We are interested in the structure of a group G of linear fractional transformations of the extended complex plane that is generated by two parabolic elements A and B, and, particularly, in the question of when such a group G is free. We shall, as usual, represent elements of G by matrices with determinant 1, which are determined up to change of sign. Two such groups G will be conjugate in the full linear fractional group, and hence isomorphic, provided they have, up to a change of sign, the same value of the invariant τ = Trace(AB) – 2. We put aside the trivial case that τ = 0, where G is abelian. In the study of these groups, two normalizations have proved convenient. Sanov (17) and Brenner (3) took the generators in the form

while Chang, Jennings, and Ree (4) took them in the form

One of the unsolved problems of plane topology is the following:

Question. What are the homogeneous bounded plane continua?

A search for the answer has been punctuated by some erroneous results. For a history of the problem see (6).

The following examples of bounded homogeneous plane continua are known : a point; a simple closed curve; a pseudo arc (2, 12); and a circle of pseudo arcs (6). Are there others?

The only one of the above examples that contains an arc is a simple closed curve. In this paper we show that there are no other such examples. We list some previous results that point in this direction. Mazurkiewicz showed (11) that the simple closed curve is the only non-degenerate homogeneous bounded plane continuum that is locally connected. Cohen showed (8) that the simple closed curve is the only homogeneous bounded plane continuum that contains a simple closed curve.

The following result is well known (as usual, [x]denotes the integral part of x):

(A) Let α and β be positive irrational numbers satisfying

1

Then the sets [nα], [nβ], n= 1, 2, …, are complementary with respect to the set of all positive integers]see, e.g. (1; 2; 4; 5; 6; 7; 8; 10; 13; 14; 15; 16). In some of these references the result, or a special case thereof, is mentioned in connection with Wythoff's game, with or without proof. It appears that Beatty (4) was the originator of the problem.

The theorem has a converse, and the following holds:

(B) Let α and β be positive. The sets [nα] and [nβ], n = 1, 2, …, are complementary with respect to the set of all positive integers if and only if α and β are irrational, and (1) holds.

Let λ(ij), i,j = 1, 2, … , m, be m2 elements in a field K of characteristic zero such that λ(ij)λ(ji) = 1 for all i and j, and X1, x2, … , xm non-commutative associative indeterminates over K. Define the elements [xi1Xi2 … xin] inductively by [xi] = xi and

Any linear combination of the elements

with coefficients in K will be called a generalized Lie elememt. Generalized Lie elements reduce to ordinary Lie elements if λ(ij) = 1 for all i and j.

The purpose of this paper is to generalize to the generalized Lie elements the following: a theorem of Friedrichs, a theorem of Dynkin-Specht-Wever (2), and the Witt formula on the dimension of the space spanned by homogeneous Lie elements of a fixed degree. The set of all generalized Lie elements will be made into an algebra which generalizes the ordinary free Lie algebra. This algebra turns out to be free in a certain sense. We shall also generalize the algebra associated with shuffles in (2).

In (8) M. V. Menon investigates the diagonal equivalence of a non-negative matrix A to one with prescribed row and column sums and shows that this equivalence holds provided there exists at least one non-negative matrix with these row and column sums and with zeros in exactly the same positions A has zeros. However, he leaves open the question of when such a matrix exists. W. B. Jurkat and H.J. Ryser in (7) study the convex set of all m × n non-negative matrices having given row and column sums.

Let

1

2

Thus p-1(n) = p(n) is just the partition function, for which Ramanujan (4) found congruence properties modulo powers of 5, 7, and 11. Ramanathan (3) considers the generalization of these congruences modulo powers of 5 and 7 for all ; unfortunately his results are incorrect, because of an error in his Lemma 4 on which his main theorems depend. This error is essentially a misquotation of the results of Watson (5), which one may readily understand in view of Watson's formidable notation.

In a recent investigation of the asymptotic behavior of the Lebesgue constants for Jacobi polynomials, we encountered the problem of obtaining an asymptotic expansion for the Jacobi polynomials , as n → ∞, which is uniformly valid for θ in . The leading term of such an expansion is provided by the following well-known formula of “Hilb's type” [13, p. 197]:

(1.1)

where α > – 1, β real and ; c and are fixed positive numbers. Note that the remainder in (1.1) is always θ1/2O(n–3/2).

The aim of this paper is to present a unified treatment of certain theorems in Combinatorial Analysis (particularly in enumerative graph theory), and their relations to various results concerning symmetric functions and the characters of the symmetric groups. In particular, it treats of the simplification that is achieved by working with S-functions in preference to other symmetric functions when dealing with combinatorial problems. In this way it helps to draw closer together the two subjects of Combinatorial Analysis and the theory of Finite Groups. The paper is mainly expository; it contains little that is really new, though it displays several old results in a new setting.

Let K and C be the closure and boundary, respectively, of the open unit disc U in the complex plane. Let be the Banach algebra whose elements are those continuous complex functions on K which are analytic in U, with norm

(f ∊ ).

Let A be an artin algebra. Let ( be a component of the stable Auslander- Reiten quiver of A. If is periodic, then the structure of G is known. Here, we are going to consider the case when is non-periodic: we will show that is isomorphic to Z with a valued quiver. In particular, there is no cyclic path in .

We study a class of generalized hypergeometric functions in several variables introduced by A. Korânyi. It is shown that the generalized Gaussian hypergeometric function is the unique solution of a system partial differential equations. Analogues of some classical results such as Kummer relations and Euler integral representations are established. Asymptotic behavior of generalized hypergeometric functions is obtained which includes some known estimates.

It is the purpose of this paper to show that many of the enumerative techniques available for counting rooted plane trees may be extended to tree-rooted maps, that is, rooted maps in which a spanning tree is distinguished as root tree. For example, tree-rooted maps are enumerated by partition, and the average number of trees in a rooted map with n edges is determined. An enumerative similarity between Hamiltonian rooted maps (that is, rooted maps with a distinguished Hamiltonian polygon) and tree-rooted maps is discussed. A 1-1 correspondence is established between treerooted maps with n edges and Hamiltonian rooted trivalent maps with 2n + 1 vertices in which the root vertex is exceptional, being divalent, both of which are in 1-1 correspondence with non-separable Hamiltonian-rooted triangularized digons with n internal vertices, where both the latter are as defined in (2).

We define sets with finitely ramified cell structure, which are generalizations of post-critically finite self-similar sets introduced by Kigami and of fractafolds introduced by Strichartz. In general, we do not assume even local self-similarity, and allow countably many cells connected at each junction point. In particular, we consider post-critically infinite fractals. We prove that if Kigami’s resistance form satisfies certain assumptions, then there exists a weak Riemannian metric such that the energy can be expressed as the integral of the norm squared of a weak gradient with respect to an energy measure. Furthermore, we prove that if such a set can be homeomorphically represented in harmonic coordinates, then for smooth functions the weak gradient can be replaced by the usual gradient. We also prove a simple formula for the energy measure Laplacian in harmonic coordinates.

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.

The difficulties with which the trace formula confronts us are manifold. Most of them arise from the non-compactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.

The object of this note is to prove the following theorem.

THEOREM. Let A be a division ring with centre Z, and suppose that for every x in A, some power (depending on x) is in . Then A is commutative.

From a symmetric balanced incomplete block design we may construct a derived design by deleting a block and its varieties. But a design with the parameters of a derived design may not be embeddable in a symmetric design. Bhattacharya (1) has such an example with λ = 3 . When λ = 1, the derived design is a finite Euclidean plane and this can always be embedded in a corresponding symmetric design which will be a finite projective plane.

In this paper we are concerned with the group of formal power series of the form

,

the coefficients being elements of a commutative ring R and the group operation being substitution. Little seems to be known of the properties of groups of this type, except in special cases, although groups of formal power series in several variables with complex coefficients have been investigated from a different point of view by Bochner and Martin (1, chap. I) and Gotô (2).

Introduction. In a recent unified theory originated by Einstein and Straus [l], the gravitational and electromagnetic fields are represented by a single nonsymmetric tensor gy which is a function of four coordinates xr(r = 1, 2, 3, 4). In addition a non-symmetric linear connection Γjki is assumed for the space and a Hamiltonian function is defined in terms of gij and Γjki. By means of a variational principle in which the gij and Γjki are allowed to vary independently the field equations are obtained and can be written

(0.1)

(0.2)

(0.3)

(0.4)

With applications in mind we establish a summation formula for the coefficients of a general Dirichlet series satisfying a suitable functional equation. Among a number of consequences we derive a generalization of an elegant divisor sum bound due to F. V. Atkinson.

We study the gap (= “projection norm” = “graph distance”) topology of the space of all (not necessarily bounded) self-adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show the surprising result that this space is connected in contrast to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.