Let $\Lambda\subseteq \{1,\ldots, N\}$ and let $\{a_{n}\}_{n\in\Lambda}$ be a sequence with $|a_n|\leq 1$ for all $n$. It is easy to see that\[ \left\|\sum_{n\in\Lambda}a_{n}e{(n\theta)}\right\|_{p}\leq \left\|\sum_{n\in\Lambda}e{(n\theta)}\right\|_{p}\] for every even integer $p$. We give an example which shows that this statement can fail rather dramatically when $p$ is not an even integer. This answers in the negative a question known as the Hardy–Littlewood majorant conjecture, thereby ruling out a certain approach to the restriction and Kakeya families of conjectures.

A cubic transformation formula for the hypergeometric function $_{2}F_{1}(\frac{1}{3},\frac{2}{3};1;z)$ is proved. As an application of this formula a number of arithmetic convolution sums are evaluated. For example, Melfi's formula, \[ \ukksum{n{-}1}{k{=}1}{k{\equiv}1 \, \mbox{(mod $3$)}}\kern-10pt\sigma(k)\sigma(n-k)=\frac{1}{9}\sigma_3(n),\;\;\;n \equiv 2\, \mbox{(mod $3$)}, \] is proved without the use of modular forms.

Let $E$ be an elliptic curve defined over $\Q$ with complex multiplication by the ring of integers of an imaginary quadratic field $K$. We apply an elliptic unit version of Ichimura–Sumida criterion for the $\Z_p$-extension $F_\infty/F$ associated to $E$ and try to determine the characteristic polynomial of the maximal unramified abelian $p$-extension of $F_\infty$.

Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p\gt 0$. Let $L/K$ be a finite Galois extension with Galois group $G\,{=}\,G_{L/K}$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. By the normal basis theorem, $L$ is always a projective $K[G]$-module. But the ring of integers $\mathcal{O}_L$ is a projective$\mathcal{O}_K[G]$-module if and only $L/K$ is tamely ramified. For wildly ramified extensions, the structure of $\mathcal{O}_L$ as an $\mathcal{O}_K[G]$-module is very complicated and quite far from understood. We propose that the ring of Witt vectors $W\s(\mathcal{O}_L)$ is a more well-behaved object. Indeed, we show that for a large class of extensions $L/K$, the pro-abelian group $H^1(G,W\s(\mathcal{O}_L))$ is zero. We conjecture that this is true in general.

An elliptic space is one whose rational homotopy and rational cohomology are both finite dimensional. David Anick conjectured that any simply connected finite CW-complex $S$ can be realized as the $k$-skeleton of some elliptic complex as long as $k\,{>}\,\dim S$, or, equivalently, that any simply connected finite Postnikov piece $S$ can be realized as the base of a fibration $F\,{\to}\,E\,{\to}\,S$ where $E$ is elliptic and $F$ is $k$-connected, as long as the $k$ is larger than the dimension of any homotopy class of $S$. This conjecture is only known in a few cases, and here we show that in particular if the Postnikov invariants of $S$ are decomposable, then the Anick conjecture holds for $S$. We also relate this conjecture with other finiteness properties of rational spaces.

The Atiyah–Singer equivariant signature formula implies that the products of isometrically inequivalent classical spherical space forms with the circle are not homeomorphic, and in fact the same conclusion holds if the circle is replaced by a torus of arbitrary dimension. These results are important in the study of group actions on manifolds. Algebraic $K$-theory yields standard classes of counterexamples for topological and smooth analogs of spherical spaceforms. The results of this paper characterize pairs of nonhomeomorphic topological spherical space forms whose products with a given torus of arbitrary dimension are homeomorphic, and the main result is that the known counterexamples are the only ones that exist. In particular, this and basic results in lower algebraic $K$-theory show that if such products are homeomorphic, then the products are already homeomorphic if one uses a 3-dimensional torus. Sharper results are established for important special cases such as fake lens spaces. The methods are basically surgery-theoretic with some input from homotopy theory. One consequence is the existence of new infinite families of manifolds in all dimensions greater than three such that the squares of the manifolds are homeomorphic although the manifolds themselves are not. Analogous results are obtained in the smooth category.

We show that the semi-direct product of a one-ended torsion-free word-hyperbolic group with $\integers$, given through an automorphism $\alpha{:}\, G \,{\to}\, G$, is a hyperbolic group relative to certain canonical subgroups of $G$ on which $\alpha$ acts periodically or with linear growth.

We study strong triviality of 2-bridge knots via Dehn surgery techniques and prove that a 2-bridge knot is strongly $n$-trivial for $n\geq 1$ if and only if it is the trivial knot, the trefoil knot or the figure-eight knot.

Given a diagram $D$ of a knot $K$, we consider the number $c(D)$ of crossings and the number $b(D)$ of overpasses of $D$. We show that, if $D$ is a diagram of a nontrivial knot $K$ whose number $c(D)$ of crossings is minimal, then $1+\sqrt{1+c(D)} \leq b(D)\leq c(D)$. These inequalities are sharp in the sense that the upper bound of $b(D)$ is achieved by alternating knots and the lower bound of $b(D)$ is achieved by torus knots. The second inequality becomes an equality only when the knot is an alternating knot. We prove that the first inequality becomes an equality only when the knot is a torus knot.

We describe probably the simplest 3-manifold which contains closed separating incompressible surfaces of arbitrarily large genus. Two applications of this observation are given. (1) For any closed, orientable 3-manifold $M$ and any integer $m\,{>}\,0$, a surgery on a link in $M$ of at most $2m\,{+}\,1$ components will provide a closed, orientable, irreducible 3-manifold containing $m$ disjoint, non-parallel, separating, incompressible surfaces of arbitrarily high genus. (2) There exists a 3-manifold $M$ containing separating incompressible surfaces $S_n$ of genus $g(S_n)$ arbitrarily large, such that the amalgamation of minimal Heegaard splittings of two resulting 3-manifolds cutting along $S_n$ can be stabilized $g(S_n)-3$ times to a minimal Heegaard splitting of $M$.

Painlevé differential equations have been an important topic of research in complex differential equations during the last century, and the last two decades in particular, with many applications not only to pure mathematics but also to physics and engineering. In this paper, we prove that any transcendental solution of the second Painlevé equation $w''\,{=}\,2w^{3}+zw+\alpha$ is of order at least $3/2$, and that any transcendental solution of the fourth Painlevé equation $2ww''\,{=}\,(w')^{2}+3w^{4}+8zw^{3}+4(z^{2}-\alpha )w^{2}+2\beta$ is of order at least $2$.

With a view to generalising rectifiability and density results to more general spaces we prove the following: let $H^s$ denote Hausdorff $s$ measure in $l^n_{\infty}$. Let $s\in(0,2]$. Let $S\subset l^n_{\infty}$ be a subset of positive locally finite Hausdorff $s$-measure with the property $$\lim_{r\rightarrow 0} \frac{H^s(B_r(x)\cap S)}{\alpha(s)2^{-s}r^s}=1\;\;\;\mathrm{for}\;\;H^{s}\;a.e.\;x\in S,$$ then $s$ is an integer and $S$ has a weak tangent at almost every point.

In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz spaces into the Colombeau algebra $\G$ are well known, but for ultradistribution and periodic hyperfunction type spaces we give new constructions. We show that the multiplication of regular enough functions (smooth, ultradifferentiable or quasianalytic), embedded into corresponding algebras, is the ordinary multiplication.

In [19], R. Kadison proved that every surjective linear isometry $\Phi{:}\, {\C A} \to {\C B}$ between two unital C*-algebras has the form $$\Phi (x) = u T (x), \hbox{ $x\in {\C A}$,}$$ where $u$ is a unitary element in ${\C B}$ and $T$ is a Jordan *-isomorphism from${\C A}$ onto ${\C B}$. This result extends the classical Banach–Stone theorem [3, 32] obtained in the 1930s to non-abelian unital C*-algebras. A. L. Paterson and A. M. Sinclair extended Kadison's result to surjective isometries between C*-algebras by replacing the unitary element $u$ by a unitary element in the multiplier C*-algebra of the range algebra [28]. Thus, every surjective linear isometry between C*-algebras preserves the triple products as $$\J xyz \,{=}\, 2^{-1} ( x y^* z + z y^* x).$$

We prove results on the asymptotic behavior of the quantization error for self-similar distributions $P$ when $n$, the number of prototypes, goes to infinity. In particular, we show that the suitably scaled quantization error converges, if $P$ is nonarithmetic and that it is asymptotically periodic in the arithmetic case. The results hold also for distributions that are absolutely continuous with respect to a self-similar distribution.

Spherically symmetric space-times have been studied in general relativity ever since the beginning when as the first exact solution of the Einstein equation the Schwarzschild space-time was given. Later on an elaborate local theory of spherically symmetric space-times was worked out with such fundamental results as Birkhoff's theorem and essential results concerning their global geometry have been achieved. Yet it seems that a general global theory is lacking even now. Some fundamental facts concerning the global geometry of spherically symmetric space-times are presented below. The concept of axis and of transverse submanifold are introduced and applied to study and classification.