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A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

Kolmogorov-Arnold-Moser (or KAM) theory was developed for conservative dynamical systems that are nearly integrable. Integrable systems in their phase space usually contain lots of invariant tori, and KAM theory establishes persistence results for such tori, which carry quasi-periodic motions. We sketch this theory, which begins with Kolmogorov's pioneering work.

In a recent paper, Chao-Ping Chen and Feng Qi (2005) established sharp upper and lower bounds for the sequence P n := 1.3 … ( 2 n − 1 ) 2.4 … 2 n P_{n}:=\frac {1.3\ldots (2n-1)}{2.4\ldots 2n} . We show that their result follows easily from a theorem of G. N Watson published in 1959. We also show that the main result of Chen and Qi’s paper is a special case of a more general inequality which admits a very short proof.

Concerns have been voiced about the correctness of certain technical points in DiPerna’s paper (Comm. Math. Phys. 91 (1983), 1–30) related to the vacuum state. In this note, we provide clarifications. Our conclusion is that these concerns mainly arise from the statement of a lemma for constructing the viscous approximate solutions and some typos; however, the gap can be either fixed by correcting the statement of the lemma and the typos or bypassed by employing the finite difference methods. In [Di], DiPerna found a global entropy solution of the isentropic Euler equations for the following exponents in the equation of state for the pressure: γ = 1 + 2/(2m+ 1), m ≥ 2 integer. (1) He divided his arguments into the following two steps. 1. Compactness framework Assume that a sequence of approximate solutions (ρ (x, t),m (x, t)), 0 ≤ t ≤ T , satisfies: (i). There exists a constant C(T ) > 0, independent of > 0, such that 0 ≤ ρ (x, t) ≤ C, |m (x, t)/ρ (x, t)| ≤ C; (ii). For all weak entropy pairs (η, q) of the isentropic Euler equations, the measure sequence η(ρ ,m )t + q(ρ ,m )x is contained in a compact subset of H −1 loc (R× [0, T ]). If γ satisfies (1), then the sequence (ρ (x, t),m (x, t)) is compact in Lloc(R× [0, T ]). The reason for the restriction on the number γ is that, in such a case, any weak entropy function is a polynomial function of the Riemann invariants (w, z). This is the key step in DiPerna’s arguments and is also his main contribution to the compensated compactness method in this aspect. Received by the editors May 16, 1996. 1991 Mathematics Subject Classification. Primary 35K55, 35L65; Secondary 76N15, 35L60, 65M06.

This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder R n × R + {{\mathbf {R}}^n} \times {{\mathbf {R}}_ + } with initial data in L r {L^r} , n ≥ 3 n \geq 3 , 1 > r > ∞ 1 > r > \infty . An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].

We give a simplified proof of an imbedding theorem by C. Fefferman [3].

Here we give a result on the postulation (i.e. the 2-normality) of nonlinearly normal embeddings of Abelian varieties. This result improves some of the results proved in a recent paper by Ch. Birkenhake.

A class of functions K(r) and the corresponding solutions are produced in the case of the differential equation d 2 u/dr 2 +1 du/r dr+K(r)e 2u(r) =1. The behavior of these solutions at an isolated singularity is also investigated

We observe that a strictly singular operator is not necessarily condensing, so that the invariant subspace problem for strictly singular operators remains open.

I regret to announce that there is a serious error in my paper in these Transactions, volume 84, pp. 430-443. The error was discovered by Louis Kokoris who found that on line 8 of page 434 the expression given as 4[g(bz)](az) should have been 4[g(az)](bz). As a consequence the computation of P(z, g, az, b) yields nothing, the proof of formula (30) is not valid, and the important Lemma 9 is not proved. Thus the paper does not give a proof of its major result stated as Theorem 1. Nevertheless, the theorems of the paper are all correct and we shall provide a revision of the proof here. This revised proof has been checked by Louis Kokoris to whom the author wishes to express his great thanks. We observe first that the equation

Introduction. In a recent paper [2] R. H. Bruck has introduced the concept of right neoring and discussed some properties of these systems. In particular, he has considered analogues of certain properties of the ring of integers. This paper is essentially a commentary on Bruck's paper and we generalize some of his results as follows. The construction of the universal right neoring in [2] is applied to the free monogenic $3-loop in any subvariety $3 of the variety of loops and a complete analogue of Theorem 4.1 of [2] is obtained for any one of these subvarieties. Then, using a result similar to those obtained in [5], it is shown that this construction yields uncountably many right neorings with an identity which generates the additive loop of the right neoring. Conversely, every right neoring with an identity which generates its additive loop can be obtained from a free monogenic $3-loop by the above construction. Each of these right neorings has some properties resembling those of the ring of integers. One possible answer is given to the question raised by Bruck concerning the existence of universal right neorings with free additive loop of arbitrary rank. A brief proof is given, using the results of [4; 5], of the cancellation properties of the monogenic universal right neoring. Finally, we discuss briefly the relationship between right neorings and the logarithmetics of Etherington.

It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian. Introduction. Let R be an associative ring equipped with the binary relation (^) defined by xay if and only if xy = x2 in R. In this paper, it is shown that ( ^ ) is an order relation on R if and only if, R has no nilpotent elements i9*0). Conditions on the binary relation (g) in order that R split into a direct product of division rings are then studied in the light of Abian's result (l, Theorem l). Using similar argumentation and using certain subdirect representation of rings with no nilpotent elements, one obtains a complete similarity with the commutative case (yet, no extra complication in the computa- tions). Conventions. R is an associative ring which is, unless otherwise stated, with no nilpotent elements (other than 0). As a result of (2), R can be embedded into a direct product of skewdomains, R—* YLiei £i (that is to say, rings R, having no one-sided divisors of zero). The former embedding is fixed throughout the paper. It is therefore legiti- mate to identify any element x in R with the family consisting of all its projections (xj.e/. Finally, all definitions in (l) are extended (verbatim) to the present case (of a noncommutative ring R) and are freely used throughout.

The reading of Ayres' interesting paper suggested to me the following remarks: 1. The order of a subset of a set S in a point p4 cannot surpass the order of S in p. Hence if S2 denotes the set of all points of S of order 2, then S2 has in each point of S the order 2, the order 1, or the order 0, where the terms "order 0" and "0-dimensional" are used synonymously. SI(M), S2(1), S" may denote the set of all points of S in which S2 has the order 0, 1, 2, respectively. The points of order 2 of S are also called the ordinary points of S, and the set S2 of all ordinarv points of S may be called the ordinary part of S. The set S" of all ordinary points of the ordinary part of S may be designated the ordinary kernel of S. WVe have

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.

The purpose of this note is to present a new proof of a generalized form of Day's fixed point theorem. The proof we give is suggested by the work of T. Mitchell in his paper, Function algebras, means, and fixed points, [2]. The version of Day's theorem which we present here has not appeared explicitly in the literature before, and seems especially well suited for application to questions concerning fixed point properties of topological semigroups. 1. Preliminaries. We adopt the terminology and notation of [2] except where otherwise specified. New terminology will be introduced as needed. Let y be a convex compactum (compact convex set in a real locally convex linear topological space E), and let A( Y) denote the Banach space of all (real) continuous affine functions on Y under the supremum norm. Observe that A(Y) contains every function of the form h=f\Y + r where fe E* and r is real; thus A(Y) separates points of Y.

We establish the relationship between certain invariant extension properties for linear functional of K. Fan when a semigroup S acts on a locally convex topological vector space and the existence of a left-invariant mean on the space of almost periodic and weakly almost periodic functions on S.

A subset A A of a topological space is said to be semi-open if there exists an open set U U such that U ⊆ A ⊆ Cl ⁡ ( U ) U \subseteq A \subseteq \operatorname {Cl} (U) where Cl ⁡ ( U ) \operatorname {Cl} (U) denotes the closure of U U . The class of semi-open sets of a given topological space ( X , T ) (X,\mathcal {T}) is denoted S .O . ( X , T ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) . In the paper Semi-open sets and semi-continuity in topological spaces, Amer. Math. Monthly 70 (1963), 36-41, Norman Levine states in Theorem 10 that if T \mathcal {T} and T ∗ {\mathcal {T}^ \ast } are two topologies for a set X X such that S .O . ( X , T ) ⊆ S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) \subseteq {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T ⊆ T ∗ \mathcal {T} \subseteq {\mathcal {T}^ \ast } . In a corollary to this theorem, Levine states that if S .O . ( X , T ) = S .O . ( X , T ∗ ) {\text {S}}{\text {.O}}{\text {.}}(X,\mathcal {T}) = {\text {S}}{\text {.O}}{\text {.}}(X,{\mathcal {T}^ \ast }) , then T = T ∗ \mathcal {T} = {\mathcal {T}^ \ast } . An example is given which shows the above-mentioned theorem and its corollary are false. This paper shows how different topologies on a set which determine the same class of semi-open subsets can arise from functions, and points out some of the implications of two topologies being related in this manner.

The object of this note is to point out and correct an error in the paper [21 of Fox, purporting to prove Fenchel's conjecture that a finitely generated. infinite Fuchsian group has a torsion-free normal subgroup of finite index. Fox divided the proof of the main result in his paper into four cases and an error occurs in the proofs of Cases (III) and (IV). A direct proof of Case (III) was given. While a direct proof of Case (IV) can be given, the author shows that it also follows indirectly from the result in the paper [1] of Burns and Solitar. Given any three integers a > 1, b > 1 and c > 1, there exist permutations A of order a and B of order b, such that AB has order c. Fox divided the proof of this lemma into the four cases: (I) ab 0 (mod 2) and max{a, b} ? c < a + b, (II) ab 0 (mod 2) and c : a + b, (III) a b c 1 (mod 2), (IV) a b 1 and c 0 (mod 2), and subcases. In the subcase m(a + b 2) + 2 ? c < (m + l)(b + 1) 1 of Case (III) where m is the largest integer not exceeding (c l)/(b + 1), and it is assumed that a ? b < c, the number of distinct symbols in the permutations [2, p. 64] nl~~~W B ( Wv-sI .. WI2 I7 (vWb B = (vl * *t*v W_ ***W P2) * (V W2 .. * ]Pi+]1)'

1. In a recent paper [2] Reich has made the observation that an example of Koebe given in 1918 [1] does not fulfill its asserted purpose. This example was to show that vanishing measure of the complement did not assure that a slit domain was minimal. Reich proceeded to fill the gap by carrying out the following somewhat more general construction. Let A be a compact perfect nowhere dense set on the x-axis in the z-plane (z=x+iy). Then there exists a compact set S in the z-plane with the properties: (i) A is the projection of S on the x-axis, (ii) S is composed of segments symmetric with respect to the x-axis and points on the x-axis, at least one segment being present, (iii) any point in S, not on the x-axis and not at the end of a segment in S, is the limit, both from the left and right, of points of S. Once this construction is performed the desired examples are easily given [2, ?4]. The object of the present paper is to give an alternative construction which is very explicit and direct.

The paper consists of two parts. In the first part a short proof of the main theorem due to A. Aziz on the location of zeros of composite polynomials is given. In the second part some properties of a fixed length continuation of a polynomial are deduced.

In [4], Phelps showed that in certain function algebras the unit ball is the closed convex hull of its extreme points. The algebra, C(X), of complex valued continuous functions on a compact Hausdorff space, will always have this property. The class of logmodular algebras which have a Gleason part which is total also was shown to have the property. In this paper we give an elementary proof of the first result (a proof which is, in theory at least, constructive). The simplest nontrivial example of a logmodular algebra with a total part is the disk algebra (i.e. the functions continuous on the closed disk and analytic in the interior). For this algebra we show that the extreme points (in fact the exposed points) of the unit ball form a dense subset of the boundary of the unit ball. Let U be the unit ball of C(X). It is well known that q in U is an extreme point of U if I q(x) I = 1 for all x in X. Now if f in U never vanishes, then f is in the closed convex hull of the extreme points; in factf is between two uniquely determined extreme points. We need only observe that for each x in X,f(x) is halfway between two uniquely determined extreme points of the disk and that these points vary continuously with x. Now it is not necessarily the case that each function in U can be approximated by a nonvanishing function. For a counterexample we need only look at h(z) = z in the algebra of continuous functions on the disk. (If If(z)-zj 1/e. Letfi, i = 1, *, N be the retract maps obtained from these rotations. Then I 1/N Efi(z)-z

where the oi are the usual generators of Bn. (See [1; 4].) A textile manufacturer asked the following question: Can one decide whether or not a braid in An is also in Qn? (It is possible to "weave" a braid in Qn while keeping its ends "tied together.") Shepperd [6] solved the above decision problem using the generators and relations of the braid group and subgroups. In this paper we give another distinct solution working in terms of automorphisms of free groups. In particular, we use formulas developed in [3; 4]. The author wishes to thank his teacher, and friend, Professor Wilhelm Magnus who has contributed more than anyone else towards the author's knowledge of mathematics.

Among other results, it is proved that if a sequence { μ n } \{ {\mu _n}\} of regular measures on a Hausdorff space, with values in a normed group, is convergent to zero for all σ \sigma -compact sets or all open sets, then there exists a maximal open set U such that μ ˙ n ( U ) → 0 , { μ ˙ n } {\dot \mu _n}(U) \to 0,\{ {\dot \mu _n}\} being the associated submeasures.

We give a simplified proof to the following theorem due to D. Quillen: if A is a commutative noetherian ring of global dimension ⩽ 1 \leqslant 1 , then finitely generated projective modules over A [ T 1 , … , T n ] A[{T_1}, \ldots ,{T_n}] are extended from A. We prove also that if A is a commutative noetherian ring of global dimension d, then finitely generated projective modules of rank > d > d over A [ T 1 , … , T n ] A[{T_1}, \ldots ,{T_n}] are extended from A.

This note deals with the correction of a small flaw in the proof of a theorem (and a lemma) in [2]. Also a simpler proof of the theorem in question is given. In the proof of Theorem 2.3 in [2], the statement "that distinct L's cannot belong to a common pure subgroup HL follows immediately from the fact that no two L's generate with G [p] the same subgroup of G" is false as it stands. One has to exercise a little more care in the selection of HL. For example, since the union [xa, Xa]aEA of all the L's (in Lemma 2.1 in [2]) is still linearly independent, there exists according to [1] a pure subgroup H of G supported by S such that HDL for each L. Hence one could choose HL=H for each L. However, all goes well if HL is chosen such that HLn [Xa, XTg aEA =L, as can easily be done. Similar remarks hold for Lemma 3.8 in [2]. A short alternate proof of Theorem 2.3 in [2] is given. In what follows let c be the cardinality of the continuum and Q be the first ordinal whose cardinality is c. The notation and terminology will be the same as that in [2].

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.