We consider the class of graph-directed constructions which are connected and have the property of finite ramification.By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplaceoperator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of theLaplace operator and provide a formula for the spectral dimension, the exponent determining the power-law scaling inthe eigenvalue counting function, and establish generic constancy for the counting-function asymptotics. In order to dothis we prove an extension of the multidimensional renewal theorem. As a result we show that it is possible for theeigenvalue counting function for fractals to require a logarithmic correction to the usual power-law growth.

AMS 2000 Mathematics subject classification: Primary 35P20; 58J50. Secondary 28A80; 60K05; 31C25

We develop a method of reducing the size of quantum minors in the algebra of quantum matrices $\mathcal{O}_q(M_n)$. We use the method to show that the quantum determinantal factor rings of $\mathcal{O}_q(M_n)c$ are maximal orders, for $q$ an element of $\mathbb{C}$ transcendental over $\mathbb{Q}$.

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

We consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determine the elasticity of $\mathcal{B}_a(n)$ based solely on the cross number. Section 4 applies the results of §3 to investigate the complete set of elasticities of Krull monoids with divisor class group $\mathbb{Z}_n$.

AMS 2000 Mathematics subject classification: Primary 20M14; 20D60; 13F05

We show that for a non-flat bornological space there is always a bornological countable enlargement; moreover, when the space is non-flat and ultrabornological the countable enlargement may be chosen to be both bornological and barrelled. It is also shown that countable enlargements for barrelled or bornological spaces are always Mackey topologies, and every quasibarrelled space that is not barrelled has a quasibarrelled countable enlargement.

AMS 2000 Mathematics subject classification: Primary 46A08; 46A20

The relative rank $\rank(S:A)$ of a subset $A$ of a semigroup $S$ is the minimum cardinality of a set $B$ such that $\langle A\cup B\rangle=S$. It follows from a result of Sierpiński that, if $X$ is infinite, the relative rank of a subset of the full transformation semigroup $\mathcal{T}_{X}$ is either uncountable or at most $2$. A similar result holds for the semigroup $\mathcal{B}_{X}$ of binary relations on $X$.

A subset $S$ of $\mathcal{T}_{\mathbb{N}}$ is dominated (by $U$) if there exists a countable subset $U$ of $\mathcal{T}_{\mathbb{N}}$ with the property that for each $\sigma$ in $S$ there exists $\mu$ in $U$ such that $i\sigma\le i\mu$ for all $i$ in $\mathbb{N}$. It is shown that every dominated subset of $\mathcal{T}_{\mathbb{N}}$ is of uncountable relative rank. As a consequence, the monoid of all contractions in $\mathcal{T}_{\mathbb{N}}$ (mappings $\alpha$ with the property that $|i\alpha-j\alpha|\le|i-j|$ for all $i$ and $j$) is of uncountable relative rank.

It is shown (among other results) that $\rank(\mathcal{B}_{X}:\mathcal{T}_{X})=1$ and that $\rank(\mathcal{B}_{X}:\mathcal{I}_{X})=1$ (where $\mathcal{I}_{X}$ is the symmetric inverse semigroup on $X$). By contrast, if $\mathcal{S}_{X}$ is the symmetric group, $\rank(\mathcal{B}_{X}:\mathcal{S}_{X})=2$.

AMS 2000 Mathematics subject classification: Primary 20M20

We introduce an action of a discrete subgroup $\varGamma$ of $SL(2,\mathbb{R})^n$ on the space of pseudodifferential operators of $n$ variables, and construct a map from the space of Hilbert modular forms for $\varGamma$ to the space of pseudodifferential operators invariant under such a $\varGamma$-action, which is a lifting of the symbol map of pseudodifferential operators. We also obtain a necessary and sufficient condition for a certain type of pseudodifferential operator to be $\varGamma$-invariant.

AMS 2000 Mathematics subject classification: Primary 11F41; 35S05

Let $X$ be a surface in $\mathbb{C}^n$ or $\mathbb{P}^n$ and let $C_{X}(X\times X)$ be the normal cone to $X$ in $X\times X$ (diagonally embedded). For a point $x\in X$, denote by $g(x):=e_x(C_X(X\times X))$ the multiplicity of $C_X(X\times X)$ at $x$. It is a former result of the authors that $g(x)$ is the degree at $x$ of the Stückrad–Vogel cycle $v(X,X)=\sum_C j(X,X;C)[C]$ of the self-intersection of $X$, that is, $g(x)=\sum_Cj(X,X;C)e_x(C)$. We prove that the stratification of $X$ by the multiplicity $g(x)$ is a Whitney stratification, the canonical one if $n=3$. The corresponding result for hypersurfaces in $\mathbb{A}^n$ or $\mathbb{P}^n$, diagonally embedded in a multiple product with itself, was conjectured by van Gastel. This is also discussed, but remains open.

AMS 2000 Mathematics subject classification: Primary 32S15. Secondary 13H15;14C17

In this paper we consider the existence of positive solutions to the boundary-value problems

\begin{align*} (p(t)u')'-q(t)u+\lambda f(t,u)\amp=0,\quad r\ltt\ltR, \\[2pt] au(r)-bp(r)u'(r)\amp=\sum^{m-2}_{i=1}\alpha_iu(\xi_i), \\ cu(R)+dp(R)u'(R)\amp=\sum^{m-2}_{i=1}\beta_iu(\xi_i), \end{align*}

where $\lambda$ is a positive parameter, $a,b,c,d\in[0,\infty)$, $\xi_i\in(r,R)$, $\alpha_i,\beta_i\in[0,\infty)$ (for $i\in\{1,\dots m-2\}$) are given constants satisfying some suitable conditions. Our results extend some of the existing literature on superlinear semipositone problems. The proofs are based on the fixed-point theorem in cones.

AMS 2000 Mathematics subject classification: Primary 34B10, 34B18, 34B15

Given a finite sequence $\bm{a}=\langle a_i\rangle_{i=1}^n$ in $\mathbb{N}$ and a sequence $\langle x_t\rangle_{t=1}^\infty$ in $\mathbb{N}$, the Milliken–Taylor system generated by $\bm{a}$ and $\langle x_t\rangle_{t=1}^\infty$ is

\begin{multline*} \qquad \mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)=\biggl\{\sum_{i=1}^na_i\cdot\sum_{t\in F_i}x_t:F_1,F_2,\dots,F_n\text{ are finite non-empty} \\[-8pt] \text{subsets of $\mathbb{N}$ with }\max F_i\lt\min F_{i+1}\text{ for }i\ltn\biggr\}.\qquad \end{multline*}

It is known that Milliken–Taylor systems are partition regular but not consistent. More precisely, if $\bm{a}$ and $\bm{b}$ are finite sequences in $\mathbb{N}$, then, except in trivial cases, there is a partition of $\mathbb{N}$ into two cells, neither of which contains $\mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)\cup \mathrm{MT}(\bm{b},\langle y_t\rangle_{t=1}^\infty)$ for any sequences $\langle x_t\rangle_{t=1}^\infty$ and $\langle y_t\rangle_{t=1}^\infty$.

Our aim in this paper is to extend the above result to allow negative entries in $\bm{a}$ and $\bm{b}$. We do so with a proof which is significantly shorter and simpler than the original proof which applied only to positive coefficients. We also derive some results concerning the existence of solutions of certain linear equations in $\beta\mathbb{Z}$. In particular, we show that the ability to guarantee the existence of $\mathrm{MT}(\bm{a},\langle x_t\rangle_{t=1}^\infty)\cup \mathrm{MT}(\bm{b},\langle y_t\rangle_{t=1}^\infty)$ in one cell of a partition is equivalent to the ability to find idempotents $p$ and $q$ in $\beta\mathbb{N}$ such that $a_1\cdot p+a_2\cdot p+\cdots+a_n\cdot p=b_1\cdot q+b_2\cdot q+\cdots+b_m\cdot q$, and thus determine exactly when the latter has a solution.

AMS 2000 Mathematics subject classification: Primary 05D10. Secondary 22A15; 54H13

The relationships between various notions of completeness of eigenvectors and root vectors of the eigenvalue problem $Af=\lambda Bf$ are investigated. Here $A$ and $B$ are self-adjoint operators in Hilbert space with $B$ bounded and positive semidefinite, and with $A$ having compact resolvent.

AMS 2000 Mathematics subject classification: Primary 47A75. Secondary 34B24; 35P10

We show that the support of a (possibly) coated anisotropic medium is uniquely determined by the electric far-field patterns corresponding to incident time-harmonic electromagnetic plane waves with arbitrary polarization and direction. Our proof avoids the use of a fundamental solution to Maxwell’s equations in an anisotropic medium and instead relies on the well-posedness and regularity properties of solutions to an interior transmission problem for Maxwell’s equations.

AMS 2000 Mathematics subject classification: Primary 35R30; 35Q60. Secondary 35P25; 78A45

It was shown by Huynh and Rizvi that a ring $R$ is semisimple artinian if and only if every continuous right $R$-moduleis injective. However, a characterization of rings, over which every finitely generated continuous rightmodule is injective, has been left open. In this note we give a partial solution for this question. Namely, we showthat for a right semi-artinian ring $R$, every finitely generated continuous right $R$-module is injective if and onlyif all simple right $R$-modules are injective.

AMS 2000 Mathematics subject classification: Primary 16D50. Secondary 16P20; 16P60

In this paper spectral properties of non-selfadjoint Jacobi operators $J$ which are compact perturbations of the operator $J_0=S+\rho S^*$, where $\rho\in(0,1)$ and $S$ is the unilateral shift operator in $\ell^2$, are studied. In the case where $J-J_0$ is in the trace class, Friedrichs’s ideas are used to prove similarity of $J$ to the rank one perturbation $T$ of $J_0$, i.e. $T=J_0+(\cdot,p)e_1$. Moreover, it is shown that the perturbation is of ‘smooth type’, i.e. $p\in\ell^2$ and

$$ \varlimsup_{n\rightarrow\infty}|p(n)|^{1/n}\le\rho^{1/2}. $$

When $J-J_0$ is not in the trace class, the Friedrichs method does not work and the transfer matrix approach is used. Finally, the point spectrum of a special class of Jacobi matrices (introduced by Atzmon and Sodin) is investigated.

AMS 2000 Mathematics subject classification: Primary 47B36. Secondary 47B37

A definition of a shock layer of thickness $\delta$ is proposed when a parabolic perturbation is applied to a scalar conservation law. The asymptotic equality $\delta\asymp\sqrt{\varepsilon}$ is established, where $\varepsilon$ denotes viscosity. This equality is proved to be optimal. Nevertheless, the equality $\delta\asymp\varepsilon$ is also proved to be valid for a class of shocks in accordance with the Mises conjecture.

AMS 2000 Mathematics subject classification: Primary 35L67. Secondary 35K65

A strip of radius $r$ in the hyperbolic plane is the set of points within distance $r$ of a given geodesic. Wedefine the density of a packing of strips of radius $r$ and prove that this density cannot exceed

$$ \mathcal{S}(r)=\frac{3}{\pi}\sinh r\mathrm{arccosh}\biggl(1+\frac{1}{2\sinh^2r}\biggr). $$

This bound is sharp for every value of $r$ and provides sharp bounds on collaring theorems for simple geodesics onsurfaces.

AMS 2000 Mathematics subject classification: Primary 51M09; 52C15. Secondary 51M04

Working on a suitable cone of continuous functions, we give new results for integral equations of the form$\lambda u(t)=\int_{G}k(t,s)f(s,u(s))\,\mathrm{d} s:=Tu(t)$, where $G$ is a compact set in $\mathbb{R}^{n}$ and $k$ is apossibly discontinuous function that is allowed to change sign. We apply our results to prove existence of eigenvaluesof some non-local boundary-value problems.

AMS 2000 Mathematics subject classification: Primary 34B10. Secondary 34B18; 47H10; 47H30

In this paper we study the existence of multiple positive solutions and the bifurcation problem for the following equation:

$$ -\Delta u+u=\biggl(\int_{\mathbb{R}^3}\frac{|u(y)|^2}{|x-y|}\,\mathrm{d}y\biggr)u+\mu f(x),\quad x\in\mathbb{R}^3, $$

where $f(x)\in H^{-1}(\mathbb{R}^3)$, $f(x)\geq0$, $f(x)\not\equiv0$. We show that there are positive constants $\mu^{*}$ and $\mu^{**}$ such that the above equation possesses at least two positive solutions for $\mu\in(0,\mu^{*})$, and no positive solution for $\mu>\mu^{**}$. Furthermore, we prove that $\mu=\mu^{*}$ is a bifurcation point for the equation under study.

AMS 2000 Mathematics subject classification: Primary 35J60; 35J70

We discuss the problem of the uniqueness of the solution to the Cauchy problem for second-order, linear, uniformly parabolic differential equations. For most uniqueness theorems the solution must be uniformly bounded with respect to the time variable $t$, but some authors have shown an interest in relaxing the growth conditions in time.

In 1997, Chung proved that, in the case of the heat equation, uniqueness holds under the restriction: $|u(x,t)|\leq C\exp[(a/t)^{\alpha}+a|x|^2]$, for some constants $C,a>0$, $0\lt\alpha\lt1$. The proof of Chung’s theorem is based on ultradistribution theory, in particular it relies heavily on the fact that the coefficients are constants and that the solution is smooth. Therefore, his method does not work for parabolic operators with arbitrary coefficients. In this paper we prove a uniqueness theorem for uniformly parabolic equations imposing the same growth condition as Chung on the solution $u(x,t)$. At the centre of the proof are the maximum principle, Gaussian-type estimates for short cylinders and a boot-strapping argument.

AMS 2000 Mathematics subject classification: Primary 35K15

By means of monotone functionals and positive linear functionals defined on suitable matrix spaces as well as new generalized Riccati transformations, oscillation criteria for self-adjoint linear Hamiltonian matrix systems are obtained. Our results are generalizations and improvements of many existing results.

AMS 2000 Mathematics subject classification: Primary 34A30; 34C10

A module of a ring of differential operators $\mathcal{D}$ over a smooth surface has order $1$ if it is isomorphic to a factor module of $\mathcal{D}$ by a cyclic ideal generated by an operator of order $1$. Let $k$ be a positive integer. We give conditions under which an indecomposable $\mathcal{D}$-module of order $1$ is GK-critical of length $k$. We also give examples of indecomposable, non-critical, $\mathcal{D}$-modules whose subfactors have order $1$.

AMS 2000 Mathematics subject classification: Primary 16S32. Secondary 37F75