t will be understood throughout this paper that G is finite. For a given transformation group or "action" (G, X) and subset HQGy we denote by F(H\ G, X) the fixed-point set of H—that is, the points x such that hx = x for h gx are of class C. When i = 0 we shall drop the manifold condition on X; every action is then of class C°. A differentiable action is a enaction. (G, X) is orthogonal if X is a euclidean sphere or an open submanifold of a euclidean space and the trans

IMMERSIONS OF MANIFOLDSC) BY MORRIS W. HIRSCH Introduction Let M and N be differentiable manifolds of dimensions k and n respec- tively, k N is called an immersion if / is of class C1 and the Jacobian matrix of/ has rank k at each point of M. Such a map is also called regular. Until recently, very little was known about the ex- istence and classification of immersions of one manifold in another. The present work addresses itself to this problem and reduces it to the problem of constructing and classifying cross-sections of fibre bundles. In 1944, Whitney [15] proved that every ^-dimensional manifold can be immersed in Euclidean space of 2k — 1 dimensions, P2*-1. The Whitney- Graustein theorem [13] classifies immersions of the circle S1 in- the plane E2 up to regular homotopy, which is a homotopy / En, k /') and (g, g') are regularly homotopic (in a sense to be defined later). Given two immersions/, g: £>*—»£ that agree on S*_1 and have the same first derivatives at points of S*_1, Q(f, g) is an element of a certain homotopy group, and has the following properties: (1) Q(f, g) =0 if and only if/and g are regularly homotopic rel S*-1, i.e., the homotopy agrees with / and g on Si_1 at each stage, up to the first derivative; (2) fl(/, g) enjoys the usual algebraic properties of a difference cochain. At this point we should like to be able to make the following statement: If / is an immersion of the Received by the editors September 29, 1958. (*) The material in this paper is essentially a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Chicago, 1958. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use