Let $X$ be a locally compact non-compact Hausdorff topological space. Consider the algebras $C\left( X \right),{{C}_{b}}\left( X \right),{{C}_{0}}\left( X \right),\,\text{and}\,{{C}_{00}}\left( X \right)$ of respectively arbitrary, bounded, vanishing at infinity, and compactly supported continuous functions on $X$. Of these, the second and third are ${{C}^{*}}$-algebras, the fourth is a normed algebra, whereas the first is only a topological algebra (it is indeed a pro-${{C}^{*}}$- algebra). The interesting fact about these algebras is that if one of them is given, the others can be obtained using functional analysis tools. For instance, given the ${{C}^{*}}$-algebra ${{C}_{0}}\left( X \right)$, one can get the other three algebras by ${{C}_{00}}\left( X \right)=K\left( {{C}_{0}}\left( X \right) \right),{{C}_{b}}\left( X \right)=M\left( {{C}_{0}}\left( X \right) \right),C\left( X \right)=\Gamma \left( K\left( {{C}_{0}}\left( X \right) \right) \right)$, where the right hand sides are the Pedersen ideal, the multiplier algebra, and the unbounded multiplier algebra of the Pedersen ideal of ${{C}_{0}}\left( X \right)$, respectively. In this article we consider the possibility of these transitions for general ${{C}^{*}}$-algebras. The difficult part is to start with a pro-${{C}^{*}}$-algebra A and to construct a ${{C}^{*}}$-algebra ${{A}_{0}}$ such that $A=\Gamma \left( K\left( {{A}_{0}} \right) \right)$. The pro-${{C}^{*}}$-algebras for which this is possible are called locally compact and we have characterized them using a concept similar to that of an approximate identity.

In this paper, we develop techniques for solving ternary Diophantine equations of the shape $A{{x}^{n}}+B{{y}^{n}}=C{{z}^{2}}$, based upon the theory of Galois representations and modular forms. We subsequently utilize these methods to completely solve such equations for various choices of the parameters $A,\,B\,\text{and}\,\text{C}$. We conclude with an application of our results to certain classical polynomial-exponential equations, such as those of Ramanujan–Nagell type.

We provide the first unconditional proof that the ring $\mathbb{Z}[\sqrt{14}]$ is a Euclidean domain. The proof is generalized to other real quadratic fields and to cyclotomic extensions of $\mathbb{Q}$. It is proved that if $K$ is a real quadratic field (modulo the existence of two special primes of $K$) or if $K$ is a cyclotomic extension of $\mathbb{Q}$ then:

The proof is a modification of the proof of a theorem of Clark and Murty giving a similar result when $K$ is a totally real extension of degree at least three. The main changes are a new Motzkin-type lemma and the addition of the large sieve to the argument. These changes allow application of a powerful theorem due to Bombieri, Friedlander and Iwaniec in order to obtain the result in the real quadratic case. The modification also allows the completion of the classification of cyclotomic extensions in terms of the Euclidean property.

Let $K$ be a finite Galois extension of the field of rational numbers with unit rank greater than 3. We prove that the ring of integers of $K$ is a Euclidean domain if and only if it is a principal ideal domain. This was previously known under the assumption of the generalized Riemann hypothesis for Dedekind zeta functions. We now prove this unconditionally.

We use the lace expansion to analyse networks of mutually-avoiding self-avoiding walks, having the topology of a graph. The networks are defined in terms of spread-out self-avoiding walks that are permitted to take large steps. We study the asymptotic behaviour of networks in the limit of widely separated network branch points, and prove Gaussian behaviour for sufficiently spread-out networks on ${{\mathbb{Z}}^{d}}$ in dimensions $d\,>\,4$.

The billiard flow in the plane has a simple geometric definition; the movement along straight lines of points except where elastic reflections are made with the boundary of the billiard domain. We consider a class of open billiards, where the billiard domain is unbounded, and the boundary is that of a finite number of strictly convex obstacles. We estimate the Hausdorff dimension of the nonwandering set ${{M}_{0}}$ of the discrete time billiard ball map, which is known to be a Cantor set and the largest invariant set. Under certain conditions on the obstacles, we use a well-known coding of ${{M}_{0}}$ [Mor91] and estimates using convex fronts related to the derivative of the billiard ball map [Sto03] to estimate the Hausdorff dimension of local unstable sets. Consideration of the local product structure then yields the desired estimates, which provide asymptotic bounds on the Hausdorff dimension's convergence to zero as the obstacles are separated.

Every norm $\mathcal{V}$ on ${{\text{C}}^{\text{n}}}$ induces two norm numerical ranges on the algebra ${{M}_{n}}\,\text{of}\,\text{all}\,\text{n}\,\times \,\text{n}$ complex matrices, the spatial numerical range

$$W\left( A \right)=\left\{ {{x}^{*}}Ay:x,y\in {{\text{C}}^{n}},{{v}^{D}}\left( x \right)=v\left( y \right)={{x}^{*}}y=1 \right\},$$

where ${{\mathcal{V}}^{D}}$ is the norm dual to $\mathcal{V}$, and the algebra numerical range

$$V\left( A \right)=\left\{ f\left( A \right):f\in S \right\},$$

where $S$ is the set of states on the normed algebra ${{M}_{n}}$ under the operator norm induced by $\mathcal{V}$. For a symmetric norm $\mathcal{V}$, we identify all linear maps on ${{M}_{n}}$ that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, i.e., linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\mathcal{V}$ is not the ${{\ell }_{1}},{{\ell }_{2}},\,\text{or}\,{{\ell }_{\infty }}$ norms, then the linear maps that preserve either numerical range or either set of states are “inner”, i.e., of the form $A\mapsto {{Q}^{*}}AQ$, where $Q$ is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the ${{\ell }_{1}}$ and the ${{\ell }_{\infty }}$ norms, the results are quite different.

Let $G=\text{S}{{\text{p}}_{\text{2n}}}$ be the symplectic group defined over a number field $F$. Let $\mathbb{A}$ be the ring of adeles. A fundamental problem in the theory of automorphic forms is to decompose the right regular representation of $G\left( \mathbb{A} \right)$ acting on the Hilbert space ${{L}^{2}}\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)$. Main contributions have been made by Langlands. He described, using his theory of Eisenstein series, an orthogonal decomposition of this space of the form: $L_{dis}^{2}\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)\,=\,{{\oplus }_{\left( M,\,\pi \right)}}L_{dis}^{2}{{\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)}_{\left( M,\,\pi \right)}},\,\text{where}\,\left( M,\,\pi \right)$ is a Levi subgroup with a cuspidal automorphic representation $\pi $ taken modulo conjugacy. (Here we normalize $\pi $ so that the action of the maximal split torus in the center of $G$ at the archimedean places is trivial.) and $L_{\text{dis}}^{2}{{\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)}_{\left( M,\pi \right)}}$ is a space of residues of Eisenstein series associated to $\left( M,\,\pi \right)$. In this paper, we will completely determine the space $L_{\text{dis}}^{2}{{\left( G\left( F \right)\backslash G\left( \mathbb{A} \right) \right)}_{\left( M,\pi \right)}}$, when $M\simeq \text{G}{{\text{L}}_{2}}\times \text{G}{{\text{L}}_{2}}$. This is the first result on the residual spectrum for non-maximal, non-Borel parabolic subgroups, other than $\text{G}{{\text{L}}_{n}}$.

Suppose $K$ is an imaginary quadratic field and $E$ is an elliptic curve over a number field $F$ with complex multiplication by the ring of integers in $K$. Let $p$ be a rational prime that splits as ${{\mathfrak{p}}_{1}}{{\mathfrak{p}}_{2}}$ in $K$. Let ${{E}_{{{p}^{n}}}}$ denote the ${{p}^{n}}$-division points on $E$. Assume that $F\left( {{E}_{{{p}^{n}}}} \right)$ is abelian over $K\,\text{for}\,\text{all}\,n\,\ge \,0$. This paper proves that the Pontrjagin dual of the $\mathfrak{p}_{1}^{\infty }$-Selmer group of $E$ over $F\left( {{E}_{{{p}^{\infty }}}} \right)$ is a finitely generated free $\wedge $-module, where $\wedge $ is the Iwasawa algebra of $\text{Gal}\left( F\left( {{E}_{{{p}^{\infty }}}} \right)/F\left( E\mathfrak{p}_{1}^{\infty }{{\mathfrak{p}}_{2}} \right) \right)$ . It also gives a simple formula for the rank of the Pontrjagin dual as a $\wedge $-module.

In [6], Walter Philipp wrote that “… the law of the iterated logarithm holds for any process for which the Borel-Cantelli Lemma, the central limit theorem with a reasonably good remainder and a certain maximal inequality are valid.” Many authors [1], [2], [4], [5], [9] have followed this plan in proving the law of the iterated logarithm for sequences (or fields) of dependent random variables.

We carry on this tradition by proving the law of the iterated logarithm for a random field whose correlations satisfy an exponential decay condition like the one obtained by Spohn [8] for certain Gibbs measures. These do not fall into the $\phi $-mixing or strong mixing cases established in the literature, but are needed for our investigations [7] into diffusions on configuration space.

The proofs are all obtained by patching together standard results from [5], [9] while keeping a careful eye on the correlations.