A key step in establishing the validity of the linear sampling method of determining an unknown scattering obstacle $D$ from a knowledge of its far-field pattern is to prove that solutions of the Helmholtz equation in $D$ can be approximated in $H^1(D)$ by Herglotz wave functions.

To this end we establish the important property that the set of Herglotz wave functions is dense in the space of solutions of the Helmholtz equation with respect to the Sobolev space $H^1(D)$ norm.

AMS 2000 Mathematics subject classification: Primary 35R30. Secondary 35P25

Let $f$ be meromorphic of finite order in the plane, such that $f^{(k)}$ has finitely many zeros, for some $k\geq2$. The author has conjectured that $f$ then has finitely many poles. In this paper, we strengthen a previous estimate for the frequency of distinct poles of $f$. Further, we show that the conjecture is true if either

1. $f$ has order less than $1+\varepsilon$, for some positive absolute constant $\varepsilon$, or

2. $f^{(m)}$, for some $0\leq m lt k$, has few zeros away from the real axis.

AMS 2000 Mathematics subject classification: Primary 30D35

Let $R$ be a commutative ring. Let $M$ respectively $A$ denote a Noetherian respectively Artinian $R$-module, and $\mathfrak{a}$ a finitely generated ideal of $R$. The main result of this note is that the sequence of sets $(\mathrm{Att}_R\mathrm{Tor}_1^R((R/\mathfrak{a}^{n}),A))_{n\in\mathbb{N}}$ is ultimately constant. As a consequence, whenever $R$ is Noetherian, we show that $\mathrm{Ass}_R\mathrm{Ext}_R^1((R/\mathfrak{a}^{n}),M)$ is ultimately constant for large $n$, which is an affirmative answer to the question that was posed by Melkersson and Schenzel in the case $i=1$.

AMS 2000 Mathematics subject classification: Primary 13E05; 13E10

Let $K$ be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value. We show that the $p$-adic main Nevanlinna Theorem holds for meromorphic functions inside an ‘open’ disc in $K$. Let $P_{n,c}$ be the Frank–Reinders’s polynomial

$$ (n-1)(n-2)X^n-2n(n-2)X^{n-1}+ n(n-1)X^{n-2}-c\qq (c\neq0,\ c\neq1,\ c\neq2) $$

and let $S_{n,c}$ be the set of its $n$ distinct zeros. For every $n\geq 7$, we show that $S_{n,c}$ is an $n$-points unique range set (counting multiplicities) for unbounded analytic functions inside an ‘open disc’, and for every $n\geq10$, we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for the same set of functions. Similar results are obtained for meromorphic functions whose characteristic function is unbounded: we obtain unique range sets ignoring multiplicities of $17$ points. A better result is obtained for an analytic or a meromorphic function $f$ when its derivative is ‘small’ comparatively to $f$. In particular, for every $n\geq5$ we show that $S_{n,c}$ is an $n$-points unique range set ignoring multiplicities for unbounded analytic functions with small derivative. Actually, in each case, results also apply to pairs of analytic functions when just one of them is supposed unbounded. The method we use is based upon the $p$-adic Nevanlinna Theory, and Frank–Reinders’s and Fujimoto’s methods used for meromorphic functions in $\mathbb{C}$. Among other results, we show that the set of functions having a bounded characteristic function is just the field of fractions of the ring of bounded analytic functions in the disc.

AMS 2000 Mathematics subject classification: Primary 12H25. Secondary 12J25; 46S10

For a two-step nilpotent locally compact group $G$, we determine the Glimm ideal space of the group $C^*$-algebra $C^*(G)$ and its topology. This leads to necessary and sufficient conditions for $C^*(G)$ to be quasi-standard. Moreover, some results about the Glimm classes of points in the primitive ideal space $\mathrm{Prim}(C^*(G))$ are obtained.

AMS 2000 Mathematics subject classification: Primary 22D25. Secondary 22D10

We study the bifurcation problem

$$ \begin{cases} -\Div(a(x)|Du|^{p-2}Du)+h(x)u^{r-1}= f(\lambda,x,u) & \text{in } \sOm\subset\RR^{N}, \\ a(x)|Du|^{p-2}Du\cdot n+b(x)u^{p-1}=\theta g(x,u) & \text{on }\sGa, \\ u\geq0,\quad u\not\equiv0 & \text{in }\sOm, \end{cases} $$

where $\sOm$ is an unbounded domain with smooth non-compact boundary $\sGa$, $n$ denotes the unit outward normal vector on $\sGa$, and $\lambda>0$, $\theta$ are real parameters. We assume throughout that $p lt r lt p^{*}=pN/ (N-p)$, $1 lt p lt N$, the functions $a$, $b$ and $h$ are positive while $f$, $g$ are subcritical nonlinearities. We show that there exist an open interval $I$ and $\lambda^\star gt 0$ such that the problem has no solution if $\theta\in I$ and $\lambda\in (0,\lambda^\star)$. Furthermore, there exist an open interval $J\subset I$ and $\lambda_0 gt 0$ such that, for any $\theta\in J$, the above problem has at least a solution if $\lambda\geq \lambda_0$, but it has no solution provided that $\lambda\in (0,\lambda_0)$.

AMS 2000 Mathematics subject classification: Primary 35J60; 35P30; 58E05; 58G28

Adapting the theory of the derived category to ordered groupoids, we prove that every ordered functor (and thus every inverse and regular semigroup homomorphism) factors as an enlargement followed by an ordered fibration. As an application, we obtain Lawson’s version of Ehresmann’s Maximum Enlargement Theorem, from which can be deduced the classical theory of idempotent-pure inverse semigroup homomorphisms and $E$-unitary inverse semigroups.

AMS 2000 Mathematics subject classification: Primary 20M18; 20L05; 20M17

We characterize the duals and biduals of the $L^p$-analogues $\mathcal{N}_\alpha^p$ of the standard Nevanlinna classes $\mathcal{N}_\alpha$, $\alpha\ge-1$ and $1\le p\lt \infty$. We adopt the convention to take $\mathcal{N}_{-1}^p$ to be the classical Smirnov class $\mathcal{N}^+$ for $p=1$, and the Hardy–Orlicz space $LH^p$ $(=(\text{Log}^+H)^p)$ for $1\lt p\lt\infty$. Our results generalize and unify earlier characterizations obtained by Eoff for $\alpha=0$ and $\alpha=-1$, and by Yanigahara for the Smirnov class.

Each $\mathcal{N}_\alpha^p$ is a complete metrizable topological vector space (in fact, even an algebra); it fails to be locally bounded and locally convex but admits a separating dual. Its bidual will be identified with a specific nuclear power series space of finite type; this turns out to be the ‘Fréchet envelope’ of $\mathcal{N}_\alpha^p$ as well.

The generating sequence of this power series space is of the form $(n^\theta)_{n\in\mathbb{N}}$ for some $0\lt\theta\lt1$. For example, the $\theta$s in the interval $(\smfr12,1)$ correspond in a bijective fashion to the Nevanlinna classes $\mathcal{N}_\alpha$, $\alpha\gt-1$, whereas the $\theta$s in the interval $(0,\smfr12)$ correspond bijectively to the Hardy–Orlicz spaces $LH^p$, $1\lt p\lt \infty$. By the work of Yanagihara, $\theta=\smfr12$ corresponds to $\mathcal{N}^+$.

As in the work by Yanagihara, we derive our results from characterizations of coefficient multipliers from $\mathcal{N}_\alpha^p$ into various smaller classical spaces of analytic functions on $\Delta$.

AMS 2000 Mathematics subject classification: Primary 46E10; 46A11; 47B38. Secondary 30D55; 46A45; 46E15\vskip-3pt

The non-local equation

$$ u_t=(u^3u_x)_x+\frac{\lambda f(u)}{(\int_{-1}^1f(u)\,\rd x)^{2}} $$

is considered, subject to some initial and Dirichlet boundary conditions. Here $f$ is taken to be either $\exp(-s^4)$ or $H(1-s)$ with $H$ the Heaviside function, which are both decreasing. It is found that there exists a critical value $\lambda^*=2$, so that for $\lambda>\lambda^{*}$ there is no stationary solution and $u$ ‘blows up’ (in some sense). If $0\lt\lambda\lt\lambda^{*}$, there is a unique stationary solution which is asymptotically stable and the solution of the IBVP is global in time.

AMS 2000 Mathematics subject classification: Primary 35B30; 35B35; 35B40; 35K20; 35K55; 35K99

Let $\rho:G\hookrightarrow\GL(n,\F)$ be a representation of a finite group $G$ over a finite field $\F$ and $f_1,\dots,f_n\in\F[V]^G$ such that the ring of invariants is a polynomial algebra $\F[f_1,\dots,f_n]$. It is known that in this case the algebra of coinvariants $\F[V]_G$ is a Poincaré duality algebra, and if, moreover, the order of $G$ is invertible in $\F$, that a fundamental class is represented by the Jacobian determinant $\mathchoice{\det\biggl[\frac{\partial f_i}{\partial z_j}\biggr]}{\det[\partial f_i/\partial z_j]}{\det[\partial f_i/\partial z_j]} {\det[\partial f_i/\partial z_j]}$, and is therefore a $\det^{-1}$-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.

AMS 2000 Mathematics subject classification: Primary 13A50; 55S10

It is shown that most properties of (bounded) completely hyperexpansive operators remain valid for unbounded $2$-hyperexpansive operators. Powers of closed $2$-hyperexpansive operators are proved to be closed and $2$-hyperexpansive. Various parts of spectra of such operators are calculated. $2$-hyperexpansive weighted shifts are investigated. Examples of unbounded closed $2$-hyperexpansive ($2$-isometric) operators with invariant dense domains are established.

AMS 2000 Mathematics subject classification: Primary 47A05; 47B37. Secondary 47B20; 47B10

We consider the following nonlinear elliptic equations

\begin{gather*} \begin{cases} \Delta u+u_{+}^{N/(N-2)}=0\amp\quad\text{in }\sOm, \\ u=\mu\amp\quad\text{on }\partial\sOm\quad(\mu\text{ is an unknown constant}), \\ \dsty\int_{\partial\sOm}\biggl(-\dsty\frac{\partial u}{\partial n}\biggr)=M, \end{cases} \end{gather*}

where $u_{+}=\max(u,0)$, $M$ is a prescribed constant, and $\sOm$ is a bounded and smooth domain in $R^N$, $N\geq3$. It is known that for $M=M_{*}^{(N)}$, $\sOm=B_R(0)$, the above problem has a continuum of solutions. The case when $M>M_{*}^{(N)}$ is referred to as supercritical in the literature. We show that for $M$ near $KM_{*}^{(N)}$, $K>1$, there exist solutions with multiple condensations in $\sOm$. These concentration points are non-degenerate critical points of a function related to the Green's function.

AMS 2000 Mathematics subject classification: Primary 35B40; 35B45. Secondary 35J40