We give a survey of old and new results concerning the expressibility of the real roots of a solvable polynomial over a real number field by real radicals. A characterization of Fermat primes is obtained in terms of solvability by real radicals for certain ploynomials.

We characterize surjective linear maps on $B\text{(X)}$ that are spectrally bounded and spectrally bounded below.

We use the homological perturbation lemma to give an algebraic proof of the cyclic Eilenberg-Zilber theorem for cylindrical modules.

We study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in ${{\mathbb{R}}^{N}}$, and (ii) when the average is taken over all dilations of star-shaped regions in ${{\mathbb{R}}^{N}}$. These cases include, respectively, the average over annuli and the average over balls centered at origin.

A basic problem in dynamics is to identify systems with positive entropy, i.e., systems which are “chaotic.” While there is a vast collection of results addressing this issue in topological dynamics, the phenomenon of positive entropy remains by and large a mystery within the broader noncommutative domain of ${{C}^{*}}$-algebraic dynamics. To shed some light on the noncommutative situation we propose a geometric perspective inspired by work of Glasner and Weiss on topological entropy. This is a written version of the author’s talk at the Winter 2002 Meeting of the Canadian Mathematical Society in Ottawa, Ontario.

We show that the product of four or five consecutive positive terms in arithmetic progression can never be a perfect power whenever the initial term is coprime to the common difference of the arithmetic progression. This is a generalization of the results of Euler and Obláth for the case of squares, and an extension of a theorem of Győry on three terms in arithmetic progressions. Several other results concerning the integral solutions of the equation of the title are also obtained. We extend results of Sander on the rational solutions of the equation in $n,\,y$ when $b\,=\,d\,=\,1$. We show that there are only finitely many solutions in $n,\,d,\,b,\,y$ when $k\,\ge \,3,\,l\,\ge \,2$ are fixed and $k\,+\,l\,>\,6$.

Let $p$ be a prime number. Let $k$ be a finite field of characteristic $p$. The subset $X\,+\,{{X}^{2}}k[[X]]$ of the ring $k\left[\!\left[ X \right]\!\right]$ is a group under the substitution law $\circ $ sometimes called the Nottingham group of $k$, it is denoted by ${{\mathcal{R}}_{k}}$. The ramification of one series $\gamma \,\in \,{{\mathcal{R}}_{k}}$ is caracterized by its lower ramification numbers: ${{i}_{m}}(\gamma )\,=\,\text{or}{{\text{d}}_{X}}({{\gamma }^{{{p}^{m}}}}\,(X)/X-1)\,$, as well as its upper ramification numbers:

$${{u}_{m}}(\gamma )\ =\ {{i}_{0}}(\gamma )+\frac{{{i}_{1}}(\gamma )-{{i}_{0}}(\gamma )}{p}\,+\,.\,.\,.\,+\,\frac{{{i}_{m}}(\gamma )-{{i}_{m-1}}(\gamma )}{{{p}^{m}}},\,\,\,\,\,\,(m\,\in \,\mathbb{N}).$$

By Sen's theorem, the ${{u}_{m}}(\gamma )$ are integers. In this paper, we determine the sequences of integers (${{u}_{m}}$) for which there exists $\gamma \,\in \,{{\mathcal{R}}_{k}}$ such that ${{u}_{m}}(\gamma )\,=\,{{u}_{m}}$ for all integer $m\,\ge \,0$.

The ${{Q}_{p}}$ spaces coincide with the Bloch space for $p\,>\,1$ and are subspaces of $\text{BMOA}$ for $0\,<\,p\,\le \,1$. We obtain lower and upper estimates for the essential norm of a composition operator from the Bloch space into ${{Q}_{p}}$, in particular from the Bloch space into $\text{BMOA}$.

Generalizing results from $[\text{MM}1]$ referring to the intersection body $\text{IK}$ and the cross-section body $CK$ of a convex body $K\,\,\subset \,\,{{\mathbb{R}}^{d}}$, $d\,\ge \,2$, we prove theorems about maximal $k$-sections of convex bodies, $k\,\in \,\{1,\,\ldots ,\,d\,-\,1\}$, and, simultaneously, statements about common maximal $(d\,-\,1)-$ and 1-transversals of families of convex bodies.

Let $g:\,{{M}^{2n}}\,\to \,{{M}^{2n}}$ be a smooth map of period $m\,>\,2$ which preserves orientation. Suppose that the cyclic action defined by $g$ is regular and that the normal bundle of the fixed point set $F$ has a $g$-equivariant complex structure. Let $F\,\pitchfork \,F$ be the transverse self-intersection of $F$ with itself. If the $g$-signature $\text{Sign(g,}\,\text{M)}$ is a rational integer and $n\,<\,\phi (m)$, then there exists a choice of orientations such that $\text{Sign}\,\text{(g,}\,\text{M)}\,\text{=}\,\text{Sign}\,\text{F}\,\text{=}\,\text{Sign}(F\,\pitchfork \,F)$.

In this paper we give an inversion formula of the Radon transform on the Heisenberg group by using the wavelets defined in [3]. In addition, we characterize a space such that the inversion formula of the Radon transform holds in the weak sense.

We show that the Prym map for 4-th cyclic étale covers of curves of genus 4 is a dominant morphism to a Shimura variety for a family of Abelian 6-folds of Weil type. According to the result of Schoen, this implies algebraicity of Weil classes for this family.

We axiomatize the main properties of the classical Turán Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

A band is a semigroup of idempotent operators. A nonnegative band $\mathcal{S}$ in $B({{L}^{2}}(X))$ having at least one element of finite rank and with rank $(S)\,>\,1$ for all $S$ in $\mathcal{S}$ is known to have a special kind of common invariant subspace which is termed a standard subspace (defined below).

Such bands are called decomposable. Decomposability has helped to understand the structure of nonnegative bands with constant finite rank. In this paper, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained which facilitates the understanding of their geometric structure.

In this paper, we give some conditions to judge when a system of Hermitian quadratic forms has a real linear combination which is positive definite or positive semi-definite. We also study some related geometric and topological properties of the moduli space.

Let $V$ be a $K3$ surface defined over a number field $k$. The Batyrev-Manin conjecture for $V$ states that for every nonempty open subset $U$ of $V$, there exists a finite set ${{Z}_{U}}$ of accumulating rational curves such that the density of rational points on $U\,-\,{{Z}_{U}}$ is strictly less than the density of rational points on ${{Z}_{U}}$. Thus, the set of rational points of $V$ conjecturally admits a stratification corresponding to the sets ${{Z}_{U}}$ for successively smaller sets $U$.

In this paper, in the case that $V$ is a Kummer surface, we prove that the Batyrev-Manin conjecture for $V$ can be reduced to the Batyrev-Manin conjecture for $V$ modulo the endomorphisms of $V$ induced by multiplication by $m$ on the associated abelian surface $A$. As an application, we use this to show that given some restrictions on $A$, the set of rational points of $V$ which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

We apply the method of complex scaling to give a natural proof of a formula relating the multiplicity of a resonance to the multiplicity of a pole of the scattering matrix.

We axiomatize the main properties of the classical Erdös-Kac Theorem in order to apply it to a general context. We provide applications in the cases of number fields, function fields, and geometrically irreducible varieties over a finite field.

This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups $G$. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of $G$ is complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced.

The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to a $G$-dynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as a $G$-dynamical system, is a factor of the dynamical local hull.

In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety $V$ which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting functions of the lines which cover the original variety $V$.