A duality formula is found for coalescing Brownian motions on the real line. It is shown that the joint distribution of a coalescing Brownian motion can be determined by another coalescing Brownian motion running backward. This duality is used to study a measure-valued process arising as the high density limit of the empirical measures of coalescing Brownian motions.

Let $T$ be a sectorial operator. It is known that the existence of a bounded (suitably scaled) ${{H}^{\infty }}$ calculus for $T$, on every sector containing the positive half-line, is equivalent to the existence of a bounded functional calculus on the Besov algebra $\Lambda _{\infty ,1}^{\alpha }({{\mathbb{R}}^{+}})$. Such an algebra includes functions defined byMikhlin-type conditions and so the Besov calculus can be seen as a result on multipliers for $T$. In this paper, we use fractional derivation to analyse in detail the relationship between $\Lambda _{\infty ,1}^{\alpha }$ and Banach algebras of Mikhlin-type. As a result, we obtain a new version of the quoted equivalence.

In the present note we introduce a straightforward algebraic generalization of inner product spaces, which we appropriately name generalized inner product (GIP) spaces. In the same fashion in which different topologies :an be introduced in inner product spaces, adequate topologies can be introduced in GIP spaces in such a manner that topological vector spaces are obtained. We enumerate and derive some fundamental properties of different topologies in GIP spaces, having primarily in mind their possible later application to quantum physics.

The desirability of having in quantum physics more general structures than Hilbert spaces (in which quantum mechanics is usually formulated) is suggested by Dirac's formalism (2), which deals with “unnormalizable” vectors. Unfortunately, although this formalism is very elegant from the point of view of the facili ty of dealing with its symbolism, it completely lacks in mathematical rigour.

1. Introduction. An incomplete latin rectangle on t symbols σ1 …, σt of size r × s is an r × s matrix in which each cell is occupied by exactly one of the symbols σ1 …, σt in such a way that no symbol occurs more than once in any row or more than once in any column. If r = t or s = t then it is a latin rectangle; if r = s < t it is an incomplete latin square; if r = s = t it is a latin square. The diagonal of a latin square consists of the cells (i, i) (1 ≦ i ≦ t) together with the symbols occupying those cells. Let an allowed sequence of length t be a sequence of length t in which no symbol occurs exactly t – 1 times. Let an allowed diagonal of length t be a diagonal occupied by an allowed sequence.

Starting with a 2-dimensional mod $p$ Galois representation, we construct a deformation to a power series ring in infinitely many variables over the $p$-adics. The image of this representation is full in the sense that it contains $S{{L}_{2}}$ of this power series ring. Furthermore, all ${{\mathbb{Z}}_{p}}$ specializations of this deformation are potentially semistable at $p$.

The following is our main result.

(A) THEOREM. Let (M, g) be a closed connected Einstein space, n = dim M ≧ 2 (with constant scalar curvature R). Let K0 be the lower bound of the sectional curvature. Then either (M, g) is isometrically diffeomorphic to a sphere and the first nonzero eigenvalue ƛ1of the Laplacian fulfils

In 1979 Edgar asked for a characterization of those completely regular Hausdorff topological spaces X which have the property that any Boolean σ-homomorphism from the Baire σ-field of X into the measure algebra of an arbitrary complete probability space can be realized by a measurable point-mapping. Those spaces X will be called homomorphism-compact or, for short, H-compact hereafter. It is wellknown that compact spaces are H-compact (cf. [4], p. 637, Proposition 3.4). We will show that the same is true for strongly measure compact spaces. On the other hand H-compact spaces are easily seen to be real-compact. Since the notions of measure-compactness and liftingcompactness (cf. [3]) also lie between strong measure-compactness and real-compactness it is natural to investigate the relations among these notions. Here the results are mainly negative (cf. Sections 4 and 6). Concerning the structural properties of H-compactness not very much can be said so far (cf. Section 7): it is, for instance, unknown whether the product of two H-compact spaces is again H-compact.

Erdös [1] has shown that, if the square-free numbers in ascending order be denoted by s1, s2, … , sn, … , then for 0 ≦ γ ≦ 2

as x → ∞. In this paper we shall extend this result by proving that the asymptotic formula in fact holds for the wider range 0 ≦ γ ≦ 3.

Similar results have been obtained previously by the author in respect of both the sequence of numbers expressible as the sum of two squares and also sequences of numbers relatively prime to given large integers, although the method used here differs from that of the earlier papers [2; 3].

Let Ω be a bounded open set in Euclidean n-space, En. Let α = (α1, … , an) be an n-tuple of non-negative integers;

and denote by Qm the set ﹛α| 0 ⩽ |α| ⩽ m}. Denote by x = (x1, … , xn) a typical point in En and put

In this paper we establish, under certain circumstances, the existence of weak and classical solutions of the quasi-linear Dirichlet problem

1

In this paper, results obtained by the author for Riemannian Spaces Vn imbedded in Euclidean Spaces EN (3; 4) are extended to Vn imbedded in VN.

The first section is introductory. In §2 the general result is obtained. This is the establishment of a certain dependency among the three basic sets of equations of the Vn with respect to the VN, namely the equations of Gauss, Codazzi and Kuehne.

In 1908, Lindelöf showed that if w = f(x) is a bounded holomorphic function in a sector S: |arg z| < θ1, and if f(z) has an asymptotic value w0 as z tends to ∞ along a half-ray in S; then f(z) tends uniformly to w0 as z tends to ∞o within any sector |arg z| ≦ θ, 0 ≦ θ < θ1. Montel (8) later replaced the condition that f(z) be bounded by the condition that f(z) be meromorphic and omitted three values. The following is an immediate consequence of the Lindelöf-Montel theorem.

THEOREM 1. Let w = f(x) be a function meromorphic in the sector | arg z| < θ1, and let f(z) tend to a value w0 as z → ∞ along the positive real axis.

Let G be a discrete group and let H = L2(G), with norm | |. Let B(H) be the ring of bounded operators on H with the norm

The right regular representation of G on H induces an injection ρ : C[G] → B(H), and W(G) is the closure of the image of ρ in the weak operator topology on B(H) (C = complex numbers). Using ρ, we identify C[G] with its image in W(G).

Using properties of skew-Hamiltonian matrices and classic connectedness results, we prove that the moduli space $M_{\text{ort}}^{0}\left( r,\,n \right)$ of stable rank $r$ orthogonal vector bundles on ${{\mathbb{P}}^{2}}$, with Chern classes $\left( {{c}_{1}},\,{{c}_{2}} \right)\,=\,\left( 0,\,n \right)$ and trivial splitting on the general line, is smooth irreducible of dimension $\left( r-2 \right)n\,-\,\left( _{2}^{r} \right)$ for $r\,=\,n$ and $n\,\ge \,4$, and $r\,=\,n-1$ and $n\,\ge \,8$. We speculate that the result holds in greater generality.

We study a relation between distinction and special values of local invariants for representations of the general linear group over a quadratic extension of $p$-adic fields. We show that the local Rankin–Selberg root number of any pair of distinguished representation is trivial, and as a corollary we obtain an analogue for the global root number of any pair of distinguished cuspidal representations. We further study the extent to which the gamma factor at 1/2 is trivial for distinguished representations as well as the converse problem.

The graphs are undirected, without loops or multiple edges. The edge set E(X) of a graph X is a set of certain unordered pairs [x, y] of distinct elements of the vertex set V(X). For x ϵ V(X) we denote by E(x; X) the edges of X incident with x. A (homo)morphism ϕ : X ⟶ Y is a function from V(X) to V(Y) which preserves edges; thus it induces ϕ# : E(X) ⟶ E(Y) by ϕ# [x, x’] = [ϕx, ϕx’].

Soient $F$ un corps commutatif localement compact non archimédien, $G=GL(n,F)$ pour un entier $n\ge 2$, et $\kappa$ un caractère de ${{F}^{\times }}$ trivial sur ${{\left( {{F}^{\times }} \right)}^{n}}$. On prouve une formule pour les $\kappa$-intégrales orbitales régulières sur $G$ permettant, si $F$ est de caractéristique $>0$, de les relever à la caractéristique nulle. On en déduit deux résultats nouveaux en caractéristique $>0$ : le “lemme fondamental” pour l’induction automorphe, et une version simple de la formule des traces tordue locale d’Arthur reliant $\kappa$-intégrales orbitales elliptiques et caractères $\kappa$-tordus. Cette formule donne en particulier, pour une série $\kappa$-discrète de $G$, les $\kappa$-intégrales orbitales elliptiques d’un pseudo-coefficient comme valeurs du caractère $\kappa$-tordu.

It is a well-known result of Sanov (5) that groups of exponent pk (p prime) satisfy the th Engel congruence (definition below). Recently, an alternative proof of this has been given by Glauberman, Krause, and Struik (3). Bruck (2) has conjectured that such groups satisfy the th Engel congruence. In this note we go some way towards proving this.

It is the purpose of this paper to establish a closer connection between the logarithmic capacity of sets and double trigonometric series. In (9), closed sets of logarithmic capacity zero were established as sets of uniqueness for a particular class of double trigonometric series under circular (C, 1) summability. By slightly changing this class of series but still maintaining closed sets of logarithmic capacity zero as sets of uniqueness, it is shown in this paper that closed sets of positive logarithmic capacity form sets of multiplicity.

We prove ${{L}^{p}}({{\mathbb{T}}^{2}})$ boundedness, $1\,<\,p\,\le \,2$, of variable coefficients singular integrals that generalize the double Hilbert transform and present two phases that may be of very rough nature. These operators are involved in problems of a.e. convergence of double Fourier series, likely in the role played by the Hilbert transform in the proofs of a.e. convergence of one dimensional Fourier series. The proof due to C.Fefferman provides a basis for our method.

Let J1 and J2 be two Jordan algebras with unit elements. We define various tensor products of J1 and J2. The first, which we call the Kronecker product, is the most obvious and is based on the tensor product of the vector spaces. We find conditions sufficient for its existence and for its non-existence. Motivated by the universal mapping property for the tensor product of associative algebras we define, in Section 2, tensor products of J1 and J2 by means of a universal mapping property. The tensor products always exist for special Jordan algebras and need not coincide with the Kronecker product when the latter exists. In Section 3 we construct a more concrete tensor product for special Jordan algebras. Here the tensor product of a special Jordan algebra and an associative Jordan algebra coincides with the Kronecker product of these algebras. We show that this "special" tensor product is the natural tensor product for some Jordan matrix algebras.