The singular boundary-value problem $(g(x'))'=\mu f(t,x,x')$, $x'(0)=0$, $x(T)=b>0$ is considered. Here $\mu$ is the parameter and $f(t,x,y)$, which satisfies local Carathéodory conditions on $[0,T]\times(\mathbb{R}\setminus\{b\})\times(\mathbb{R}\setminus\{0\})$, may be singular at the values $x=b$ and $y=0$ of the phase variables $x$ and $y$, respectively. Conditions guaranteeing the existence of a positive solution to the above problem for suitable positive values of $\mu$ are given. The proofs are based on regularization and sequential techniques and use the topological transversality theorem.

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

We construct a Bousfield–Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum $E$ with unit and which is related to the homotopy groups of a certain unstable $E$ completion $X_E^{\wedge}$ of a space $X$. For $E$ an $\mathbb{S}$-algebra this completion agrees with that of the first author and Thompson. We also establish in detail the Hopf algebra structure of the unstable cooperations (the coalgebraic module) $E_*(\underline{E}_*)$ for an arbitrary Landweber exact spectrum $E$, extending work of the second author with Hopkins and with Turner and giving basis-free descriptions of the modules of primitives and indecomposables. Taken together, these results enable us to give a simple description of the $E_2$-page of the $E$-theory Bousfield–Kan spectral sequence when $E$ is any Landweber exact ring spectrum with unit. This extends work of the first author and others and gives a tractable unstable Adams spectral sequence based on a $v_n$-periodic theory for all $n$.

AMS 2000 Mathematics subject classification: Primary 55P60; 55Q51; 55S25; 55T15. Secondary 55P47

In this paper we prove the existence of at least three classical solutions for the problem

$$ \left\{ \begin{aligned} \amp-(|u'|^{p-2}u')'=\lambda f(t,u)h(u'), \\ \ampu(a)=u(b)=0, \end{aligned} \right. $$

when $\lambda$ lies in an explicitly determined open interval.

Our main tool is a very recent three-critical-points theorem stated in a paper by D. Averna and G. Bonanno (Topolog. Meth. Nonlin. Analysis22 (2003), 93–103).

AMS 2000 Mathematics subject classification: Primary 34B15

Let $1\leq p,q\leq\infty$, $s\in\mathbb{R}$ and let $X$ be a Banach space. We show that the analogue of Marcinkiewicz’s Fourier multiplier theorem on $L^p(\mathbb{T})$ holds for the Besov space $B_{p,q}^s(\mathbb{T};X)$ if and only if $1\ltp\lt\infty$ and $X$ is a UMD-space. Introducing stronger conditions we obtain a periodic Fourier multiplier theorem which is valid without restriction on the indices or the space (which is analogous to Amann’s result (Math. Nachr.186 (1997), 5–56) on the real line). It is used to characterize maximal regularity of periodic Cauchy problems.

AMS 2000 Mathematics subject classification: Primary 47D06; 42A45

We consider Sturm–Liouville boundary-value problems on the interval $[0,1]$ of the form $-y''+qy=\lambda y$ with boundary conditions $y'(0)\sin\alpha=y(0)\cos\alpha$ and $y'(1)=(a\lambda+b)y(1)$, where $a\lt0$. We show that via multiple Crum–Darboux transformations, this boundary-value problem can be transformed ‘almost’ isospectrally to a boundary-value problem of the same form, but with the boundary condition at $x=1$ replaced by $y'(1)\sin\beta=y(1)\cos\beta$, for some $\beta$.

AMS 2000 Mathematics subject classification: Primary 34B07; 47E05; 34L05

We construct some series of polyhedral schemata which represent orientable closed connected 3-manifolds. We show that these manifolds have spines corresponding to certain balanced presentations of their fundamental groups. Then we study some covering properties of such manifolds and prove that many of them are cyclic branched coverings of lens spaces. Our theorems contain a number of published results from various sources as particular cases.

AMS 2000 Mathematics subject classification: Primary 57M12; 57M50; 57M60

We construct certain elements in the motivic cohomology group $H^3_{\mathcal{M}}(E\times E',\mathbb{Q}(2))$, where $E$ and $E'$ are elliptic curves over $\mathbb{Q}$. When $E$ is not isogenous to $E'$ these elements are analogous to circular units in real quadratic fields, as they come from modular parametrizations of the elliptic curves. We then find an analogue of the class-number formula for real quadratic fields, which specializes to the usual quadratic class-number formula when $E$ and $E'$ are quadratic twists.

AMS 2000 Mathematics subject classification: Primary 11F67; 14G35. Secondary 11F11; 11E45; 14G10

We study families of integer polynomials having roots very close to each other.

AMS 2000 Mathematics subject classification: Primary 11C08

In this work we present a generalization of an exact sequence of normal bordism groups given in a paper by H. A. Salomonsen (Math. Scand.32 (1973), 87–111). This is applied to prove that if $h:M^n\to X^{n+k}$, $5\leq n\lt2k$, is a continuous map between two manifolds and $g:M^n\to BO$ is the classifying map of the stable normal bundle of $h$ such that $(h,g)_*:H_i(M,\mathbb{Z}_2)\to H_i(X\times BO,\mathbb{Z}_2)$ is an isomorphism for $i\lt n-k$ and an epimorphism for $i=n-k$, then $h$ bordant to an immersion implies that $h$ is homotopic to an immersion. The second remark complements the result of C. Biasi, D. L. Gonçalves and A. K. M. Libardi (Topology Applic.116 (2001), 293–303) and it concerns conditions for which there exist immersions in the metastable dimension range. Some applications and examples for the main results are also given.

AMS 2000 Mathematics subject classification: Primary 57R42. Secondary 55Q10; 55P60

We obtain heat-kernel estimates for higher-order operators with measurable coefficients that can be singular or degenerate. Precise constants are given, which are sharp for small times.

AMS 2000 Mathematics subject classification: Primary 35K30; 47D06. Secondary 35K25; 35K35

Let $D$ be a division ring whose group of units satisfies a non-trivial group identity $w$. Let $\alpha$ be the sum of positive degrees of indeterminates occurring in $w$. If the centre of $D$ contains more than $3\alpha$ elements, then $D$ is commutative.

AMS 2000 Mathematics subject classification: Primary 16R50. Secondary 16K20

We formulate and prove a free quantum analogue of the first fundamental theorems of invariant theory. More precisely, the polynomial function algebras on matrices are replaced by free algebras, while the universal cosovereign Hopf algebras play the role of the general linear group.

AMS 2000 Mathematics subject classification: Primary 16W30

In this paper we compute the non-abelian tensor square for the free 2-Engel group of rank $n>3$. The non-abelian tensor square for this group is a direct product of a free abelian group and a nilpotent group of class 2 whose derived subgroup has exponent 3. We also compute the non-abelian tensor square for one of the group’s finite homomorphic images, namely, the Burnside group of rank $n$ and exponent 3.

AMS 2000 Mathematics subject classification: Primary 20F05; 20F45. Secondary 20F18

The series

\begin{alignat*}{2} L_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \\ M_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma_3(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \\ N_{r,4}(q)\amp=\sum_{n=0}^\infty\sigma_5(4n+r)q^{4n+r},\amp\quad r\amp=0,1,2,3, \end{alignat*}

are evaluated and used to prove convolution formulae such as

$$ \sum_{m\le n}\sigma(4m-3)\sigma(4n-(4m-3))=4\sigma_3(n)-4\sigma_3(\tfrac12n). $$

AMS 2000 Mathematics subject classification: Primary 11A25

We introduce an invariant, called the contact number, associated with each Euclidean submanifold. We show that this invariant is, surprisingly, closely related to the notions of isotropic submanifolds and holomorphic curves. We are able to establish a simple criterion for a submanifold to have any given contact number. Moreover, we completely classify codimension-$2$ submanifolds with contact number ${\geq}3$. We also study surfaces in $\mathbb{E}^6$ with contact number ${\geq}4$. As an immediate consequence, we obtain the first explicit examples of non-spherical pseudo-umbilical surfaces in Euclidean spaces.

AMS 2000 Mathematics subject classification: Primary 53C40; 53A10. Secondary 53B25; 53C42

We present a wavelet characterization of anisotropic Besov spaces $B_{p,q}^{\bm{\alpha}}(\mathbb{R}^n)$, valid for the whole range $0\ltp,q\lt\infty$, and in terms of multi-resolution analyses with dilation adapted to the anisotropy of the space. Our proofs combine classical techniques based on Bernstein and Jackson-type inequalities, and nonlinear methods for the cases $p\lt1$. Among the consequences of our results, we characterize $B_{p,q}^{\bm{\alpha}}$ as a linear approximation space, and derive embeddings and interpolation formulae for $B_{p,q}^{\bm{\alpha}}$, which appear to be new in the literature when $p\lt1$.

AMS 2000 Mathematics subject classification: Primary 42B35; 42C40; 41A17

In two previous papers we established the structure of the normal closure of a cyclic permutable subgroup $A$ of a finite group, first when $A$ has odd order and second when $A$ has even order, but with an extra hypothesis that was unnecessary in the odd case. Here we describe the most general situation without any restrictions on $A$.

AMS 2000 Mathematics subject classification: Primary 20D35; 20D40

This paper is a further contribution to the extensive study by a number of authors of the subalgebra lattice of a Lie algebra. We give some necessary and some sufficient conditions for a subalgebra to be upper modular. For algebraically closed fields of any characteristic these enable us to determine the structure of Lie algebras having abelian upper-modular subalgebras which are not ideals. We then study the structure of solvable Lie algebras having an abelian upper-modular subalgebra which is not an ideal and which has trivial intersection with the derived algebra; in particular, the structure is determined for algebras over the real field. Next we classify non-solvable Lie algebras over fields of characteristic zero having an upper-modular atom which is not an ideal. Finally, it is shown that every Lie algebra over a field of characteristic different from two and three in which every atom is upper modular is either quasi-abelian or a $\mu$-algebra.

AMS 2000 Mathematics subject classification: Primary 17B05; 17B50; 17B30; 17B20

We prove that the determination of all $M^*$-groups is essentially equivalent to the determination of finite groups generated by an element of order 3 and an element of order 2 or 3 that admit a particular automorphism. We also show how the second commutator subgroup of an $M^*$-group $G$ can often be used to construct $M^*$-groups which are direct products with $G$ as one factor. Several applications of both methods are given.

AMS 2000 Mathematics subject classification: Primary 20D45; 20E36. Secondary 14H37; 30F50

We prove that the Bergman projection on $L^p(w)$ $(p\neq 2)$, where $w(r)=(1-r^2)^A\textrm{e}^{-B/(1-r^2)^{\alpha}}$, is not bounded.

AMS 2000 Mathematics subject classification: Primary 47B10