The relationship of the Lp (Rn) estimate and two endpoint estimates for the multi-linear singular integral operators is considered. Some conditions are given under which one of these estimates implies the others.

We rigorously approach the Schrödinger equation of derivative type qt + iqxx + λ |q|2qx + μq2q̄x = 0, λ ∈ R, μ ∈ C, by the cubic nonlinear Schrödinger equation AT + i AXX + i k0 (λ − μ) |A|2A = 0. We also study the case of the KdV-like equation qt + i qxx + aqxxx + i |q|2q + λ̃ (|q|2q)x + μ̃ |q|2qx = 0, λ̃ μ̃ ∈ R, arising in optical physics.

Convergent expansions are derived for three types of orthogonal polynomials: Charlier, Laguerre and Jacobi. The expansions have asymptotic properties for large values of the degree. The expansions are given in terms of functions that are special cases of the given polynomials. The method is based on expanding integrals in one or two points of the complex plane, these points being saddle points of the phase functions of the integrands.

Given an infinite-dimensional vector space V, we consider the semigroup GS (m, n) consisting of all injective linear α: V → V for which codim ran α = n, where dim V = m ≥ n ≥ ℵ0. This is a linear version of the well-known Baer–Levi semigroup BL (p, q) defined on an infinite set X, where |X| = p ≥ q ≥ ℵ0. We show that, although the basic properties of GS (m, n) are the same as those of BL (p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS (m, n) and some of its maximal subsemigroups; in this, we follow previous work on BL (p, q) by Sutov and Sullivan as well as Levi and Wood.

We study a boundary-value problem for a particular semilinear elliptic equation on Rn (n ≥ 2), whose solutions represent generalized stream functions for steady axisymmetric ideal fluid flows. Solutions are shown to exist that generalize those already known in dimensions 2 and 3. An isoperimetric characterization is given for our solutions, which represent generalized spherical vortex-rings. As a corollary, a sharp Sobolev-type inequality is obtained.

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 $k$ 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

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

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

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 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

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

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

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

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

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

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