Let A be a simple C*-algebra with real rank zero, stable rank one and weakly unperforated K0(A) of countable rank. We show that a monomorphism Φ: C(S2) → A can be approximated pointwise by homomorphisms from C(S2) into A with finite dimensional range if and only if certain index vanishes. In particular,we show that every homomorphism ϕ from C(S2) into a UHF-algebra can be approximated pointwise by homomorphisms from C(S2) into the UHF-algebra with finite dimensional range.As an application, we show that if A is a simple C*-algebra of real rank zero and is an inductive limit of matrices over C(S2) then A is an AF-algebra. Similar results for tori are also obtained. Classification of Hom (C(X), A) for lower dimensional spaces is also studied.

M. Sion and T. Traynor investigated ([15]-[17]), measures and integrals having values in topological groups or semigroups. Their definition of integrability was a modification of Phillips-Rickart bilinear vector integrals, in locally convex topological vector spaces.

The purpose of this paper is to develop a good notion of an integration process in partially ordered groups, based on their order structure. The results obtained generalize some of the results of J. D. M. Wright ([19]-[22]) where the measurable functions are real-valued and the measures take values in partially ordered vector spaces.

Let if be a σ-algebra of subsets of T, X a lattice group, Y, Z partially ordered groups and m : H → F a F-valued measure on H. By F(T, X), M(T, X), E(T, X) and S(T, X) are denoted the lattice group of functions with domain T and with range X, the lattice group of (H, m)-measurable functions of F(T, X) and the lattice group of (H, m)-elementary measurable functions of F(T, X) and the lattice group of (H, m)-simple measurable functions of F(T, X) respectively.

In this work, we obtain a local maximum principle along the Ricci flow $g(t)$ under the condition that $\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$ for $t>0$ for some constant ${\alpha }>0$ . As an application, we will prove that under this condition, various kinds of curvatures will still be nonnegative for $t>0$ , provided they are non-negative initially. These extend the corresponding known results for Ricci flows on compact manifolds or on complete noncompact manifolds with bounded curvature. By combining the above maximum principle with the Dirichlet heat kernel estimates, we also give a more direct proof of Hochard’s [15] localized version of a maximum principle by Bamler et al. [1] on the lower bound of different kinds of curvatures along the Ricci flows for $t>0$ .

Let ${{\mathcal{H}}_{n}}$ be the real linear space of $n\,\times \,n$ complex Hermitian matrices. The unitary (similarity) orbit $\mathcal{U}\left( C \right)$ of $C\,\in \,{{\mathcal{H}}_{n}}$ is the collection of all matrices unitarily similar to $C$. We characterize those $C\,\in \,{{\mathcal{H}}_{n}}$ such that every matrix in the convex hull of $\mathcal{U}\left( C \right)$ can be written as the average of two matrices in $\mathcal{U}\left( C \right)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}\left( C \right)$, the convexity of images of $\mathcal{U}\left( C \right)$ under linear transformations, and some related questions concerning the joint $C$-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

The nonlinear Sturm-Liouville equation

$$-{{\left( p{y}' \right)}^{\prime }}\,+\,qy\,=\,\text{ }\!\!\lambda\!\!\text{ }\left( 1-f \right)ry\,on\,[0,\,1]$$

is considered subject to the boundary conditions

$$\left( {{\text{a}}_{j}}\text{ }\lambda \text{ }\text{+}{{\text{b}}_{j}} \right)y\left( j \right)=\left( {{c}_{j}}\text{ }\lambda \text{ }+{{d}_{j}} \right)\left( p{y}' \right)\left( j \right),j=0,1$$

Here ${{\text{a}}_{0}}\,=\,0\,=\,{{c}_{0}}$ and $p,\,r\,>\,0$ and $q$ are functions depending on the independent variable $x$ alone, while $f$ depends on $x,\,y\,and\,{y}'$. Results are given on existence and location of sets of $(\lambda ,\,y)$ bifurcating from the linearized eigenvalues, and for which $y$ has prescribed oscillation count, and on completeness of the $y$ in an appropriate sense.

Let Ω be a metric space with metric ρ, let C be a class of closed sets from Ω and let τ be a non-negative real-valued set function on C. We assume that the empty set ϕ is in C and that τ(I)= 0 if and only if I = ϕ. For each set A in Ω, we define φ(A), 0 ≤ φ(A) ≤ ∞ by:

where the infimum is taken for all possible countable collections of sets I(n) from C such that:

and the diameter of I(n), d(I(n)), is less than ∈ for every n.

L'objet de cet article est essentiellement de poursuivre la généralisation entreprise par Sabidussi [4] du concept d'ensemble primitif de sommets d'un graphe, introduit par Halin [1].

Dans le présent article, surface de Riemann ou plus simplement surface désignera une variété analytique complexe R, connexe, sans bord et de dimension 1. En terme d'une variable locale, une fonction harmonique h, définie sur R, possédant une singularité isolée au point z0, peut s'écrire comme la somme d'une fonction harmonique

et d'une partie singulière

avec αn, βn ∈ C, γ ∈ R; les deux séries sont supposées convergentes, la première pour |z – z0| suffisamment petit, la seconde pour tout z ≠ z0. Alors h(z) = u(z) + s(z). Nous dirons que la singularité de h est non-essentielle si s(z) est de la forme

et newtonienne ou logarithmique si elle s'écrit

Let A1, A2,… , An be a finite collection of subsets (not necessarily distinct) of a set A. By a transversal of A1, A2,… , An we shall mean a set of n distinct elements a1, a2,… , an of A such that, for some permutation i1 i2, … , in of the integers 1, 2, … , n,

More generally, we shall say that the set {a1, a2, … , ar}, (r ≤ n) is a partial transversal oi A1, A2, … An of length r if (i) a1, a2, … , ar are distinct elements of A and (ii) there exists a set of distinct integers i1, i2… , irsuch that

A well-known theorem of P. Hall (2) states that the sets A1, A2… , An have a transversal (of length n) if, and only if, every k of them contain collectively at least k distinct elements (k = 1, 2, … , n).

Consider a finite sequence of linear contractions ${{S}_{j}}(x)\,=\,\rho x\,+{{d}_{j}}$ and probabilities $pj\,>\,0$ with $\Sigma \,{{p}_{j}}\,=\,1$. We are interested in the self-similar measure $\mu \,=\,\Sigma \,{{p}_{j}}\,\mu \,{}^\circ \,s_{j}^{-1}$, of finite type. In this paper we study the multi-fractal analysis of such measures, extending the theory to measures arising from non-regular probabilities and whose support is not necessarily an interval.

Under some mild technical assumptions, we prove that there exists a subset of supp $\mu $ of full $\mu $ and Hausdorff measure, called the truly essential class, for which the set of (upper or lower) local dimensions is a closed interval. Within the truly essential class we show that there exists a point with local dimension exactly equal to the dimension of the support. We give an example where the set of local dimensions is a two element set, with all the elements of the truly essential class giving the same local dimension. We give general criteria for these measures to be absolutely continuous with respect to the associated Hausdorff measure of their support, and we show that the dimension of the support can be computed using only information about the essential class.

To conclude, we present a detailed study of three examples. First, we show that the set of local dimensions of the biased Bernoulli convolution with contraction ratio the inverse of a simple Pisot number always admits an isolated point. We give a precise description of the essential class of a generalized Cantor set of finite type, and show that the $k$-th convolution of the associated Cantor measure has local dimension at $x\,\in \,(0,\,1)$ tending to 1 as $k$ tends to infinity. Lastly, we show that within a maximal loop class that is not truly essential, the set of upper local dimensions need not be an interval. This is in contrast to the case for finite type measures with regular probabilities and full interval support.

Let r(T) denote the spectral radius of the operator T acting on a complex Hilbert space H. Let S be a multiplicative semigroup of operators on H. We say that r is permutable on 𝓢 if r(ABC) = r(BAC), for every A,B,C ∈ 𝓢. We say that r is submultiplicative on 𝓢 if r(AB) ≤ r(A)r(B), for every A, B ∈ 𝓢. It is known that, if r is permutable on 𝓢, then it is submultiplicative. We show that the converse holds in each of the following cases: (i) 𝓢 consists of compact operators (ii) 𝓢 consists of normal operators (iii) 𝓢 is generated by orthogonal projections.

In this paper, we study the boundary quotient $\text{C}^{\ast }$-algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{\ast }$-algebra of the standard product of $k$ odometers over $n_{i}$-letter alphabets $(1\leqslant i\leqslant k)$ is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_{i}:1\leqslant i\leqslant k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph $\text{C}^{\ast }$-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz–Pimsner $\text{C}^{\ast }$-algebra is isomorphic to the boundary quotient $\text{C}^{\ast }$-algebra. Some relations between the boundary quotient $\text{C}^{\ast }$-algebra and the $\text{C}^{\ast }$-algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.

All spaces considered are completely regular. C*(X) denotes the set of bounded continuous real-valued functions in X. A subspace S of X is called C*-embedded in X if for every f ∈ C*(S) there is ϕ ∈ C*(X) with ϕ⥤S = f.

A space X is called almost compact if |βX – X| ≦ 1; basically disconnected if every cozero-set has open closure; extremally disconnected if every open set has open closure; an F-space if every cozero-set is C*-embedded; small if |C*(X)| = 2ω; and weakly Lindelöf if every open cover has a subfamily with and ⋃ dense. A point p of a space X is called a P-point of X if every Gδ-set in X which contains p is a neighborhood of p.

ω(X) denotes the weight of X.

We introduce some non-linear transformations from the set of Hausdorff moment sequences into itself; among them is the one defined by the formula: $T\left( {{\left( {{a}_{n}} \right)}_{n}} \right)\,=\,1/\left( {{a}_{0}}\,+\cdots +\,{{a}_{n}} \right).$ We give some examples of Hausdorff moment sequences arising from the transformations and provide the corresponding measures: one of these sequences is the reciprocal of the harmonic numbers ${{\left( 1+1/2\,+\cdots +\,1/\left( n+1 \right) \right)}^{-1}}.$

Fix a positive integer r. Let AF be the ring of adeles of a number field F. For a parabolic subgroup P of SLr, we fix a Levi decomposition P = MN, and we let

Let be the Weyl group of . It follows from a recent work of James Arthur [1,2] (also cf. [3]) that, among the terms appearing in the trace formula for SLr(AF), coming from the Eisenstein series, are those which are a constant multiple (depending only on M and w) of

1

where σ is a cusp form on M(AF) satisfying wσ ≅ σ,

and

in the notation of [2, 3]).

Forelli in [1] has described the isometries of Hp(U) into Hp(U) for p≠2, 0 < p < ∞. We shall extend his methods to characterize the isometries of Hp(Un) onto Hp(Un).

The notation we shall use can be found in Rudin [3].

Let II represent a permutation that induces a map on functions of n complex variables by

Clearly II is an isometry of Hp(Un) onto Hp(Un).

In this paper we give some new characterizations of the projection property in Archimedean Riesz spaces. Our approach primarily explores the interrelationships between such things as the band structure or the prime ideal structure of an Archimedean vector lattice and corresponding structures of its Dedekind completion. Our results show that, in general, there is a ‘strong“ relationship if and only if the original vector lattice has the projection property. The main result of this paper is Theorem 2.6 which both summarizes and extends all of the results we obtain prior to it.

A connected Lie group G is generated by one-parameter subgroups exp(tX1), … , exp(tXk) if every element of G can be written as a finite product of elements chosen from these subgroups. This happens just in case the Lie algebra of G is generated by the corresponding infinitesimal transformations X1, … , Xk ; indeed the set of all such finite products is an arcwise connected subgroup of G, and hence a Lie subgroup by Yamabe's theorem [9]. If there is a positive integer n such that every element of G possesses such a representation of length at most n, G is said to be uniformly finitely generated by the one-parameter subgroups.

We prove dispersive and Strichartz inequalities for the solution of the wave equation related to the full Laplacian on the Heisenberg group, by means of Besov spaces defined by a Littlewood–Paley decomposition related to the spectral resolution of the full Laplacian. This requires a careful analysis due also to the non-homogeneous nature of the full Laplacian. This result has to be compared to a previous one by Bahouri, Gérard and Xu concerning the solution of the wave equation related to the Kohn Laplacian.

Given a point set E on an open Riemann surface V we denote by H(E) the space of functions u harmonic in open sets O(u) containing E. Let V0 be a regular region of V with border α, and consider restrictions f to α of functions in H(α). For V1 = V — ⊽0, an operator L from H(α) to H(⊽1) is, by definition, normal if

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