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

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 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 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

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

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

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

Let $k$ be a field, and let $\alpha$ and $\alpha'$ be two algebraic numbers conjugate over $k$. We prove a result which implies that if $L\subset k(\alpha,\alpha')$ is an abelian or Hamiltonian extension of $k$, then $[L:k]\leq[k(\alpha):k]$. This is related to a certain question concerning the degree of an algebraic number and the degree of a quotient of its two conjugates provided that the quotient is a root of unity, which was raised (and answered) earlier by Cantor. Moreover, we introduce a new notion of the non-torsion power of an algebraic number and prove that a monic polynomial in $X$—irreducible over a real field and having $m$ roots of equal modulus, at least one of which is real—is a polynomial in $X^m$.

AMS 2000 Mathematics subject classification: Primary 11R04; 11R20; 11R32; 12F10

It is shown that a condition on the order of a meromorphic function in a result of A. A. Gol’dberg cannot be relaxed.

AMS 2000 Mathematics subject classification: Primary 30D35

We study the differential geometry of hypersurfaces in hyperbolic space. As an application of the theory of Lagrangian singularities, we investigate the contact of hypersurfaces with families of hyperspheres or equidistant hyperplanes.

AMS 2000 Mathematics subject classification: Primary 53A35. Secondary 58C27

It is well known that $\varOmega^2S^{2n+1}$ is approximated by $\textrm{Rat}_{k}(\mathbb{C}P^{n})$, the space of based holomorphic maps of degree $k$ from $S^2$ to $\mathbb{C}P^{n}$. In this paper we construct a space $G_{k}^{n}$ which is an analogue of $\textrm{Rat}_{k}(\mathbb{C}P^{n})$, and prove that under the natural map $j_k:G_{k}^{n}\to\varOmega^2S^{2n}$, $G_{k}^{n}$ approximates $\varOmega^2S^{2n}$.

AMS 2000 Mathematics subject classification: Primary 55P35

It is known that, for generic $q$, the $\mathcal{H}$-invariant prime ideals in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ are generated by quantum minors (see S. Launois, Les idéaux premiers invariants de $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$, J. Alg., in press). In this paper, $m$ and $p$ being given, we construct an algorithm which computes a generating set of quantum minors for each $\mathcal{H}$-invariant prime ideal in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$. We also describe, in the general case, an explicit generating set of quantum minors for some particular $\mathcal{H}$-invariant prime ideals in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$. In particular, if $(Y_{i,\alpha})_{(i,\alpha)\in[[1,m]]\times[[1,p]]}$ denotes the matrix of the canonical generators of $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$, we prove that, if $u\geq3$, the ideal in $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ generated by $Y_{1,p}$ and the $u\times u$ quantum minors is prime. This result allows Lenagan and Rigal to show that the quantum determinantal factor rings of $O_{q}(\mathcal{M}_{m,p}(\mathbb{C}))$ are maximal orders (see T. H. Lenagan and L. Rigal, Proc. Edinb. Math. Soc.46 (2003), 513–529).

AMS 2000 Mathematics subject classification: Primary 16P40. Secondary 16W35; 20G42

Let $u$ be a solution of a generalized Cauchy–Riemann system in $\mathbb{R}^n$. Suppose that $|u|\le1$ in the unit ball and $|u|\le\varepsilon$ on some closed set $E$. Classical results say that if $E$ is a set of positive Lebesgue measure, then $|u|\le C\varepsilon^\alpha$ on any compact subset of the unit ball. In the present work the same estimate is proved provided that $E$ is a subset of a hyperplane and the (capacitary) dimension of $E$ is greater than $n-2$. The proof gives control of constants $C$ and $\alpha$.

AMS 2000 Mathematics subject classification: Primary 31B35. Secondary 35B35; 35J45

Explicit expressions are derived for the inverses of operators of a particular class that includes the operator corresponding to a system of coupled integral equations having weighted difference kernels. The inverses are expressed in terms of a finite number of functions and a systematic way of generating different sets of these functions is devised. The theory generalizes those previously derived for a single integral equation and an integral-equation system with pure difference kernels. The connection is made between the finite generation of inverses and embedding.

AMS 2000 Mathematics subject classification: Primary 45A05

Let $w(z)$ be an arbitrary transcendental solution of the fourth (respectively, second) Painlevé equation. Concerning the frequency of poles in $|z|\le r$, it is shown that $n(r,w)\gg r^2$ (respectively, $n(r,w)\gg r^{3/2}$), from which the growth estimate $T(r,w)\gg r^2$ (respectively, $T(r,w)\gg r^{3/2}$) immediately follows.

AMS 2000 Mathematics subject classification: Primary 34M55; 34M10