Let 0 < wbe a smooth function on a complete Riemannian manifold Mn, and define L = — Δ — ▽ (log w) and Rw =Ric - w -1 Hess w.In this paper we show that if Rw ≥ —nK, (K ≥0), then the positive solutions of (L + ∂/∂t)u —0 satisfy a gradient estimate of the same form as that obtained by Li and Yau ([LY]) when Lis the Laplacian. This is used to obtain a parabolic Harnack inequality, which in turn, yields upper and lower Gaussian estimates for the heat kernel of L.The results obtained are applied to study the LPmapping properties of t→ e-tL μfor measures μ which are α-dimensional in a sense that generalises the local uniform α-dimensionality introduced by R. S. Strichartz ([St2], [St3]).

Let $A$ be an amenable separable ${{C}^{*}}$-algebra and $B$ be a non-unital but $\sigma $-unital simple ${{C}^{*}}$- algebra with continuous scale. We show that two essential extensions ${{\tau }_{1}}$ and ${{\tau }_{2}}$ of $A$ by $B$ are approximately unitarily equivalent if and only if

$$\left[ {{\tau }_{1}} \right]\,=\,\left[ {{\tau }_{2}} \right]\,\text{in}\,KL\left( A,\,M\left( B \right)/B \right).$$

If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL\left( A,\,M\left( B \right)/B \right)$. Using $KL\left( A,\,M\left( B \right)/B \right)$, we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.

Dans ce travail, R désignera toujours un anneau commuta tif ayant un élément unité 1. Les R-algèbres considérées seront supposées associatives. Si A est une R-algèbre, nous supposerons toujours 1 · a = a quel que soit a ∊ A. Si B ⊂ A, nous désignerons par Ann(B) l'ensemble des r ∊ R tels que rB = {0}.

Soit A une R-algèbre commutative (avec ou sans élément unité). Nous désignerons par EA l'ensemble des éléments idempotents de A. Si l'on définit pour e, f ∊ EA,

alors EA devient un treillis distributif relativement complémenté dont 0 est le plus petit élément [2].

Nous nous intéresserons aux R-algèbre commutatives engendrées par leurs éléments idempotents.

In 1972, Lelek introduced the notion of Class (W) in his seminar at the University of Houston [see below for definitions of concepts mentioned here]. Since then there has been much interest in classifying and characterizing continua in Class (W). For example, Cook has a result [5, Theorem 4] which implies that any hereditarily indecomposible continuum is in Class (W) Read [21, Theorem 4] showed that all chainable continua are in Class (W), and Feuerbacher proved the following result:

(1.1) THEOREM [7, Theorem 7]. A non-chainable circle-like continuum is in Class (W) if and only if it is not weakly chainable

In [14, 4.2 and section 6], a covering property (denoted here and in [18] by CP) was defined and studied primarily for the purpose of proving that indecomposability is a Whitney property for the class of chainable continua [14, 4.3].

We prove comparison theorems for the probability of life in a Lipschitz domain between Brownian motion and random walks.

1. Introduction. Our interest here lies in the following theorem:

THEOREM 1. Assume there is defined on Rn (n ≧ 3) a “square-distance” of the form

where (gij) is a given symmetric non-singular matrix over the reals and x = (x1, …, xn), y = (y1, …, yn) ∈ Rn. Assume further that f is a bijection ofRnwhich preserves a given fixed square-distance ρ, i.e. d(x, y) = ρ if and only if d(ƒ(x),ƒ(y)) = ρ. Then (unless ρ = 0 and (gij) is positive or negative definite) ƒ(x) = Lx + ƒ(0), where L is a linear bijection ofRnsatisfying d(Lx, Ly) = ±d(x, y) for all x, y ∈ Rn (the – sign is possible if and only if ρ = 0 and (gij) has signature 0).

Let denote the 2-fold cyclic covering space branched over a link L in S3. We wish to describe an infinite family of prime knots and links in which each member L exhibits two minimal 6-plat representations, where the associated Heegaard splittings of are minimal and inequivalent. Thus each knot or link of that family admits at least two equivalence classes of 6-plat representations which are minimal.

It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szegö polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szegö polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients etc. are derived.

This paper uses the notion of the quantum dimension to obtain new results on the cohomology and representation theory of quantum groups at a root of unity. In particular, we consider the elementary theory of support varieties for quantum groups.

1. Introduction. After establishing a duplication principle (§2) which enables us to derive a group of order 2h with k + 1 involutory generators from any group of order h with k generators, we shall prove the following combination principle :

Let and be two groups, each having k generators, such that the rth generators of and have relatively prime periods for each r from 1 to k.

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute $\mathbb{Z}/2$–Betti numbers of these spaces.

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

In this paper a K-theoretic classification is given of the C*-dynamical systems where An is finite-dimensional. Corresponding to the trivial action is the K-theoretic classification for AF algebras obtained in [3] (also see [1]).

Let $a$ be a natural number greater than 1. Let ${{f}_{a}}\left( n \right)$ be the order of $a\,\bmod \,n$. Denote by $\omega \left( n \right)$ the number of distinct prime factors of $n$. Assuming a weak form of the generalised Riemann hypothesis, we prove the following conjecture of Erdös and Pomerance:

The number of $n\,\le \,x$ coprime to a satisfying

$$\alpha \le \frac{\omega \left( {{f}_{a}}\left( n \right) \right)-{{\left( \log \,\log \,n \right)}^{2}}/2}{{{\left( \log \,\log \,n \right)}^{3/2}}/\sqrt{3}}\le \beta $$

is asymptotic to $\left( \frac{1}{\sqrt{2\pi }}\int_{\alpha }^{\beta }{{{e}^{-{{t}^{2}}/2}}}dt \right)\frac{x\phi \left( a \right)}{a}$ as $x$ tends to infinity.

Consider the second order nonlinear differential equation

y" + a(t)f(y) = 0

where a(t) ∈ C[0,∞),f(y) GC1 (-∞, ∞),ƒ'(y) ≥ 0 and yf(y) > 0 for y ≠ 0. Furthermore, f(y) also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function f(y) = |γ|γ sgny with γ > 1 and 0 < γ < 1 known as the Emden-Fowler case. The coefficient a(t) is allowed to be negative for arbitrarily large values of t. Oscillation criteria involving integral averages of a(t) due to Wintner, Hartman, and recently Butler, Erbe and Mingarelli for the linear equation are shown to remain valid for the general equation, subject to certain nonlinear conditions on f(y). In particular, these results are therefore valid for the Emden-Fowler equation.

In 1851 Prouhet (2) stated that any bk+1 consecutive positive integers (b a positive integer ⩾ 2) can be separated into b sets o, …, b-1 each with bk members in such a way that where designates the sum of the tth powers of the numbers in j. In 1947 Lehmer (1) generalized this result.

We shall consider methods of summation A, B, … defined by matrices of real elements (amn), (bmn), (m, n = 1, 2, …) which are regular, that is, have the three well-known properties of Toeplitz (4, p. 43). A method A is said to be core-consistent with the methodBfor bounded sequences if the A-core (3, p. 137; and 4, p. 55) of each real bounded sequence is contained in its B-core. B is totally included in A, B ≪A, if each real sequence which is B-summable to a definite limit (this limit may be finite or infinite of a definite sign) is also A-summable to the same limit.

As an analog of a recently established minimax equality for directed graphs [1], I. Simon has suggested that the following be investigated.

1.1. For a finite acyclic directed graph G, a minimum collection of directed coboundaries whose union is the edge set of G has cardinality equal to that of a maximum strong matching of G.

This minimax equality is here proved, using a characterization of a maximum strong matching of an acyclic graph as the set of edges of a longest directed path in the graph.

The terms employed in the above theorem are defined as follows. Let G be a finite directed graph with vertex set VG and edge set eG

In this article, we examine the relationship between the notions of thick groups and that of essentially finitely indecomposable (e.f.i.) groups. These groups are indecomposable objects in the quotient category of the category of abelian groups induced by the Serre-class of bounded groups.

We show that if $u$ and $v$ are any two unitaries in a unital ${{C}^{*}}$–algebra such that $\left\| uv\,-\,vu \right\|\,<\,2$ and $uv{{u}^{*}}{{v}^{*}}$ commutes with $u$ and $v$, then the ${{C}^{*}}$–subalgebra ${{A}_{u,v}}$ generated by $u$ and $v$ is isomorphic to a quotient of some rotation algebra ${{A}_{\theta }}$, provided that ${{A}_{u,v}}$ has a unique tracial state. We also show that the Exel trace formula holds in any unital ${{C}^{*}}$–algebra. Let $\theta \,\in \,\left( -1/2,\,1/2 \right)$ be a real number. For any $\in \,>\,0$, we prove that there exists $\delta \,>\,0$ satisfying the following: if $u$ and $v$ are two unitaries in any unital simple ${{C}^{*}}$–algebra $A$ with tracial rank zero such that

$$\left\| uv\,-\,{{e}^{2\pi i\theta }}vu \right\|\,<\,\delta \,\,\,\text{and}\,\,\frac{1}{2\pi i}\tau \left( \log \left( uv{{u}^{*}}{{v}^{*}} \right) \right)\,=\,\theta ,$$

for all tracial states $\tau$ of $A$, then there exists a pair of unitaries $\widetilde{u}$ and $\widetilde{v}$ in $A$ such that

$$\widetilde{u}\widetilde{v}\,=\,{{e}^{2\pi i\theta }}\widetilde{v}\widetilde{u},\,\,\,\,\,\,\,\left\| u\,-\,\widetilde{u} \right\|\,<\,\in \,\,\,\text{and}\,\,\left\| v\,-\,\widetilde{v} \right\|\,<\,\in.$$