A one-to-one correspondence is described between the set ${\cal J}(m)$ of numerical semigroups with multiplicity $m$ and the set of non-negative integer solutions of a system of linear Diophantine inequalities. This correspondence infers in ${\cal J}(m)$ a semigroup structure and the resulting semigroup is isomorphic to a subsemigroup of ${\bb N}^{m-1}$ . Finally, this result is particularized to the symmetric case.

Let g [ratio ] M → N be a quasiconformal harmonic diffeomorphism between noncompact Riemann surfaces M and N. In this paper we study the relation between the map g and the complex structures given on M and N. In the case when M and N are of finite analytic type we derive a precise estimate which relates the map g and the Teichmüller distance between complex structures given on M and N. As a corollary we derive a result that every two quasiconformally related finitely generated Kleinian groups are also related by a harmonic diffeomorphism. In addition, we study the question of whether every quasisymmetric selfmap of the unit circle has a quasiconformal harmonic extension to the unit disk. We give a partial answer to this problem. We show the existence of the harmonic quasiconformal extensions for a large class of quasisymmetric maps. In particular it is proved that all symmetric selfmaps of the unit circle have a unique quasiconformal harmonic extension to the unit disk.

First, the strong Stone–Weierstrass conjecture for postliminal ${\rm JB}^*$ -triples is proved. This is combined with the recent classification of ${\rm JB}^*$ -triples $A$ for which $\partial_e(A^*_1)$ is ${\rm weak}^*$ dense in $A^*_1$ , and the weak Stone–Weierstrass theorem for ${\rm JB}^*$ -triples is obtained. If $B$ is a ${\rm JB}^*$ -subtriple of a ${\rm JB}^*$ -triple $A$ such that $B$ separates $\overline{\partial_e(A^*_1)}\cup\{0\}$ , then $B=A$ .

We consider the Sturm–Liouville equation \renewcommand{\theequation}{1.\arabic{equation}} \begin{equation} y^{\prime\prime}(x)+\{\lambda - q(x)\}y(x) = 0\quad (0 \le x < \infty) \end{equation} with a boundary condition at $x = 0$ which can be either the Dirichlet condition \begin{equation} y(0) = 0 \end{equation} or the Neumann condition \begin{equation} y^\prime(0) = 0. \end{equation} As usual, $\lambda$ is the complex spectral parameter with $0 \le \arg \lambda < 2\pi$ , and the potential $q$ is real-valued and locally integrable in $[0, \infty)$ .

Some almost sharp sufficient conditions of oscillation and nonoscillation are obtained for the superlinear delay differential equation

A linear autonomous system of differential equations $\dot{x}=Ax$ can be transformed to its Jordan normal form, that is, the transformed system is in block diagonal form and the blocks correspond to different eigenvalues. This result is generalized to arbitrary nonautonomous linear systems $\dot{x}=A(t)x$ with a locally integrable matrix function $A:{\bb R}\longrightarrow {\bb R}^{N\times N}$ .

Second-order elliptic operators realized under Dirichlet boundary conditions in bounded domains are shown to generate strongly continuous semigroups in the topology of uniform convergence. The stochastic representation and integral representation with kernel for the strongly continuous semigroups are given. Furthermore, several useful formulae for the principal eigenvalue of second-order elliptic operators are obtained.

The paper studies the existence of multiple solutions to the following p-Laplacian type elliptic problem (p > 1):

where Ω is a bounded domain in ℝN(N [ges ] 1) with smooth boundary ∂Ω, and f(x, u) goes asymptotically in u to [mid ]u[mid ]p−2u at infinity. It is well known that this kind of nonlinear term creates some difficulties in the application of the mountain pass theorem because of the lack of an Ambrosetti–Rabinowitz type superlinear condition on f(x, u). An improved mountain pass theorem is used to prove that the above problem possesses multiple solutions under some natural conditions on f(x, u), and some known results are generalized.

Fuglede's conjecture states that a set $\Omega\subset{\mathbf R}^n$ tiles ${\mathbf R}$ by translations if and only if $L^2(\Omega)$ has an orthogonal basis of exponentials. New partial results are obtained supporting the conjecture in dimension 1.

It is determined which Bloch-type conditions on a function $f\in \bigcap_{0 < p <\infty}H^p$ ensure that $f\in {\rm BMOA}$ .

Multilinear commutators with vector symbol $\vec{b}=(b_1,\ldots,b_m)$ defined by \[ T_{\vec{b}}(f)(x)=\int_{{\bb R}^n}\Bigg[\prod\limits^m_{j=1}(b_j(x)-b_j(y))\Bigg]K(x,y)f(y)dy \] are considered, where $K$ is a Calderón–Zygmund kernel. The following a priori estimates are proved for $w\in A_\infty$ . For $0 < p < \infty$ , there exists a constant $C$ such that \[ \|\dot{T}_{{\vec{b}}}(f)\|_{L^P(w)}\le C\|\vec{b}\|\|M_{L(\log\,L)^{1/r}}(f)\|_{L^P(w)} \] and \[ \sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:|T_{\vec{b}}f(y)|>t\})\le C\sup_{t>0}\frac{1}{\Phi(\frac{1}{t})}w(\{y\in{\bb R}^n:M_{L(\log\,L)^{1/r}}(\|\vec{b}\|f)(y)>t\}), \] where \begin{eqnarray*} &\|\vec{b}\|=\prod\limits^m_{j=1}\|b_j\|_{osc_{\exp L}^r j},\\ &\Phi(t)=t\log^{1/r}(e+t),\quad \frac{1}{r}=\frac{1}{r_1}+\cdots+\frac{1}{r_m}, \end{eqnarray*} and $M_{L(\log L)^{\alpha}}$ is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman.

As a corollary, it is deduced that the operators $T_{\vec{b}}$ are bounded on $L^p(w)$ when $w\in A_p$ , and that they satisfy corresponding weighted $L(\log\,L)^{1/r}$ -type estimates with $w\in A_1$ .

Let $F$ be a germ of a holomorphic function at $0$ in ${\bb C}^{n+1}$ , having $0$ as a critical point not necessarily isolated, and let $\tilde{X}:= \sum^n_{j=0} X^j(\partial/\partial z_j)$ be a germ of a holomorphic vector field at $0$ in ${\bb C}^{n+1}$ with an isolated zero at $0$ , and tangent to $V := F^{-1}(0)$ . Consider the ${\cal O}_{V,0}$ -complex obtained by contracting the germs of Kähler differential forms of $V$ at $0$ \renewcommand{\theequation}{0.\arabic{equation}} \begin{equation} \Omega^i_{V,0}:=\frac{\Omega^i_{{\bb C}^{n+1},0}}{F\Omega^i_{{\bb C}^{n+1},0}+dF\wedge{\Omega^{i-1}}_{{\bb C}^{n+1}},0} \end{equation} with the vector field <formula form="inline" disc="math" id="frm14"><formtex notation="AMSTeX"> $X:=\tilde{X}|_V$ on $V$ : \begin{equation} 0\longleftarrow {\cal O}_{V,0} {\buildrel X\over\longleftarrow}\,\Omega_{V,0}^1\,{\buildrel X\over\longleftarrow}\, \cdots \,{\buildrel X\over\longleftarrow}\, \Omega_{V,0}^n\, {\buildrel X\over\longleftarrow}\, \Omega_{V,0}^{n+1}\longleftarrow 0. \end{equation}

The problem of studying the largest rearrangement invariant space of functions with almost everywhere convergent Fourier series is considered. A quasi-Banach space QA of functions is defined with almost everywhere convergent Fourier series that strictly contains Antonov and Soria's spaces ([Lscr ] log [Lscr ] log log log [Lscr ] and B*ϕ1([ ])).

Let $G$ be a separable locally compact group and let $\hat{G}$ be its dual space with Fell's topology. It is well known that the set $P(G)$ of continuous positive-definite functions on $G$ can be identified with the set of positive linear functionals on the group $C^*$ -algebra $C^*(G)$ . We show that if $\pi$ is discrete in $\hat{G}$ , then there exists a nonzero positive-definite function $\phi_{\pi}$ associated with $\pi$ such that $\phi_{\pi}$ is a $w^*$ -strongly exposed point of $P(G)_0$ , where $P(G)_0=\{f\in p(G):f(e)\le 1\}$ . Conversely, if some nonzero positive-definite function $\phi_\pi$ associated with $\pi$ is a $w^*$ -strongly exposed point of $P(G)_0$ , then $\pi$ is isolated in $\hat{G}$ . Consequently, $G$ is compact if and only if, for every $\pi\in\hat{G}$ , there exists a nonzero positive-definite function associated with $\pi$ that is a $w^*$ -strongly exposed point of $P(G)_0$ . If, in addition, $G$ is unimodular and $\pi\in\hat{G}_\rho$ , then $\pi$ is isolated in $\hat{G}_{\rho}$ if and only if some nonzero positive-definite function associated with $\pi$ is a $w^*$ -strongly exposed point of $P_{\rho}(G)_0$ , where $\rho$ is the left regular representation of $G$ and $\hat{G}_{\rho}$ is the reduced dual space of $G$ . We prove that if $B_{\rho}(G)$ has the Radon–Nikodym property, then the set of isolated points of $\hat{G}_{\rho}$ (so square-integrable if $G$ is unimodular) is dense in $\hat{G}_{\rho}$ . It is also proved that if $G$ is a separable SIN-group, then $G$ is amenable if and only if there exists a closed point in $\hat{G}_{\rho}$ . In particular, for a countable discrete non-amenable group $G$ (for example the free group $F_2$ on two generators), there is no closed point in its reduced dual space $\hat{G}_{\rho}$ .

Periodic neutral functional differential equations are considered. Sufficient conditions for existence, uniqueness and global attractivity of periodic solutions are established by combining the theory of monotone semiflows generated by neutral functional differential equations and Krasnosel'skii's fixed-point theorem. These results are applied to a concrete neutral functional differential equation that can model single-species growth, the spread of epidemics, and the dynamics of capital stocks in a periodic environment.

A shadow of a subset A of ℝn is the image of A under a projection onto a hyperplane. Let C be a closed nonconvex set in ℝn such that the closures of all its shadows are convex. If, moreover, there are n independent directions such that the closures of the shadows of C in those directions are proper subsets of the respective hyperplanes then it is shown that C contains a copy of ℝn−2. Also for every closed convex set B ‘minimal imitations’ C of B are constructed, that is, closed subsets C of B that have the same shadows as B and that are minimal with respect to dimension.

The closed 3-manifolds of constant positive curvature were classified long ago by Seifert and Threlfall. Using well-known information about the orthogonal group O(4), their full isometry groups Isom(M) are calculated. It is determined which elliptic 3-manifolds admit Seifert fiberings that are invariant under all isometries, and it is verified that the inclusion of Isom(M) to Diff(M) is a bijection on path components.

The paper first gives sufficient conditions on the critical points and the Schwarzian derivative of a real rational function $R$ such that the Julia set of $R$ is $\bar{{\bb C}}$ . Further, it is shown that under mild conditions on another real rational function $\tilde{R}$ with possibly non-empty Fatou set, the Julia set of $\tilde{R} \circ R$ is the whole Riemann sphere again. Then families of rational functions are given whose Julia set is $\bar{{\bb C}}$ and whose critical points are not necessarily preperiodic. Concrete examples were previously available only for the preperiodic case. Finally, it is demonstrated that the methods presented also apply to the construction of polynomials whose Julia sets are dendrites and whose critical points in the Julia set are not necessarily preperiodic.

The Beurling algebras $l^1({\cal D},\omega)\;({\cal D}={\bb N},{\bb Z})$ that are semi-simple, with compact Gelfand transform, are considered. The paper gives a necessary and sufficient condition (on $\omega$ ) such that $l^1({\cal D},\omega)$ possesses a uniform quantitative version of Wiener's theorem in the sense that there exists a function $\phi:]0,+\infty[\longrightarrow ]0,+\infty[$ such that, for every invertible element $x$ in the unit ball of $l^1({\cal D},\omega)$ , we have \[ \|x^{-1}\|\le \phi(r(x^{-1}))\quad r(x^{-1})\hbox{ is the spectral radius of }x^{-1}. \]

For a discrete group G there are two well known completions. The first is the Malcev (or unipotent) completion. This is a prounipotent group [Uscr ], defined over ℚ, together with a homomorphism ψ : G → [Uscr ] that is universal among maps from G into prounipotent ℚ-groups. To construct [Uscr ], it suffices for us to consider the case where G is nilpotent; the general case is handled by taking the inverse limit of the Malcev completions of the G/ΓrG, where Γ[bull ]G denotes the lower central series of G. If G is abelian, then [Uscr ] = G [otimes ] ℚ. We review this construction in Section 2.