Let E2(T) denote the error term in the asymptotic formula for ∫T0[mid ]ζ(½+it)[mid ]4dt. It is proved that there exist constants A>0, B>1 such that for T[ges ]T0>0 every interval [T, BT] contains points T1, T2 for which

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and that ∫T0[mid ]E2(t) [mid ]adt[Gt ]T1+(a/2) for any fixed a[ges ]1. These results complement earlier results of Motohashi and Ivić that ∫T0E2(t) dt[Lt ]T3/2 and that ∫T0E22(t) dt[Lt ]T2logCT for some C>0. Omega-results analogous to the above ones are obtained also for the error term in the asymptotic formula for the Laplace transform of [mid ]ζ(½+it)[mid ]4.

Three questions concerning the distribution of the numbers of points on elliptic curves over a finite prime field are considered. First, the previously published bounds for the distribution are tightened slightly. Within these bounds, there are wild fluctuations in the distribution, and some heuristics are discussed (supported by numerical evidence) which suggest that numbers of points with no large prime divisors are unusually prevalent. Finally, allowing the prime field to vary while fixing the field of fractions of the endomorphism ring of the curve, the order of magnitude of the average order of the number of divisors of the number of points is determined, subject to assumptions about primes in quadratic progressions.

There are implications for factoring integers by Lenstra's elliptic curve method. The heuristics suggest that (i) the subtleties in the distribution actually favour the elliptic curve method, and (ii) this gain is transient, dying away as the factors to be found tend to infinity.

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.

In this paper we consider a generalization to algebraic number fields of the classical Hermite constant γn. For this constant we extend the well-known Minkowski bound and study the notion of extreme forms in this setting.

We say that a bounded linear operator T acting on a Banach space B is antisupercyclic if for any $x\in B$ either $T^nx=0$ for some positive integer n or the sequence $\{T^nx/\|T^nx\|\}$ weakly converges to zero in B. Antisupercyclicity of T means that the angle criterion of supercyclicity is not satisfied for T in the strongest possible way. Normal antisupercyclic operators and antisupercyclic bilateral weighted shifts are characterized.

As for the Volterra operator V, it is proved that if $1\leq p\leq\infty$ and any $f\in L_p[0,1]$ then the limit $\lim_{n\to\infty} (n!\|V^nf\|_p)^{1/n}$ does exist and equals $1-\inf\,\hbox{\rm supp}\,(f)$. Upon using this asymptotic formula it is proved that the operator V acting on the Banach space $L_p[0,1]$ is antisupercyclic for any $p\in(1,\infty)$. The same statement for $p=1$ or $p=\infty$ is false. The analogous results are proved for operators $V^zf(x)=(1/{\Gamma(z)})\int_0^x f(t) (x-t)^{z-1}\,dt$ when the real part of $z\in{\mathbb C}$ is positive.

In 1903 Minkowski showed that, given pairwise different unit vectors μ1, …, μm in Euclidean n-space ℝn which span ℝn, and positive reals μ1, …, μm such that [sum ]mi=1μiμi = 0, there exists a polytope P in ℝn, unique up to translation, with outer unit facet normals μ1, …, μm and corresponding facet volumes μ1, …, μm. This paper deals with the computational complexity of the underlying reconstruction problem, to determine a presentation of P as the intersection of its facet halfspaces. After a natural reformulation that reflects the fact that the binary Turing-machine model of computation is employed, it is shown that this reconstruction problem can be solved in polynomial time when the dimension is fixed but is #ℙ-hard when the dimension is part of the input.

The problem of ‘Minkowski reconstruction’ has various applications in image processing, and the underlying data structure is relevant for other algorithmic questions in computational convexity.

The paper shows that any graph G with the maximum degree Δ(G) [ges ] 8, which is embeddable in a surface Σ of Euler characteristic χ(Σ) [ges ] 0, is totally (Δ(G)+2)-colorable. In general, it is shown that any graph G which is embeddable in a surface Σ and satisfies the maximum degree Δ(G) [ges ] (20/9) (3−χ(Σ))+1 is totally (Δ(G)+2)-colorable.

Leaning on a remarkable paper of Pryce, the paper studies two independent classes of topological Abelian groups which are strictly angelic when endowed with their Bohr topology. Some extensions are given of the Eberlein–šmulyan theorem for the class of topological Abelian groups, and finally, for a large subclass of the latter, Bohr angelicity is related to the Schur property.

The spectrum and essential spectrum of a self-adjoint operator in a real Hilbert space are characterized in terms of Palais–Smale conditions on its quadratic form and Rayleigh quotient respectively.

The subject is the 2-point boundary value problem for a second order ordinary differential equation

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the simplest and oldest of all boundary value problems, which arose in Euler's work on calculus of variations in the eighteenth century. Accordingly, there is a vast literature, mostly in the context of applied mathematics, where one frequently uses the methods of functional analysis.

Here we adopt a different point of view: we look for the number of solutions of problem (1), if any, and how this number varies with the endpoints (t1, x1) and (t2, x2). The concept of focal decomposition, introduced in [15] and developed by Peixoto and Thom in [19], expresses precisely this point of view (see §2). It was further developed by Kupka and Peixoto in [10] in the context of geodesics. From there, one is led naturally to relationships with the arithmetic of positive definite quadratic forms, a line that is considered in [16]. In both [10] and [16] attention is drawn to the close formal relationship between focal decomposition and the Brillouin zones of solid state physics. In [17] this line is pursued further and it is pointed out that the focal decomposition associated to (1) appears naturally as a prerequisite for the semiclassical quantization of this equation via the Feynman path integral method.

The main goal of the present paper is to prove Theorem 2, stated at the end of §4. There we give a functional ∫t2t1L (t, x, x˙) dt and consider its Euler equation. To this second order differential equation we associate the corresponding focal decomposition. Theorem 2 gives a criterion on the Lagrangian L from which we get a rough geometric description of this focal decomposition. It turns out to be somewhat similar to the one associated to the pendulum equation x¨+sin x = 0, worked out by Peixoto and Thom in [19, pp. 631, 197]. In the case L = (x˙2/2)−V(x) this type of focal decomposition is generic in some topological sense.

In this paper we study the metric geometry of the space $\Sigma$ of positive invertible elements of a von Neumann algebra ${\mathcal A}$ with a finite, normal and faithful tracial state $\tau$. The trace induces an incomplete Riemannian metric $\langle x,y\rangle_a=\tau (ya^{-1}xa^{-1})$, and, though the techniques involved are quite different, the situation here resembles in many relevant aspects that of the $n\times n$ matrices when they are regarded as a symmetric space. For instance, we prove that geodesics are the shortest paths for the metric induced, and that the geodesic distance is a convex function; we give an intrinsic (algebraic) characterization of the geodesically convex submanifolds $M$ of $\Sigma$; and under a suitable hypothesis we prove a factorization theorem for elements in the algebra that resembles the Iwasawa decomposition for matrices. This factorization is obtained via a nonlinear orthogonal projection $\Pi_M:\Sigma\to M$, a map which turns out to be contractive for the geodesic distance.

It is proved that, for each pair (m, n) of non-negative integers, there is a Banach space [Xfr ] for which K0([Bscr ]([Xfr ]))≅ℤm and K1([Bscr ]([Xfr ]))≅ℤn. The K-groups of all closed ideals of operators contained in the ideal of strictly singular operators are computed, and some results about the existence of splittings of certain short exact sequences are derived.

The study of the pentagon equation leads to results on the structure and classification of finite quantum groups. It is proved that $L$ is a finite-dimensional Hopf algebra if and only if there exists an invertible matrix $R$, solution of the pentagon equation $R^{12}R^{13}R^{23}=R^{23}R^{12}$, such that $L\cong P(n, R)$; the Hopf algebra structure of $P(n, R)$ is explicitly described using generators and relations. Finally, it is proved that there exists a one-to-one correspondence between the set of types of $n$-dimensional Hopf algebras and the set of orbits of the action ${\rm GL}_n(k)\times (M_n(k)\otimes M_n(k)) \,{\longrightarrow}\, M_n(k)\otimes M_n(k), (u, R)\,{\longrightarrow}\, (u\otimes u)R(u\otimes u)^{-1}$, the representatives of which are invertible solutions of length $n$ for the pentagon equation.

the paper considers the eigenvalue problem \[ -\delta u-\alpha u+\lambda g(x)u=0\quad \mbox{with}u\in h^{1}(\mathbb{r}^{n}),\;u\neq0, \] where $\alpha,\lambda\in\mathbb{r}$ and \[g(x)\equiv0\mbox{on}\overline{\omega},\quad\ g(x)\in(0,1]\mbox{on}{\mathbb{r}^{n}}\setminus{\overline{\omega}}\quad\mbox{and}\quad\lim_{|x|\rightarrow+\infty}g(x)=1 \] for some bounded open set $\omega\in\mathbb{r}^{n}$.

given $\alpha>0$, does there exist a value of $\lambda>0$ for which the problem has a positive solution? it is shown that this occurs if and only if $\alpha$ lies in a certain interval $(\gamma,\xi_{1})$ and that in this case the value of $\lambda$ is unique, $\lambda=\lambda(\alpha)$. the properties of the function $\lambda(\alpha)$ are also discussed.

It is shown that the invariant subspace of the Bergman space $A^p$ of the unit disk, generated by either an $A^p$-interpolating sequence or a singular inner function with a single point mass on the unit circle, is complemented in $A^p$.

Let v be a measure on a separable metric space. For t, q ∈ R, the centred Hausdorff measures μh with the gauge function h(x, r) = rt(vB(x, r))q is studied. The dimension defined by these measures plays an important role in the study of multifractals. It is shown that if v is a doubling measure, then μh is equivalent to the usual spherical measure, and thus they define the same dimension. Moreover, it is shown that this is true even without the doubling condition, if q [ges ] 1 and t [ges ] 0 or if q [les ] 0. An example in R2 is also given to show the surprising fact that the above assertion is not necessarily true if 0 < q < 1. Another interesting question, which has been asked several times about the centred Hausdorff measure, is whether it is Borel regular. A positive answer is given, using the above equivalence for all gauge functions mentioned above.

In this study of the behaviour of the number of conjugacy classes in finite $p$-groups using pro-$p$ groups, the conjugacy growth function $r_n(G)= \max\{r(G/N)\,{|}\,N\triangleleft_o G,|G\,{:}\,N|=n\}$ is introduced. It is proved that there are no infinite pro-$p$ groups of linear conjugacy growth (that is, there is no $c$ such that $r_n(G)\le c\log_2 n$ for all $n>1$) and it is shown that many known pro-$p$ groups are of exponential conjugacy growth (that is, there exists $\epsilon\,{>}\,0$ such that $r_n(G)\,{\ge}\,n^\epsilon$ for infinitely many values of $n$).

A family of Hecke $C^*$ -algebras can be realised as crossed products by semigroups of endomorphisms. It is shown by dilating representations of the semigroup crossed product that the category of representations of the Hecke algebra is equivalent to the category of continuous unitary representations of a totally disconnected locally compact group.

The paper gives the first examples of C∞ measure preserving flows and diffeomorphisms on the torus that are partially mixing and have a simple spectrum.