In this paper we shall work in the category of countable CW-complexes with base point and base point preserving maps. All homotopies shall also respect base points. For simplicity, we shall frequently use the same symbol for a map and its homotopy class. Given spaces X, Y, we denote the set of homotopy classes of maps from X to Y by [X, Y]. We have an isomorphism τ: [∑X, Y] → [X, Ω Y] taking each map to its adjoint, where ∑ is the suspension functor and Ω is the loop functor. We shall denote τ(1 ∑x) by e′ and τ-1(1Ωx) by e.

An n-knot (Sn+2, Sn) is amphicheiral if there is an orientation reversing autohomeomorphism of Sn+2 leaving Sn invariant as a set. It is invertible if there is an orientation preserving autohomeomorphism of Sn+2 whose restriction to Sn is an orientation reversing autohomeomorphism of Sn onto itself. In 1961 Fox [8, Problem 35] asked if there exist non-amphicheiral locally flat 2-knots. We will prove the following THEOREM 1. For any integer n there are smooth n-knots which are neither amphicheiral nor invertible.

A knot (Sn+2, Sn) is + amphicheiral (resp. —amphicheiral) if there is an orientation reversing autohomeomorphism f of Sn+2 leaving Sn invariant such that f| Snpreserves (resp. reverses) orientation.

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).

The main purpose of this note is to prove a special case of the following conjecture.

Conjecture. If F is holomorphic on the unit ball Bn in Cn and has positive real part, then F is in Hp(Bn) for 0 < p < ½(n + 1).

Here Hp(Bn) (0 < p < ∞) denote the usual Hardy spaces of holomorphic functions on Bn. See below for definitions. We remark that the conjecture is known for 0 < p < 1 and that some evidence for it already exists in the literature; for example [1, Theorems 3.11 and 3.15] where it is shown that a particular extreme element of the convex cone of functions

is in Hp(B2) for 0 < p < 3/2.

In [5] Pall denned a partitioning of a quadratic space over a field of characteristic not 2 to be a collection of disjoint (except for ﹛0﹜ ) maximal totally isotropic subspaces whose union formed the set of isotropic vectors. Clearly isometric quadratic spaces simultaneously do or do not have partitionings. Pall exhibited the existence of partitionings for the spaces associated with

over formally real fields for n = 1, 2, 4, 8 and over Z/(p), p prime, for n = 1, 2. Using the latter, he was able to find a new proof for Jacobi's formula for the number of representations of a positive integer as the sum of four integral squares.

In this article, we are concerned with the tempered fractional parabolic problem

$$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$

where $-\left (\Delta +\lambda \right )^{\frac {\alpha }{2}}$ is a tempered fractional operator with $\alpha \in (0,2)$ and $\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators.

In a recent paper (5), the author has established the Euclidean spaces of least dimension in which the topological products of finite collections of k-cell-like continua can be embedded. Specifically, it was shown that, for each pair of positive integers k and n, the topological product of any collection of nk-cell-like continua can be embedded in Euclidean space of dimension k(n + 1). This result includes a theorem of Bennett (1) that the topological product of any finite collection of n snakelike continua can be embedded in Euclidean space of dimension n + 1.

B. Banaschewski [1] has produced a counterexample to [2, Theorem 6]. As noted in [1, Remark 1], our error occurs in the final paragraph of the purported proof of [2, Theorem 6], for P need not be a subpresheaf of I. Accordingly, it remains an open problem to find an analogue of [2, Proposition 1] in the context of Boolean spaces. We hope that attacks on this problem will be facilitated by the (valid) initial three paragraphs of the argument given for [2, Theorem 6].

The following alterations to [2] are in order. Example 5, being a corollary of Theorem 6, remains doubtful, although the special case noted on pp. 1034-1035 is not affected. In Corollary 7, the assertion that j preserves infectives remains doubtful, although the proof for divisibility of j(M)(G) is valid.

A quantization of a fixed classical mechanical system is firstly an association between quantum mechanical observables (preferably self-adjoint operators on Hilbert space) and classical mechanical observables (i.e. real-valued functions on phase space). Secondly, a quantization should permit an interpretation of the correspondence principle that ‘classical mechanics is the limit of quantum mechanics as Planck's constant approaches zero'. With these two underlying precepts, Section 2 states the four basic requirements, I to IV, of a quantization along with an additional requirement V that characterizes the subclass of special quantizations.

Let G be a separable locally compact group (separable in the sense that the topology of G has a countable base). Let S be a standard Borel space on which G acts on the right such that:

(1) s · g1g2 = (s · g1) · g2;

(2) s · e = s;

(3) (s, g) → s · g is a Borel function from S × G to S.

If μ is a Borel measure on S, let μg be the Borel measure on S defined by μg(E) = μ(E · g).

Let μ be a Borel measure on S which is quasi-invariant under the action of G; i.e., μg and μ are absolutely continuous (g ∈ G). The triple (G, S, μ) is called a dynamical system [11; 8].

Consider the following general problem. Let (G, S, μ) be a dynamical system.

Let F be a local field of characteristic zero, with q elements in its residue field, ring of integers uniformizer ωF and maximal ideal . Let GF = GL2(F). We fix Haar measures dg and dz on GF and ZF, the centre of GF, so that

meas(K) = meas

where K = GL2() is a maximal compact subgroup of GF. If T is a torus in GF we take dt to be the Haar measure on T such that

means(TM)=1

where TM denotes the maximal compact subgroup of T.

If X is a space and , the kth deleted product space Xk* of X is the topological product X ⨯ X ⨯ . . . ⨯ X of copies of X minus the set of all points of the form (x, x, . . . , x), where x ∊ X. In (4), the author shows that the homology groups of Xk*, where X is a tree, produce as much information about trees as counting the orders of vertices.

Two examples of topological embeddings of S2 in S4 are constructed. The first has the unusual property that the fundamental group of the complement is isomorphic to the integers while the second homotopy group of the complement is nontrivial. The second example is a non-locally flat embedding whose complement exhibits this property locally.

Two theorems are proved. The first answers the question of just when good π1 implies the vanishing of the higher homotopy groups for knot complements in S4. The second theorem characterizes local flatness for 2-spheres in S4 in terms of a local π1 condition.

We consider the Randers space $({{V}^{n}},\,{{F}_{b}})$ obtained by perturbing the Euclidean metric by a translation, ${{F}_{b}}\,=\,\alpha \,+\,\beta $, where $\alpha $ is the Euclidean metric and $\beta $ is a 1-form with norm $b,\,0\,\le \,b\,<\,1$. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces $({{V}^{3}},\,{{F}_{b}})$ of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when $b\,=\,0$. It also reduces to the equation that characterizes the minimal rotational surfaces in $({{V}^{3}},\,{{F}_{b}})$ when $H\,=\,0$, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for $0\,<\,b\,<\,\frac{\sqrt{3}}{3}$. Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical systemand by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.

By employing the external Kasparov product, in [18], Hawkins, Skalski, White, and Zacharias constructed spectral triples on crossed product C$^\ast $-algebras by equicontinuous actions of discrete groups. They further raised the question for whether their construction turns the respective crossed product into a compact quantum metric space in the sense of Rieffel. By introducing the concept of groups separated with respect to a given length function, we give an affirmative answer in the case of virtually Abelian groups equipped with certain orbit metric length functions. We further complement our results with a discussion of natural examples such as generalized Bunce-Deddens algebras and higher-dimensional noncommutative tori.

We show that if $m,n\ge 0,\lambda >1$, and $R$ is a rational function with numerator, denominator of degree $\le m,n$, respectively, then there exists a set $S\subset [0,1]$ of linear measure $\ge \,\frac{1}{4}\,\exp \left( -\frac{13}{\log \,\text{ }\!\!\lambda\!\!\text{ }} \right)$ such that for $r\in S$,

1

$$_{|z|=r\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,|z|=r}^{\max |R(z)|/min|R(z)|\le {{\lambda }^{m+n}}.}$$

Here, one may not replace $\frac{1}{4}\exp (-\frac{13}{\log \lambda })$ by $\exp (-\frac{2-\varepsilon }{\log \lambda })$, for any $\varepsilon >0$. As our motivating application, we prove a convergence result for diagonal Padé approximants for functions meromorphic in the unit ball.

Let $E/F$ be a quadratic extension of number fields. In this paper, we show that the genus formula for Hilbert kernels, proved by M. Kolster and A. Movahhedi, gives the 2-rank of the Hilbert kernel of $E$ provided that the 2-primary Hilbert kernel of $F$ is trivial. However, since the original genus formula is not explicit enough in a very particular case, we first develop a refinement of this formula in order to employ it in the calculation of the 2-rank of $E$ whenever $F$ is totally real with trivial 2-primary Hilbert kernel. Finally, we apply our results to quadratic, bi-quadratic, and tri-quadratic fields which include a complete 2-rank formula for the family of fields $\mathbb{Q}(\sqrt{2},\sqrt{\delta )}$ where $\delta $ is a squarefree integer.

Let p denote the sequence {pn} and set wherever this series converges. (Where no limits are stated, sums are throughout to be taken from n = 0 to n = ∞.) We use a similar notation with other letters in place of p. Given two sequences p, q, the convolution p*q is defined by

it is familiar, and easily verified, that the operation of convolution is commutative and associative. We write Pn = (p*l)n (where 1 denotes the sequence {1} ), and take P-1 to mean 0. If, for all n ≦ 0, Pn ≠ 0, then we define the Nörlund mean (N, p) of the sequence s as σn, where

and (σ-1 = 0. If σn → λ as n → ∞, then 5 is said to be limitable (N, p) to the number λ

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).