The singular boundary-value problem $(g(x'(t)))'=\mu f(t,x(t),x'(t))$, $x(0)=x(T)=0$ and $\max\{x(t):0\le t\le T\}=A$ is considered. Here $\mu$ is the parameter and the negative function $f(t,u,v)$ satisfying local Carathéodory conditions on $[0,T]\times(0,\infty)\times(\mathbb{R}\setminus\{0\})$ may be singular at the values $u=0$ and $v=0$ of the phase variables $u$ and $v$. The paper presents conditions which guarantee that for any $A>0$ there exists $\mu_A>0$ such that the above problem with $\mu=\mu_A$ has a positive solution on $(0,T)$. The proofs are based on the regularization and sequential techniques and use the Leray–Schauder degree and Vitali’s convergence theorem.

AMS 2000 Mathematics subject classification: Primary 34B16; 34B18

In this paper we extract some conclusions about Newton non-degenerate ideals and the computation of Łojasiewicz exponents relative to this kind of ideal. This motivates us to study the Newton non-degeneracy condition on the Jacobian ideal of a given analytic function germ $f:(\mathbb{C}^n,0)\to(\mathbb{C},0)$. In particular, we establish a connection between Newton non-degenerate functions and functions whose Jacobian ideal is Newton non-degenerate.

AMS 2000 Mathematics subject classification: Primary 32S05. Secondary 57R45

We consider elementary operators on centrally closed prime algebras that are Lie (or Jordan) homomorphisms or commutativity preservers.

AMS 2000 Mathematics subject classification: Primary 16N60. Secondary 16R50; 47B47

A new asymptotic approach is suggested for studying spectra of linear differential operators with periodic coefficients. The resulting formal recurrent procedure and its rigorous justification allow us to prove a classical theorem on the density of states in one dimension.

AMS 2000 Mathematics subject classification: Primary 34L20. Secondary 74J99; 78M35

We give a closed formula for the Conway function of a splice in terms of the Conway function of its splice components. As corollaries, we refine and generalize results of Seifert, Torres and Sumners-Woods.

AMS 2000 Mathematics subject classification: Primary 57M25

In this article we extend the results about Gorenstein modules and Foxby duality to a non-commutative setting. This is done in §3 of the paper, where we characterize the Auslander and Bass classes which arise whenever we have a dualizing module associated with a pair of rings. In this situation it is known that flat modules have finite projective dimension. Since this property of a ring is of interest in its own right, we devote §2 of the paper to a consideration of such rings. Finally, in the paper’s final section, we consider a natural generalization of the notions of Gorenstein modules which arises when we are in the situation of §3, i.e. when we have a dualizing module.

AMS 2000 Mathematics subject classification: Primary 16D20

For a large class of subsets $\varOmega\subset\mathbb{R}^{N}$ (including unbounded domains), we discuss the Fredholm and properness properties of second-order quasilinear elliptic operators viewed as mappings from $W^{2,p}(\varOmega;\mathbb{R}^{m})$ to $L^{p}(\varOmega;\mathbb{R}^{m})$ with $N\ltp\lt\infty$ and $m\geq1$. These operators arise in the study of elliptic systems of $m$ equations on $\varOmega$. A study in the case of a single equation ($m=1$) on $\mathbb{R}^{N}$ was carried out by Rabier and Stuart.

AMS 2000 Mathematics subject classification: Primary 35J45; 35J60. Secondary 47A53; 47F05

Let $L$ be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let $f:L\to\{\pm1\}$ be a homomorphism and, for $\alpha=\sum\alpha_\ell\ell\in\mathbb{Z}L$, define $\alpha^f=\sum f(\ell)\alpha_\ell\ell^{-1}$. Call $\alpha$ f-unitary if $\alpha^f=\alpha^{-1}$ or $\alpha^f=-\alpha^{-1}$. In this paper, we identify the RA loops $L$ with the property that all units in $\mathbb{Z}L$ are $f$-unitary. Along the way, we extend a famous theorem of Higman to a case still undecided in group rings.

AMS 2000 Mathematics subject classification: Primary 20N05. Secondary 17D05; 16S34; 16U60

The imaginary powers of the Laplace operator over the circle give a $C_0$ group of bounded linear operators on $\mathsf{L}_{\theta}^p(0,2\pi)$ ($1\ltp\lt\infty$). Whereas the group is unbounded on $\mathsf{L}^4$, this paper shows that the $\mathsf{L}^4$ long-time averages of each $f$ in $\mathsf{L}^2$ are bounded. This is a Fourier restriction phenomenon.

AMS 2000 Mathematics subject classification: Primary 42A15; 42B15

In this paper we study the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded. Apart from the standard operator norm on $B(H)$, we consider a rich variety of symmetric operator norms and spaces of operator-Lipschitz functions with respect to these norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of operator-Lipschitz functions.

AMS 2000 Mathematics subject classification: Primary 47A56. Secondary 47L20

This paper continue to study the interrelation and hierarchy of the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded and operator stable. It also examines various properties of symmetrically normed ideals, introduces new classes of ideals: regular and Fuglede, and investigates them.

AMS 2000 Mathematics subject classification: Primary 47A56; 47L20

We study a nonlinear second-order periodic problem driven by the scalar $p$-Laplacian with a non-smooth potential. We consider the so-called doubly resonant situation allowing complete interaction (resonance) with both ends of the spectral interval. Using variational methods based on the non-smooth critical-point theory for locally Lipschitz functions and an abstract minimax principle concerning linking sets we establish the solvability of the problem.

AMS 2000 Mathematics subject classification: Primary 34B15; 34C25

For a subset $E\subseteq\mathbb{R}^{d}$ and $x\in\mathbb{R}^{d}$, the local Hausdorff dimension function of $E$ at $x$ is defined by

$$ \mathrm{dim}_{\mathrm{loc}}(x,E)=\lim_{r\searrow0}\mathrm{dim}(E\cap B(x,r)), $$

where ‘dim’ denotes the Hausdorff dimension. Using some of our earlier results on so-called multifractal divergence points we give a short proof of the following result: any continuous function $f:\mathbb{R}^{d}\to[0,d]$ is the local dimension function of some set $E\subseteq\mathbb{R}^{d}$. In fact, our result also provides information about the rate at which the dimension $\mathrm{dim}(E\cap B(x,r))$ converges to $f(x)$ as $r\searrow0$.

AMS 2000 Mathematics subject classification: Primary 28A80

Let $\mathcal{A}$ be a closed, point-separating sub-algebra of $C_0(X)$, where $X$ is a locally compact Hausdorff space. Assume that $X$ is the maximal ideal space of $\mathcal{A}$. If $f\in\mathcal{A}$, the set $f(X)\cup\{0\}$ is denoted by $\sigma(f)$. After characterizing the points of the Choquet boundary as strong boundary points, we use this equivalence to provide a natural extension of the theorem in [10], which, in turn, was inspired by the main result in [6], by proving the ‘Main Theorem’: if $\varPhi:\mathcal{A}\rightarrow\mathcal{A}$ is a surjective map with the property that $\sigma(fg)=\sigma(\varPhi(f)\varPhi(g))$ for every pair of functions $f,g\in\mathcal{A}$, then there is an onto homeomorphism $\varLambda:X\rightarrow X$ and a signum function $\epsilon(x)$ on $X$ such that

$$ \varPhi(f)(\varLambda(x))=\epsilon(x)f(x) $$

for all $x\in X$ and $f\in\mathcal{A}$.

AMS 2000 Mathematics subject classification: Primary 46J10; 46J20

Some results are presented relating to questions raised in a recent paper by Anderson, Hayman and Pommerenke regarding the size of the set of boundary points of the unit disc at which a univalent function has a prescribed radial growth.

AMS 2000 Mathematics subject classification: Primary 30C55

This paper deals with a reaction–diffusion model with inner absorptions and coupled nonlinear boundary conditions of exponential type. The critical exponents are described via a pair of parameters that satisfy a certain matrix equation containing all the six nonlinear exponents of the system. Whether the solutions blow up or not is determined by the signs of the two parameters. A more precise analysis, depending on the geometry of $\varOmega$ and the absorption coefficients, is proposed for the critical sign of the parameters.

AMS 2000 Mathematics subject classification: Primary 35K55; 35B33