In this paper the germ of neighborhood of a compact leaf in a Lagrangian foliation is symplectically classified when the compact leaf is ${{\mathbb{T}}^{2}}$, the affine structure induced by the Lagrangian foliation on the leaf is complete, and the holonomy of ${{\mathbb{T}}^{2}}$ in the foliation linearizes. The germ of neighborhood is classified by a function, depending on one transverse coordinate, this function is related to the affine structure of the nearly compact leaves.

Let $F$ be a totally real number field and let $\text{G}{{\text{L}}_{n}}$ be the general linear group of rank $n$ over $F$. Let $\mathfrak{p}$ be a prime ideal of $F$ and ${{F}_{\mathfrak{p}}}$ the completion of $F$ with respect to the valuation induced by $\mathfrak{p}$. We will consider a finite quotient of the affine building of the group $\text{G}{{\text{L}}_{n}}$ over the field ${{F}_{\mathfrak{p}}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph.

The generating degree $\text{g}\deg \left( A \right)$ of a topological commutative ring $A$ with char $A\,=\,0$ is the cardinality of the smallest subset $M$ of $A$ for which the subring $\mathbb{Z}\left[ M \right]$ is dense in $A$. For a prime number $p$, ${{\mathbb{C}}_{p}}$ denotes the topological completion of an algebraic closure of the field ${{\mathbb{Q}}_{p}}$ of $p$-adic numbers. We prove that $\text{g}\deg \left( {{\mathbb{C}}_{p}} \right)\,=\,1$, i.e., there exists $t$ in ${{\mathbb{C}}_{p}}$ such that $\mathbb{Z}\left[ t \right]$ is dense in ${{\mathbb{C}}_{p}}$. We also compute $\text{gdeg}\left( A\left( U \right) \right)$ where $A\left( U \right)$ is the ring of rigid analytic functions defined on a ball $U$ in ${{\mathbb{C}}_{p}}$. If $U$ is a closed ball then $\text{gdeg}\left( A\left( U \right) \right)\,=\,2$ while if $U$ is an open ball then $\text{gdeg}\left( A\left( U \right) \right)$ is infinite. We show more generally that $\text{gdeg}\left( A\left( U \right) \right)$ is finite for any affinoid$U$ in ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$ and $\text{gdeg}\left( A\left( U \right) \right)$ is infinite for any wide open subset $U$ of ${{\mathbb{P}}^{1}}\left( {{\mathbb{C}}_{p}} \right)$.

Let $X$ be a reduced nonsingular quasiprojective scheme over $\mathbb{R}$ such that the set of real rational points $X\left( \mathbb{R} \right)$ is dense in $X$ and compact. Then $X\left( \mathbb{R} \right)$ is a real algebraic variety. Denote by $H_{k}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ the group of homology classes represented by Zariski closed $k$-dimensional subvarieties of $X\left( \mathbb{R} \right)$. In this note we show that $H_{1}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ is a proper subgroup of ${{H}_{1}}\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ for a nonorientable hyperelliptic surface $X$. We also determine all possible groups $H_{1}^{a\lg }\left( X\left( \mathbb{R} \right),\,\mathbb{Z}/2 \right)$ for a real ruled surface $X$ in connection with the previously known description of all possible topological configurations of $X$.

We show that every compactum has cohomological dimension 1 with respect to a finitely generated nilpotent group $G$ whenever it has cohomological dimension 1 with respect to the abelianization of $G$. This is applied to the extension theory to obtain a cohomological dimension theory condition for a finite-dimensional compactum $X$ for extendability of every map from a closed subset of $X$ into a nilpotent $\text{CW}$-complex $M$ with finitely generated homotopy groups over all of $X$.

We find a lower bound on the number of imaginary quadratic extensions of the function field ${{\mathbb{F}}_{q}}\left( T \right)$ whose class groups have an element of a fixed order.

More precisely, let $q\,\ge \,5$ be a power of an odd prime and let $g$ be a fixed positive integer $\ge \,3$. There are $\gg \,{{q}^{\ell \left( \frac{1}{2}+\frac{1}{g} \right)}}$ polynomials $D\,\in \,{{\mathbb{F}}_{q}}\left[ T \right]$ with $\deg \left( D \right)\,\le \,\ell $ such that the class groups of the quadratic extensions ${{\mathbb{F}}_{q}}\left( T,\,\sqrt{D} \right)$ have an element of order $g$.

We prove an algebraic “no-go theorem” to the effect that a nontrivial Poisson algebra cannot be realized as an associative algebra with the commutator bracket. Using it, we show that there is an obstruction to quantizing the Poisson algebra of polynomials generated by a nilpotent basic algebra on a symplectic manifold. This result generalizes Groenewold’s famous theorem on the impossibility of quantizing the Poisson algebra of polynomials on ${{\mathbf{R}}^{2n}}$. Finally, we explicitly construct a polynomial quantization of a symplectic manifold with a solvable basic algebra, thereby showing that the obstruction in the nilpotent case does not extend to the solvable case.

This paper is concernedwith the structure of the arithmetic sum of a finite number of central Cantor sets. The technique used to study this consists of a duality between central Cantor sets and sets of subsums of certain infinite series. One consequence is that the sum of a finite number of central Cantor sets is one of the following: a finite union of closed intervals, homeomorphic to the Cantor ternary set or an $M$-Cantorval.

Let ${{B}_{N}}$ be the unit ball in ${{\mathbb{C}}^{N}}$ and let $f$ be a function holomorphic and ${{L}^{2}}$-integrable in ${{B}_{N}}$. Denote by $E\left( {{B}_{N}},\,f \right)$ the set of all slices of the form $\Pi \,=\,L\,\cap \,{{B}_{N}}$, where $L$ is a complex one-dimensional subspace of ${{\mathbb{C}}^{N}}$, for which $f{{|}_{\Pi }}$ is not ${{L}^{2}}$-integrable (with respect to the Lebesgue measure on L). Call this set the exceptional set for $f$. We give a characterization of exceptional sets which are closed in the natural topology of slices.

Let $c\,=\,\left( {{c}_{1}},\ldots ,{{c}_{n}} \right)$ be such that ${{c}_{1}}\,\ge \,\cdots \,\ge \,{{c}_{n}}$. The $c$-numerical range of an $n\,\times \,n$ matrix $A$ is defined by

$${{W}_{c}}\left( A \right)\,=\,\left\{ \sum\limits_{j=1}^{n}{{{c}_{j}}\left( A{{x}_{j}},\,{{x}_{j}} \right)\,:\,\left\{ {{x}_{1}},\ldots ,{{x}_{n}} \right\}\,\text{an}\,\text{orthonormal basis for }{{\mathbf{C}}^{n}}} \right\}\,,$$

and the $c$-numerical radius of $A$ is defined by ${{r}_{c}}\left( A \right)\,=\,\max \left\{ \left| z \right|\,:\,z\,\in \,{{W}_{c}}\left( A \right) \right\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying

$${{W}_{c}}\left( \phi \left( A \right) \right)={{W}_{c}}\left( A \right)\text{for}\,\,\text{all}\,\,A\,\text{or}\,\,{{r}_{c}}\left( \phi \left( A \right) \right)={{r}_{c}}\left( A \right)\text{for}\,\,\text{all}\,A.$$

We classify finite subgroups of $\text{SO}\left( 4 \right)$ generated by anti-unitary involutions. They correspond to involutions fixing pointwise a Lagrangian plane. Explicit descriptions of the finite groups and the configurations of Lagrangian planes are obtained.

In this note the multiplicities of binary recurrences over algebraic number fields are investigated under some natural assumptions.

In May, 1999 James Greig Arthur, University Professor at the University of Toronto was awarded the Canada Gold Medal by the National Science and Engineering Research Council. This is a high honour for a Canadian scientist, instituted in 1991 and awarded annually, but not previously to a mathematician, and the choice of Arthur, although certainly a recognition of his greatmerits, is also a recognition of the vigour of contemporary Canadian mathematics.

Jacobi-like forms for a discrete subgroup $\Gamma \,\subset \,\text{SL}\left( 2,\,\mathbb{R} \right)$ are formal power series with coefficients in the space of functions on the Poincaré upper half plane satisfying a certain functional equation, and they correspond to sequences of certain modular forms. We introduce Hecke operators acting on the space of Jacobi-like forms and obtain an explicit formula for such an action in terms of modular forms. We also prove that those Hecke operator actions on Jacobi-like forms are compatible with the usual Hecke operator actions on modular forms.

Functions defined on closed sets are simultaneously approximated and interpolated by meromorphic functions with prescribed poles and zeros outside the set of approximation.

Precise, elegant evaluations are given for Gauss sums of orders six and twelve.

In answer to a question posed in [3], we give sufficient conditions on a Lie nilmanifold so that any ergodic rotation of the nilmanifold is metrically conjugate to its inverse. The condition is that the Lie algebra be what we call quasi-graded, and is weaker than the property of being graded. Furthermore, the conjugating map can be chosen to be an involution. It is shown that for a special class of groups, the condition of quasi-graded is also necessary. In certain examples there is a continuum of conjugacies.

We construct unbounded positive ${{C}^{2}}$-solutions of the equation $\Delta u\,+\,K{{u}^{\left( n+2 \right)/\left( n-2 \right)}}\,=\,0$ in ${{\mathbb{R}}^{n}}$ (equipped with Euclidean metric ${{g}_{0}}$) such that $K$ is bounded between two positive numbers in ${{\mathbb{R}}^{n}}$, the conformal metric $g\,=\,{{u}^{4/\left( n-2 \right)}}{{g}_{0}}$ is complete, and the volume growth of $g$ can be arbitrarily fast or reasonably slow according to the constructions. By imposing natural conditions on $u$, we obtain growth estimate on the ${{L}^{2n/\left( n-2 \right)}}$-norm of the solution and show that it has slow decay.

There is a theorem, usually attributed to Napoleon, which states that if one takes any triangle in the Euclidean Plane, constructs equilateral triangles on each of its sides, and connects the midpoints of the three equilateral triangles, one will obtain an equilateral triangle. We consider an analogue of this problem in the hyperbolic plane.

We show that an $\text{RA}$ loop has a torsion-free normal complement in the loop of normalized units of its integral loop ring. We also investigate whether an $\text{RA}$ loop can be normal in its unit loop. Over fields, this can never happen.