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Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

Let Em and En be orthogonal Euclidean spaces of dimensions m and n respectively and with the origin of each as their only common point. In a previous paper (3) I gave what was intended to be a proof of the relationwhere the dimension of A, dim A, is the Besicovitch dimension, i.e. the number s such that the Hausdorff measure in any dimension greater than s is zero whilst that in any dimension less than s is infinite, where A and B are subsets of En and Em respectively and where A × B is the Cartesian product of A with B.

The aim of this short note is to improve some recent results of B. Grunbaum by some remarks. We use Grunbaum's notations.

The dynamic response of an infinite, elastic plate to a steadily moving transverse force is studied under the so-called improved plate theory. Analytic solutions for the subsonic and the intersonic cases are obtained using the method of integral transforms. The usual difficulty of the nonuniqueness of the steady-state solutions is circumvented by formulating the problems as initial value problems and then seeking the large time limits of the solutions of such initial value problems. The subsonic and intersonic solutions obtained here are similar to that obtained previously for the supersonic case. However, in contrast with the supersonic solution, which contains two families of Mach lines, the subsonic solution contains no Mach lines and the intersonic solution contains only one family of Mach lines.

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Let H be a real or complex Hilbert space and let L p(1≦p

Within the class of offspring distributions with given meanm>1 and probability of no offspringp o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp 0

M. Rosenblum in [2] presented a most ingenious proof of the Fuglede and Putnam Theorems by means of entire vector valued functions [1, p. 59]. We will demonstrate that some curious properties of bounded Hilbert space operators can be derived from Rosenblum's argument and similar arguments. Throughout this text we mean by an "operator" a bounded linear transformation of a Hilbert space into itself. Given an operator A we mean by "exp A " the uniform limit of the series I+A +A 2/2 1 +A3/3! +A4/4! + * * * . We let A * denote the adjoint of the operator A, and let z* denote the complex conjugate of the complex number z. A "normal" operator is an operator which commutes with its adjoint. A critical fact in the Rosenblum proof is that given a normal operator A and any complex number z, exp (izA) exp (iz*A *) exp (izA +iz*A *) = exp (iz*A *) exp (izA), and this operator is unitary because i(zA +z*A *) is skew hermitian. Our first result states, among other things, that the converse is true; if the above equations hold for a fixed operator A and all complex numbers z, then A is normal.

Let f(x) and F(x) be polynomials which are supposed to be decomposed into a series of n terms asFurther let θ be a distributive operation and φ(x) be a given function. Then the functional equationhas a solution of the formwhere yν (x) (ν = 1, 2, … n) is the solution of the equation

PROOF. By a theorem of Rudin a function gEH' in U whose boundary values are real a.e. on I can be analytically continued to D [3, p. 59]. The lemma follows on applying Rudin's result to gi= (1/2) (fl+f2) and g2=(i/2) (fi-f2). PROOF OF THEOREM 1. By a well-known decomposition theorem [2, p. 87], f(z)=B(z)F2(Z), where B(z) is a Blaschke product and F(z) EH1. Since the boundary values of B (z) have absolute value one a.e. on K, we have a.e. on I, f(ei0)= |f(eio)I, or B(ei0)F2(ei0) = F2(ei0) |, and hence

In this short paper we present an existence (an unicity) result for a first order differential equation in Hilbert spaces with right-hand side almost-periodic in the sense of Stepanoff.

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.

In order to treat certain "non-symmetric" potential theories, It6 I-4] and Bliedtner [1] have generalized Beurling and Deny's theory of Dirichlet spaces by replacing the inner product in the Dirichlet space with a bilinear form defined on a real, regular functional space. This bilinear form, called a Dirichlet form, is supposed to be continuous and coercive, and furthermore to satisfy a "contraction"-condition. When the underlying space is a locally compact abelian group, there is a complete characterization of translation invariant Dirichlet spaces in terms of real, negative definite functions on the dual group. The main purpose of the paper is to obtain an analogous characterization (Theorem 3.7) of translation invariant Dirichlet forms, in terms of complex, negative definite functions on the dual group. The contents of the paper may be summarized in the following way: In w 1 we study positive closed sesquilinear forms on an abstract complex Hilbert space, and using the Lax-Milgram theorem, it is shown, that with every such form is associated a resolvent. The theory is specialized in w 2 to sesquilinear forms on the Hilbert space L 2 (X, ~), where some relations with normal contractions are discussed. In particular an example of a positive closed form fl is given, such that the unit contraction operates with respect to fl, but not with respect to the form fl*, which is adjoint to ft. However, in the case of a translation invariant, positive closed form fl on L 2 (G), where G is a locally compact abelian group, it is shown in w 3, that the unit contraction operates with respect to fl if and only if it operates with respect to fl*, and we finish by establishing the above mentioned characterization of translation invariant Dirichlet forms.

The theory of *-representations of Banach *-algebras on Hilbert space is one of the most useful and most successful parts of the theory of Banach algebras. However, there are only scattered results concerning the representations of general Banach algebras on Banach spaces. It may be that a comprehensive representation theory is impossible. Nevertheless, for some special algebras interesting and worthwhile results can be proved. This is true for (Y), the algebra of all bounded operators on a Banach space Y, and for (Y), the subalgebra of (Y) consisting of operators with finite dimensional range. The representations of (Y) are studied in a recent paper by H. Porta and E. Berkson (6), and in another recent paper (8), P. Chernoff determines the structure of the representations of (Y) (and also of some more general algebras of operators). In both these papers, (Y), which is the socle of the algebras under consideration, plays an important role in the theory. This suggests the possibility that a more general representation theory can be formulated in the case of a normed algebra with a nontrivial socle. This we attempt to do in this paper.

Ergodic theory is a relatively new branch of mathematics which from a mathematical point of view may be regarded as generated by the interaction of measure theory and the theory of transformation groups. Its basic concept of "metric transitivity" or "ergodicity" was introduced in 1928 in a paper of Paul Smith and G. D. Birkhoff on dynamical systems. However, the significance of this concept was not appreciated until late 1931 when J. yon Neumann and G. D. Birkhoff proved the celebrated mean and pointwise ergodic theorems, and one may regard the nearly simultaneous appearance of these papers as marking the birth of the subject. Birkhoff's proof of the much more difficult pointwise ergodic theorem was stimulated by yon Neumann's theorem and yon Neumann, in turn, was stimulated by a key observation of B. O. Koopman. Let De be a surface of constant energy E in the phase space D of some Hamiltonian dynamical system. Let V,(~o) denote the point of phase space representing the "state" of the system t time units after it was represented by ~o. Then, for each t, oJ -. Vt(oJ ) is a one-to-one transformation of f2 e onto itself which conserves the natural volume element ~e in f2e induced in £2 e by the Liouville measure dql .." dqn dpl "'" d p n . Moreover, Vq+t~ = VqVt~ for all real numbers t 1 and tz. Koopman's observation (not so obvious 40 years ago as now) was that we may obtain a unitary representation't --+ Ut of the additive group of the real line in the Hilbert space 5°2(f2e, ~e) by defining Ut( f ) (co ) = f (V, ( , -o) ) .

For a sufficiently general immersion of a smooth or polyhedral closed 2-dimensional surface into Euclidean 3-space, the number of triple points is congruent modulo 2 to the Euler characteristic. The approach of this paper involves elementary notions of modification of surfaces by surgery. Let f: M2 2E3 be an immersion in general position of a closed surface M2 into Euclidean 3-space so that there are finitely many N(f) triple points of /. The purpose of this note is to give a direct and elementary proof of the fact that N(f) x(M) (mod 2). The techniques used in this paper are related to the notion of surgery for surfaces, by which a pair of discs is replaced by a cylinder having the same boundary. The approach used in this paper was reported on at the American Mathematical Society Annual Meeting in Las Vegas, January 1972. With the added assumption of differentiability, it is possible to give a new proof of this same result using normal characteristic classes and singularities of projections, as in [l]. For an immersion /: M2 -. E3, set Gr(f) = lx in E3 f| -1(x) consists of precisely r points of M2}. The condition that f is in general position implies that if x e Gr(f) and f1(x) = tp1, P2,..., P,i then there are disjoint disc neighborhoods D(p1), D(p2),.9., D(pr) of these points in M and a homeomorphism w: B D3 of a ball neighborhood B of x to the unit ball in E3 so that (w of) (D(pi)) is the intersection of D3 with the plane orthogonal to the ith coordinate vector. If the immersion f is from a particular category, for example, differentiable or piecewise-linear, then we may assume that the homeomorphism w is in the same category, and in fact all the constructions which will be described can be altered in fairly standard ways so Presented to the Society January 18, 1972; received by the editors September 10, 1973. AMS(MOS) subject classfications(1970). Primary 57C35, 57D40, 57D65; Secondary 55A20.

In this paper, extreme value theory is considered for stationary sequences ζn satisfying dependence restrictions significantly weaker than strong mixing. The aims of the paper are: (i) To prove the basic theorem of Gnedenko concerning the existence of three possible non-degenerate asymptotic forms for the distribution of the maximum Mn = max(ξ1...ξn), for such sequences. (ii) To obtain limiting laws of the form $$\mathop {\lim }\limits_{n \to \infty } \Pr \{ M_n^{(r)} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } u\} = e^{ - \tau } \sum\limits_{s = 0}^{r - 1} {\tau ^s /S!} $$ where Mn(r)is the r-th largest of ξ1...ξn, and Prξ1>un∼Τ/n. Poisson properties (akin to those known for the upcrossings of a high level by a stationary normal process) are developed and used to obtain these results. (iii) As a consequence of (ii), to show that the asymptotic distribution of Mn(r)(normalized) is the same as if the {ξn} were i.i.d. (iv) To show that the assumptions used are satisfied, in particular by stationary normal sequences, under mild covariance conditions.

The countable box product of ordinals is examined in the paper for normality and paracompactness. The continuum hypothesis is used to prove that the box product of countably many a-compact ordinals is paracompact and that the box product of another class of ordinals is normal. A third class trivially has a nonnormal product. Because I have found a countable box product of ordinals useful in the past [1], this class of spaces particularly interests me. The purpose of this paper is to tell what I know about which of these spaces is paracompact or normal. In [2] I prove that the continuum hypothesis implies the box product of countably many a-compact, locally compact, metric spaces is paracompact. I prove here that the continuum hypothesis implies the box product of countably many a-compact ordinals is paracompact (Theorem 1) and the box product of another class of ordinals is normal (Theorem 2). The proof of Theorems 1 and 2 is a quite messy join of the techniques of [1] and [2] which raises some doubt in my mind as to whether these theorems are worth proving. Because I care, because I think these spaces are set theoretically interesting and topologically useful, because I think these theorems are best possible, the theorems are worth the mess to me. A. If {XXAOA)\ is a family of topological spaces, a box in HIAeA XA is a set TheA UA where each UA is open in XA. The box product of {XAIA A iS HAA XA topologized by using the set of all boxes in it as a basis. Throughout the paper the following notation is used. An ordinal a is the set of all ordinals less than a and a is topologized by the interval topology. The statement that a is a cardinal means that a is an ordinal and no smaller ordinal has the same cardinality as a. The notation IAnA BAA is used to mean the ordinary Cartesian product of the f,3's and never the cardinal or ordinal arithmetic product. Similarly a#9 means the set of all functions from ,B into a rather than an arithmetic operation. If a is an ordinal, let cf(a) denote the cofinality of a; that is cf(a) is the smallest ordinal 8 such that there is a subset A of a, order isomorphic with 8, such that /3 < a implies there is a y El A with /3 < y. Observe that a is a a-compact ordinal if and only if a is compact or cf(a) =w. Received by the editors December 10, 1971 and, in revised form, October 1, 1972. AMS (MOS) subject classiflcations (1970). Primary 54B10, 54A25, 54D15, 54D20, 54D30, 02K25.

For n ⩾ 2 n \geqslant 2 , let G ( n ) G(n) denote the two-fold covering group of SO e ( 1 , n ) {\text {SO} _e}(1,n) . In case n ⩾ 3 , G ( n ) n \geqslant 3,G(n) is isomorphic to Spin ⁡ ( 1 , n ) \operatorname {Spin} (1,n) and is simply connected. In a previous paper we determined all the irreducible quasi-simple representations of these groups, up to infinitesimal equivalence. The main purpose of the present paper is to determine which of these representations are unitarizable. Thus, with the aid of some results of Harish-Chandra and Nelson we determine all the irreducible unitary representations of G ( n ) G(n) , up to unitary equivalence. One by-product of our analysis is the explicit construction of the infinitesimal equivalences, which are known to exist from our previous work, between the various subquotient representations and certain subrepresentations in the nonirreducible cases of the nonunitary principal series representations of G ( n ) G(n) .

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than 2(36n+57)2n when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form eO(m10gm) should hold, but I have not so far succeeded in proving such a result.

A detailed probabilistic treatment is given of a birth-and-death process proposed by Williams (1969) in which the elements of the process bear up to s inanimate marks. Equations for the second-order moments of the process, and approximate marginal univariate solutions, are derived. The exact bivariate solution is given for the case s = 1. For general s the variance of the mark population is also derived. MARKED BIRTH-AND-DEATH PROCESS; MULTIVARIATE LAGRANGE DIFFERENTIAL EQUATIONS; MULTITYPE BRANCHING PROCESS Introduction In a recent paper Williams (1969) has considered the following process. We have a population of elements developing as a linear birth-and-death process and each element bears a number of inanimate marks. When a marked element dies then it and its marks are removed from the population; when a marked element divides, its marks segregate independently and without loss, each with probability I into one of the two daughter elements. It is assumed that marking an element does not affect its birth and death rates (cf. Clifford and Sudbury (1972)). Williams' process was proposed as a model of an experiment conducted on bacteria (the elements) infected by phage (the inanimate marks), by Meynell (1959). (The bacteria are lysogenic with respect to the phage, which means that the phage do not interfere with the behaviour of the bacteria.) His object was to use the marking process in order to separate the birth and death rates of the bacteria; if we denote birth and death rates by A and p respectively, then the mean-value measures on the bacterial population (assumed to develop as a linear birth-and-death process) will estimate the difference (A p). As the marks are inanimate, however, they will only be affected by p and so mean-value measures on the mark population enable us to estimate p (see Williams (1969)). The papers of Meynell and Williams taken together provide a most interesting example of a virological experiment and the probability model of that experiment. Received in revised form 10 August 1973 Research sponsored by a Science Research Council grant and a World Health Organisation research traineeship. 423 This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms 424 BYRON J. T. MORGAN Williams considered the mean-value solution of his model; this was adequate for Meynell's experiment in which population sizes were large. Here we shall give a fuller probabilistic statement of the model and make some contributions to its solution. Two differences from Williams' work lie in the adoption of an upper bound, s, to the number of marks any element can have, and the assumption (except where stated) of birth and death rates independent of time. We 'shall assume general initial conditions which, in Meynell's experiment, correspond to the distribution of phage over bacteria once these have been initially exposed to a phage population. Williams took this distribution to be Poisson; the distribution of phage over bacteria has, however, been the subject of much probabilistic study (e.g., see Gani (1965) and Morgan (1971)). Finally, let us note that the value s = I, on which we shall particularly concentrate, probably gives the most realistic description of Meynell's experiment. 2. The probability model The model is a continuous-time Markov process with discrete states. Let N,(t) be a random variable denoting the number of elements with i marks at time t, i = 0, 1,2, ..., s. Let n = (n0,n1, l , n.., ), Iu = (Uo, u1, *,us), N(t) = (No(t), N(t), --,N,(t)), and suppose N,(O) = ai, i = 0, 1,2, --, s. Let P(n; t) = P(N(t) = n), then corresponding to death we have the transitions, (No, ..., Ni, ---, N,) (No, --, Ni 1, , N,) at probability rate pNi, i = 0, 1,2, ,s, while corresponding to birth we have the following transitions (No, N,) ,(No + 1, .., N,) at probability rate, 2 =1 N 2-' +,ANo, (No, , Ni ,N2i,..., X (No, Ni + 2, , N2iI, at probability rate, This content downloaded from 207.46.13.180 on Thu, 08 Sep 2016 05:01:40 UTC All use subject to http://about.jstor.org/terms On the distribution of inanimate marks over a linear birth-and-death process 425 where we use [is] to denote the integral part of is, and (No, " , Ni, * * *, Nk i Nk, "', Ns) -*(No, ..., Nj + 1, ..., Nk-i + 1, ..., Nj 1, ..., Nl) at probability rate 2N 2-k' i = 1,2,...,C[(k 1)], k = 3,4,...,s (smaller k give transitions already considered above). It is readily verified that the sum of the transition rates is (A + yp) X = oN,, as required in the overall birth-and-death process. Introducing the operator Di, which increases the argument ni of a function of n by 1, we then deduce that the Kolmogorov forward equations (for all possible n) are, d S S (1) +(n; t) P(n; t)(A+ t)1 (n2 + D,P(n;t)(n +1) dt ==0 1=0

Jack Edmonds developed a new way of looking at extremal combinatorial problems and applied his technique with a great success to the problems of the maximal-weight degreeconstrained subgraphs. Professor C. St. J.A. Nash-Williams suggested to use Edmonds' approach in the context of hamiltonian graphs. In the present paper, we determine a new set of inequalities (the “comb inequalities”) which are satisfied by the characteristic functions of hamiltonian circuits but are not explicit in the straightforward integer programming formulation. A direct application of the linear programming duality theorem then leads to a new necessary condition for the existence of hamiltonian circuits; this condition appears to be stronger than the ones previously known. Relating linear programming to hamiltonian circuits, the present paper can also be seen as a continuation of the work of Dantzig, Fulkerson and Johnson on the traveling salesman problem.

This paper addresses the problem of selecting the parameter in a family of algorithms for unconstrained minimization known as Self Scaling Variable Metric (SSVM) Algorithms. This family, that has some very attractive properties, is based on a two parameter formula for updating the inverse Hessian approximation, in which the parameters can take any values between zero and one. Earlier results obtained for SSVM algorithms apply to the entire family and give no indication of how the choice of parameter may affect the algorithm's performance. In this paper, we examine empirically the effect of varying the parameters and relaxing the line-search. Theoretical consideration also leads to a switching tule for these parameters. Numerical results obtained for the SSVM algorithm indicate that with proper parameter selection it is superior to the DFP algorithm, particularly for high-dimensional problems.

In this paper we show that a ring R is von Neumann regular (or a V-ring) if and only if every central localization of R at a maximal ideal of its center is von Neumann regular (or a V-ring). Strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. Also we consider whether regular PI-rings can be characterized by the property that all central localizations at maximal ideals of the center are simple. Commutative von Neumann regular rings have been characterized in various ways. However, very few of these characterizations extend to noncommutative rings. The results in this paper arose from attempting to extend to noncommutative rings a well-known theorem of Kaplansky [11, Theorem 6] which states that commutative regular rings are characterized by the property that all localizations at maximal ideals are fields. It turns out that the obvious extension of this theorem to the noncommutative case is valid. That being, a ring is regular if and only. if all central localizations at maximal ideals of the center are regular. An analogous theorem is obtained for Vrings. Also we show that strongly regular rings are characterized by the property that all central localizations at maximal ideals of the center are division rings. With an eye to the commutative theory, we consider whether regular PIrings can be characterized by the property that all central localizations at maximal ideals of the center are simple. We provide an example to show that this is not the case. However, it is true if and only if contraction provides a 1:1 correspondence between maximal ideals of the ring and maximal ideals

In a previously published paper [9], the authors proved an approximation theorem for geodesically convex functions on Riemannian manifolds (Theorem 2 of [91). Applications of this theorem to complex function theory on noncompact K~ihler manifolds have been announced in [8, II and III] . The main result of the present paper is an application of this approximation theorem to Riemannian geometry proper. To describe this principal theorem, let M be a Riemannian manifold and let Z(M) be the closure of the set of all C ~176 subharmonic functions in C~ the algebra of continuous functions on M equipped with the compact open topology. (A C ~ function f is subharmonic if Af>O, where the sign

A dissectible polyhedron is a natural extension of a concept whose history dates back to at least 1758 and Euler [7]—the concept of a dissection of a polygon. An interesting historical survey of dissections of a polygon is given by Brown [4]. Some approaches to the classical problem have been given by Moon and Moser [9] and by Guy [8] ; the latter provides an approach which is the basis of the work in this paper. A summary of enumeration results on dissections of polygons and polyhedra by automorphism groups has been given by the authors [2].Recent extensions of the problem have been investigated in a series of papers by Brown and Tutte [3; 5; 14; 15] and b y Takeo [10; 11; 12; 13].

In his paper [2] H. Petersson gives a formula which contains information about the “free zeros” of modular forms for subgroups of the classical modular group. Numerical values can be obtained from this formula only by the calculation of all roots of a polynomical, the degree of which is the number k of the free zeros, while the coefficients are rational with nominators and denumerators growing very rapidly with k. This paper contains a table of J(zi) where zi (i=1, ..., k) are the free zeros of the Eisenstein series for the classical modular group with k≤10.

In a recent note [6] the first author has announced the discovery of a family of transformation groups (G,), > o which have the following property: G, acts on the n! elements of the symmetric group ~ , and the number of its orbits is equal to the n-th tangent or secant number, according as n is odd or even. The purpose of this paper is to give a complete description of these groups. Applications to enumeration problems will appear in a subsequent paper. The tangent (or Euler) number are defined by the series expansion of tan u

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.

In this expository paper, we shall analyze a particularly important class of examples of surfaces and hypersurfaces in Euclidean 4-space, namely those which arise by considering real 4-space as the space of twocomplex variablesz andw and by taking geometric loci of the formf(z,w)=0 or hypersurfaces associated with such loci. Such surfaces and hypersurfaces are important in the study of the singularities of algebraic curves, as described for example in the book of Milnor [3], and they have been used recently in the construction of foliations of the 3-dimensional sphere by Lawson [2]. The examples of this paper were first presented at the International Symposium of Dynamical Systems and Foliations at Salvador in the summer of 1971, and the author expresses his gratitude for the opportunity to participate in that conference.

In this paper the methods of M. Karoubi (MR 41 #6205) are generalized to the case of equivariant K-theory. The sets of Fredholm operators in certain (Hilbert) spaces of representations of finite groups G are described which are classifying spaces for equivariant K-functors. The results were announced in the paper MR 46 #2702. Bibliography: 16 items.

In this paper the mathematical methods of quantum field theory are applied to some problems that arise in the statistical description of mechanical systems with very many (in the idealized case, infinitely many) degrees of freedom. This application is based on a graphical representation of the individual terms of the formal perturbation series in powers of the coupling constant in the form of Feynman diagrams. A variety of properties of such diagrams makes it possible to sum partially the perturbation series with a view to obtaining closed integral equations that contain the required quantities as unknowns. The approach is treated in more detail in connection with the statistical hydrodynamics of a developed turbulent flow, which is similar to the theory of a quantum Bose field with strong interaction. The functional formulation of statistical hydrodynamics makes it possible to obtain integral equations of turbulence theory, which can also be derived by means of diagram methods. At the end of the paper, some closed equations of statistical hydrodynamics are considered.