Using properties of the Frobenius eigenvalues, we show that, in a precise sense, ‘most’ isomorphism classes of (principally polarized) simple abelian varieties over a finite field are characterized, up to isogeny, by the sequence of their division fields, and we show a similar result for ‘most’ isogeny classes. Some global cases are also treated.

Benford's law (to base $B$) for an infinite sequence $\{x_k: k \ge 1\}$ of positive quantities $x_k$ is the assertion that $\{ \log_B x_k : k \ge 1\}$ is uniformly distributed $(\bmod\ 1)$. The $3x+1$ function $T(n)$ is given by $T(n)=(3n+1)/{2}$ if $n$ is odd, and $T(n)= n/2$ if $n$ is even. This paper studies the initial iterates $x_k= T^{(k)}(x_0)$ for $1 \le k \le N$ of the $3x+1$ function, where $N$ is fixed. It shows that for most initial values $x_0$, such sequences approximately satisfy Benford's law, in the sense that the discrepancy of the finite sequence $\{\log_B x_k: 1 \le k \le N \}$ is small.

Smooth complete algebraic varieties defined over an algebraically closed field are divided into two classes according to the behaviour of the albanese map. Criteria are given for when a variety belongs to either class. The relationship with the Tate conjecture for abelian varieties is discussed.

We prove a general theorem showing that iterated skew polynomial extensions of the type that fit the conditions needed by Cauchon's deleting derivations theory and by the Goodearl–Letzter stratification theory are unique factorisation rings in the sense of Chatters and Jordan. This general result applies to many quantum algebras; in particular, generic quantum matrices and quantized enveloping algebras of the nilpotent part of a semisimple Lie algebra are unique factorisation domains in the sense of Chatters. The result also extends to generic quantum grassmannians (by using noncommutative dehomogenisation) and to the quantum groups ${\mathcal O}_q(\text{GL}_n)$ and ${\mathcal O}_q(\text{SL}_n)$.

In this paper we construct new obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group. We also determine the structure of the cokernel of the Johnson homomorphism for degrees 2 and 3.

Let $G$ be a finite (non-abelian) irreducible linear subgroup over the complex numbers, and let $g$ be an element of $G$ of prime order $p$. Suppose that $g$ does not belong to a proper normal subgroup of $G$. We show that the number of distinct eigenvalues of $g$ can only be $p,p-1,p-2,(p+1)/2$ or $(p-1)/2$. Moreover, we provide a full classification of such groups $G $ provided that $g$ has at most $p-2$ distinct eigenvalues.

We consider the cohomology of Mackey functors for infinite groups and define the Mackey-cohomological dimension ${\rm cd}_{{\mathfrak M}} G$ of a group $G$. We will relate this dimension to other cohomological dimensions such as the Bredon-cohomological dimension ${\rm cd}_{{\mathfrak F}}G$ and the relative cohomological dimension ${{\mathfrak F}}\text{-}{\rm cd} G$. In particular, we show that for virtually torsion free groups the Mackey-cohomological dimension is equal to both ${{\mathfrak F}}\text{-}{\rm cd} G$ and the virtual cohomological dimension.

A formula for the interior $\varepsilon$-neighborhood of the classical von Koch snowflake curve is computed in detail. This function of $\varepsilon$ is shown to match quite closely with earlier predictions of what it should be, but is also much more precise. The resulting ‘tube formula’ is expressed in terms of the Fourier coefficients of a suitable nonlinear and periodic analog of the standard Cantor staircase function and reflects the self-similarity of the Koch curve. As a consequence, the possible complex dimensions of the Koch snowflake are computed explicitly.

We study concentration phenomena for the system

\[\varepsilon^2 \Delta v - v - \delta \phi v + \gamma v^{p} =0,\quad \Delta\phi+ \delta v^2=0 \]

in the unit ball $B_1$ of $\mathbb{R}^3$ with Dirichlet boundary conditions. Here $\varepsilon,\ \delta,\ \gamma >0$ and $p>1$. We prove the existence of positive radial solutions $(v_{\varepsilon}, \phi_{\varepsilon})$ such that $v_{\varepsilon}$ concentrates at a distance $({\varepsilon}/{2}) |{\rm log}\, {\varepsilon} |$ away from the boundary $\partial B_1$ as the parameter $\varepsilon$ tends to 0. The approach is based on a combination of Lyapunov–Schmidt reduction procedure together with a variational method.

This paper deals with the lack of compactness in the nonlinear elliptic problem $-\Delta u+u=|u|^{p-2}u$ in $\Omega,\ u>0$ in $\Omega,\ u=0$ on $\partial \Omega$, when $\Omega$ is un unbounded domain in $\mathbb{R}^n$ and $2<p<2n/(n-2)$.

In particular, a domain $\widetilde\Omega$ is provided where non-converging Palais–Smale sequences exist at every energy level. Nevertheless, it is proved that the problem has infinitely many solutions on $\widetilde\Omega$.

This paper estimates from below the attractor dimension of the dynamical system determined from a chemotaxis growth model which was presented by Mimura and Tsujikawa. It is already known that the dynamical system has exponential attractors and it is also known by numerical computations that the model contains various pattern solutions. This paper is then devoted to estimating the attractor dimension from below and in fact to showing that, as the parameter of chemotaxis increases and tends to infinity, so does the attractor dimension. Such a result is in a good correlation with the numerical results.

We study the following question: given an open set $\Omega$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $\Omega$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied so far mostly for convex domains $\Omega$ in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and ${\mathbb Z}^d$. We exhibit upper bounds for $\int f$ assuming geometric properties of $\Omega$ of two types: (a) packing properties of $\Omega$ and (b) spectral properties of $\Omega$. Several examples and applications of the main theorems are shown. In particular, we recover and extend several known results concerning convex domains in Euclidean space. Also, we investigate the question of estimating $\int_{\Omega}f$ over possibly dispersed sets solely in dependence of the given measure $m:=|\Omega|$ of $\Omega$. In this respect we show that in ${\mathbb R}$ and ${\mathbb Z}$ the integral is maximal for intervals.

We give necessary and sufficient conditions in order that inequalities of the type

\[ \| T_K f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \quad f \in L^p(d\sigma), \]

hold for a class of integral operators $T_K f(x) = \int_{R^n} K(x,y) f(y)\,d \sigma(y)$ with nonnegative kernels, and measures $d \mu$ and $d\sigma$ on $\mathbb{R}^n$, in the case where $p>q>0$ and $p>1$.

An important model is provided by the dyadic integral operator with kernel $K_{\mathcal D}(x, y)= \sum_{Q\in{\mathcal D}} K(Q)\chi_Q(x) \chi_Q(y)$, where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\mathbb{R}^n$, and $K(Q)$ are arbitrary nonnegative constants associated with $Q \in{\mathcal D}$.

The corresponding continuous versions are deduced from their dyadic counterparts. In particular, we show that, for the convolution operator $T_k f = k \star f$ with positive radially decreasing kernel $k(|x-y|)$, the trace inequality

\[\| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d x)}, \quad f \in L^p(dx),\]

holds if and only if ${\mathcal W}_{k}[\mu] \in L^s (d\mu)$, where $s = q(p-1)/(p-q)$. Here ${\mathcal W}_{k}[\mu]$ is a nonlinear Wolff potential defined by ${\mathcal W}_{k}[\mu](x)=\int_0^{+\infty} k(r) \bar{k}(r)^{1/(p-1)} \mu(B(x,r))^{1/(p-1)} r^{n-1} \, dr$, and $\bar{k}(r)=(1/r^n)\int_0^r k(t) t^{n-1} \, dt$. Analogous inequalities for $1\le q < p$ were characterized earlier by the authors using a different method which is not applicable when $q<1$.

We describe complemented copies of $\ell_2$ both in $C(K_1)\hat{\otimes}_{\pi} C(K_2)$ when at least one of the compact spaces $K_i$ is not scattered and in $L_1(\mu_1)\hat{\otimes}_{\epsilon} L_1(\mu_2)$ when at least one of the measures is not atomic. The corresponding local construction gives uniformly complemented copies of the $\ell_2^n$ in $c_0\hat{\otimes}_{\pi} c_0$. We continue the study of $c_0\hat{\otimes}_{\pi} c_0$ showing that it contains a complemented copy of Stegall's space $c_0(\ell_2^n)$ and proving that $(c_0\ppi c_0)''$ is isomorphic to $\ell_\infty(\ell_\infty^n\hat{\otimes}_{\pi} \ell_\infty^n)$, together with other results. In the last section we use Hardy spaces to find an isomorphic copy of $L_p$ in the space of compact operators from $L_q$ to $L_r$, where $1<p,q,r<\infty$ and $1/r=1/p+1/q$.

We produce examples of Legendrian isotopy classes in $ST^*\mathbb{R}^n$ (the space of cooriented contact elements of $\mathbb{R}^n$, endowed with its standard contact structure) such that any element of those classes has a singular (non-immersed) wave front, that is, the projection to the base of the fibration $ST^*\mathbb{R}^n \rightarrow \mathbb{R}^n$, when restricted to such a Legendrian embedding, has local singularities. Furthermore, our examples are such that the underlying Legendrian homotopy classes contain some Legendrian embeddings whose wave fronts are immersed. These examples are motivated by a question of Arnold about the removability of singularities of wave fronts by means of Legendrian isotopy. The two key points, which are of independent interest, are as follows.

(1) A duality argument which allows the question to be translated into a problem about Legendrian submanifolds of the one-jets space of the sphere, and hence to use the technology of generating families.

(2) Several independent constructions of families of functions on compact manifolds such that the ‘critical sets’ of these families are connected. For example, we prove that given two compact, connected manifolds $M$ and $N$ of positive dimension, there exists

\[f~: N \times M \rightarrow \mathbb{R}, (x,y) \rightarrow f(x,y),\quad x\in N, \ y \in M\]

such that the equation $(\partial f/\partial y)(x,y)=0$ is regular and defines a smooth and connected submanifold of $M \times N$.