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

We investigate the multivariate sampling theory associated with multiparameter eigenvalue problems. A several-variable counterpart of the classical sampling theorem of Whittaker, Kotel’nikov and Shannon is given. It arose when the multiparameter system has order one. Two-dimensional sampling theorems associated with two-parameter systems of second-order differential operators will be established. The sampling formulae are of multivariate non-uniform Lagrange interpolation type. Unlike many of the known formulae, the interpolating functions are not necessarily products of single variable functions.

By means of $M$-structure and dimension theory, we generalize some known results and obtain some new ones on almost transitivity in $\mathcal{C}_0(L,X)$. For instance, if $X$ has the strong Banach–Stone property, then almost transitivity of $\mathcal{C}_0(L,X)$ is divided into two weaker properties, one of them depending only on topological properties of $L$ and the other being closely related to the covering dimensions of $L$ and $X$. This leads to some non-trivial examples of almost transitive $\mathcal{C}_0(L,X)$ spaces.

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 the two-parameter Sturm–Liouville system

$$ -y_1''+q_1y_1=(\lambda r_{11}+\mu r_{12})y_1\quad\text{on }[0,1], $$

with the boundary conditions

$$ \frac{y_1'(0)}{y_1(0)}=\cot\alpha_1\quad\text{and}\quad\frac{y_1'(1)}{y_1(1)}=\frac{a_1\lambda+b_1}{c_1\lambda+d_1}, $$

and

$$ -y_2''+q_2y_2=(\lambda r_{21}+\mu r_{22})y_2\quad\text{on }[0,1], $$

with the boundary conditions

$$ \frac{y_2'(0)}{y_2(0)} =\cot\alpha_2\quad\text{and}\quad\frac{y_2'(1)}{y_2(1)}=\frac{a_2\mu+b_2}{c_2\mu+d_2}, $$

subject to the uniform-left-definite and uniform-ellipticity conditions; where $q_{i}$ and $r_{ij}$ are continuous real valued functions on $[0,1]$, the angle $\alpha_{i}$ is in $[0,\pi)$ and $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ are real numbers with $\delta_{i}=a_{i}d_{i}-b_{i}c_{i}>0$ and $c_{i}\neq0$ for $i,j=1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.

We investigate whether differential polynomials in real transcendental meromorphic functions have non-real zeros. For example, we show that if $g$ is a real transcendental meromorphic function, $c\in\mathbb{R}\setminus\{0\}$ and $n\geq3$ is an integer, then $g'g^n-c$ has infinitely many non-real zeros. If $g$ has only finitely many poles, then this holds for $n\geq2$. Related results for rational functions $g$ are also considered.

Isometric deformations of immersed surfaces in Euclidean 3-space are studied by means of the drehriss. When the immersion is of constant mean curvature and the deformation preserves the mean curvature, we determine the drehriss explicitly in terms of the immersion and its Gauss map. These methods are applied to obtain an alternative classification of the Smyth surfaces, i.e. constant mean curvature immersions of the plane into Euclidean 3-space which admit the action of $S^1$ as a non-trivial group of internal isometries.

A probabilistic representation formula for general systems of linear parabolic equations, coupled only through the zero-order term, is given. On this basis, an implicit probabilistic representation for the vorticity in a three-dimensional viscous fluid (described by the Navier–Stokes equations) is carefully analysed, and a theorem of local existence and uniqueness is proved. The aim of the probabilistic representation is to provide an extension of the Lagrangian formalism from the non-viscous (Euler equations) to the viscous case. As an application, a continuation principle, similar to the Beale–Kato–Majda blow-up criterion, is proved.

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

One of the most fundamental problems in the study of Lagrangian submanifolds from a Riemannian geometric point of view is the classification of Lagrangian immersions of real-space forms into complex-space forms. In this article, we solve this problem for the most basic case; namely, we classify Lagrangian surfaces of constant curvature in the complex Euclidean plane $\mathbb{C}^2$. Our main result states that there exist 19 families of Lagrangian surfaces of constant curvature in $\mathbb{C}^2$. Twelve of the 19 families are obtained via Legendre curves. Conversely, Lagrangian surfaces of constant curvature in $\mathbb{C}^2$ can be obtained locally from the 19 families.

Let $A$ be a commutative comodule algebra over a commutative bialgebra $H$. The group of invertible relative Hopf modules maps to the Picard group of $A$, and the kernel is described as a quotient group of the group of invertible group-like elements of the coring $A\otimes H$, or as a Harrison cohomology group. Our methods are based on elementary $K$-theory. The Hilbert 90 theorem follows as a corollary. The part of the Picard group of the coinvariants that becomes trivial after base extension embeds in the Harrison cohomology group, and the image is contained in a well-defined subgroup $E$. It equals $E$ if $H$ is a cosemisimple Hopf algebra over a field.

Universal Taylor series are defined on simply connected domains, but they do not exist on an annulus. Instead we introduce universal Laurent or Laurent–Faber series on finitely connected domains in $\mathbb{C}$. These are generic universalities. Furthermore, we study some properties of universal Laurent series on an annulus.

In this paper we study the existence and multiplicity of positive radial solutions for a class of semilinear elliptic systems in bounded annular domains with non-homogeneous boundary conditions by the use of a fixed-point theorem of cone expansion/compression type.

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

The homogeneous Riemannian structures on the three-dimensional Berger spheres, their corresponding reductive decompositions and the associated groups of isometries are obtained. The Berger 3-spheres are also considered as homogeneous almost contact metric manifolds.

Let $X$, $Y$ be compact Hausdorff spaces and let $T:C(X,\mathbb{R})\to C(Y,\mathbb{R})$ be an invertible linear operator. Non-standard analysis is used to give a new intuitive proof of the Amir–Cambern result that if $\|T\|\hskip1pt\|T^{-1}\|\lt2$, then there is a homeomorphism $\psi:Y\to X$. The approach provides a proof of the following representation theorem for such near-isometries:

$$ Tf=(T1_X)(f\circ\psi)+Sf, $$

with $\|S\|\leq2(\|T\|-(1/\|T^{-1}\|))$, so $\|S\|\lt\|T\|$. If $\|T\|\hskip1pt\|T^{-1}\|=1$, then $S=0$, giving the well-known representation for isometries.

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

We pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.

Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class $c$ nilpotent groups admit degree $c+1$ polynomial isoperimetric functions.

Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions—the analogues of non-deterministic space and time.

In 1940 Paul Erdős introduced the ‘rational Hilbert space’, which consists of all vectors in the real Hilbert space $\ell^2$ that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces $\mathcal{E}$ of the Banach spaces $\ell^p$ that are constructed as ‘products’ of zero-dimensional subsets $E_n$ of $\mathbb{R}$. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an $\mathcal{E}$ is closed in $\ell^p$, then it is homeomorphic to complete Erdős space if and only if $\dim\mathcal{E}>0$ and every $E_n$ is zero dimensional.