In this paper we consider the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. Using new exponential sums in tandem with a sieve method we are able to provide stronger “minor arc” estimates than previous authors, thereby improving the saving obtained in the exponent by a factor 8/7.

Let $F$ be a free group, freely generated by a non-empty set $X$ and let $R$ be a cyclically reduced word in $F$. Let $X_0$ be the subset of elements of $X$ which occur in $R$ or in $R^{-1}$. Let $G$ be the quotient of $F$ by the normal closure of $R$. By the classical Dehn–Magnus Freiheitssatz, if $Y$ is a subset of $X$ which does not contain $X_0$, then the subgroup of $G$ generated by the image of $Y$ in $G$ is freely generated by it.

Free products of groups can be viewed as generalisations of free groups, by replacing the infinite cyclic groups generated by individual elements of $X$, by arbitrary groups. It is known that in special cases the corresponding version of the Freiheitssatz holds true for one-relator quotients of free products of two groups (components), for example if the relator satisfies a small cancellation condition.

In this work we extend Magnus' Freiheitssatz to an arbitrary number of components and where the subset corresponding to $Y$ above contains arbitrary number of components, provided that the relator satisfies the small cancellation condition C(8) and some further conditions. As an application we imitate, under certain conditions, the result of Magnus stating that one-relator groups have the structure of HNN extensions.

The method of proof includes a combinatorial analysis of van Kampen diagrams, relying on the structure theorem for small cancellation diagrams.

Let $\varphi\,{:}\,G \,{\rightarrow}\,{\rm O}(d)$ be an orthogonal representation of a group $G$. A finite subset $X$ of a Euclidean space ${\bf R}^d$ is said to be subtransitive under$(G, \varphi)$ when it is a subset of an orbit under the action of ${\varphi }(G)$; furthermore, we say that $X$ is finitely subtransitive when it is subtransitive under $(G, \varphi)$ for some finite group $G$. If $X$ is subtransitive under a representation $\varphi\,{:}\,G\,{\rightarrow}\, {\rm O}(d)$ then it is clear that $X$ is spherical; that is, contained in a sphere in the representation space ${\bf R}^d$. In connection with Euclidean Ramsey theory, I.B. Leader [6] has posed the converse question:

Is is true that every finite spherical set is finitely subtransitive?

The paper deals with analytic families of planar vector fields, studying methods to detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit cycles that can locally bifurcate from it. It is known that this multi-parameter problem can be reduced to a single-parameter one, in the sense that there exist analytic curves in parameter space along which the maximal cyclicity can be attained. In that case one speaks about a maximal cyclicity curve (mcc) in case only the number is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is also taken into account. In view of obtaining efficient algorithms for detecting the cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear depending on certain general properties of the families or of their associated Bautin ideal. In any case by well chosen examples we show that prudence is appropriate.

Let $E(1)_K$ denote the homotopy rational elliptic surface corresponding to a knot $K$ in $S^3$ constructed by R. Fintushel and R. J. Stern. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive 2-dimensional homology class in $E(1)_K$ when $K$ is any nontrivial fibred knot in $S^3$. We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.

Arithmetic and non arithmetic hyperbolic 3-manifolds having infinitely many fibrations over $S^{1}$ of arbitrary high genus are constructed. A concrete example is studied in detail.

We study different localization techniques both for classical and motivic spectra. In particular, we show that the motivic model structures of Morel and Voevodsky are cellular in the sense of Hirschhorn and thus may be further localized. As a possible application, we discuss a motivic version of the chromatic filtration and the chromatic spectral sequence.

We define equivariant Chern–Schwartz–MacPherson classes of a possibly singular algebraic $G$-variety over the base field $\mathbb{C}$, or more generally over a field of characteristic 0. In fact, we construct a natural transformation $C^G_*$ from the $G$-equivariant constructible function functor $\cal{F}^G$ to the $G$-equivariant homology functor $H^G_*$ or $A^G_*$ (in the sense of Totaro–Edidin–Graham). This $C^G_*$ may be regarded as MacPherson's transformation for (certain) quotient stacks. The Verdier–Riemann–Roch formula takes a key role throughout.

The classical divergence theorem for an $n$-dimensional domain $A$ and a smooth vector field $F$ in $n$-space \[\int_{\partial A} F \cdot n = \int_A div F\] requires that a normal vector field $n(p)$ be defined a.e. $p \in \partial A$. In this paper we give a new proof and extension of this theorem by replacing $n$ with a limit $\star \partial A$ of 1-dimensional polyhedral chains taken with respect to a norm. The operator $\star$ is a geometric dual to the Hodge star operator and is defined on a large class of $k$-dimensional domains of integration $A$ in $n$-space called chainlets. Chainlets include a broad range of domains, from smooth manifolds to soap bubbles and fractals. We prove as our main result the Star theorem \[ \displaystyle\int_{\star A} \omega = (-1)^{k(n-k)}\int_A \star \omega \] where $\omega$ is a $k$-form and $A$ is an $(n-k)$-chainlet. When combined with the general Stokes' theorem ([H1, H2]) \[\int_{\partial A} \omega = \int_A d \omega\] this result yields optimal and concise forms of Gauss' divergence theorem \[\int_{\star \partial A}\omega = (-1)^{(k)(n-k)} \int_A d\star \omega\] and Green's curl theorem \[\int_{\partial A} \omega = \int_{\star A} \star d\omega.\]

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps $f: K\,{\rightarrow}\, K$, where $K$ is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of $f$ converges to a periodic orbit and, moreover, the period of each periodic point of $f$ is bounded by \[ \beta_N = \max_{q+r+s=N}\frac{N!}{q!r!s!}= \frac{N!}{\big\lfloor\frac{N}{3}\big\rfloor!\big\lfloor\frac{N\,{+}\,1}{3}\big\rfloor! \big\lfloor\frac{N\,{+}\,2}{3}\big\rfloor!}\sim \frac{3^{N+1}\sqrt{3}}{2\pi N}, \] where $N$ is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in $\mathbb{R}^n$, we show that the upper bound is asymptotically sharp.

These results are an extension of work by Lemmens and Scheutzow concerning periodic orbits in the interior of the standard positive cone in $\mathbb{R}^n$.

We investigate spherically symmetric equilibrium states of the Vlasov–Poisson system, relevant in galactic dynamics. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a region with compact closure. Based on a dynamical systems analysis we derive theorems that guarantee that the steady state solutions have finite mass and compact support.