Quadratic forms and systems of forms in many variables

Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$ variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for $V(F_1,\dotsc,F_R)$. Previous work in this direction required $n$ to grow at least quadratically with $R$. We give a similar result for $R$ forms of degree $d$, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as $n> d2^dR$. In the case that $d\geq 3$, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients.Comment: 29 pages, in revie

Systems of cubic forms in many variables

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular we can handle systems of forms in $O(R)$ variables, previous work having required that $n \gg R^2$. One conjectures that $n \geq 6R+1$ should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.Comment: 23 pages, submitte

Strichartz estimates for the Schroedinger equation on non-rectangular two-dimensional tori

We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) flat tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the number of solutions of Diophantine problems.Comment: 28 page

The elliptic sieve and Brauer groups

A theorem of Serre states that almost all plane conics over Q ${{\mathbb {Q}}}$ have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations

Bounds for spectral projectors on the Euclidean cylinder

We prove essentially optimal bounds for norms of spectral projectors on thin spherical shells for the Laplacian on the cylinder $(\mathbb{R} / \mathbb{Z}) \times \mathbb{R}$. In contrast to previous investigations into spectral projectors on tori, having one unbounded dimension available permits a compact self-contained proof

Bounds for spectral projectors on tori

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on tori. This is closely related to the boundedness of resolvents of the Laplacian and the boundedness of Lp norms of eigenfunctions of the Laplacian. We formulate a conjecture and partially prove it

Locked nailing for the treatment of displaced articular fractures of the calcaneus: description of a new procedure with calcanail®

Although open reduction and internal fixation is considered the best method for treating displaced articular fractures of the calcaneus, lateral approach is at high risk for wound healing complications. For this reason, the authors developed a posterior approach and a new implant to perform both intrafocal reduction and internal fixation. The aim of this technical note is to describe this method of treatment for displaced articular fractures of the calcaneus, which offered the following advantages: (a) the creation of a working channel that provides also a significant bone autograft, (b) the intrafocal reduction of the displaced articular surface, (c) the insertion of a locking nail that maintains the reduced articular surface at the right height, (d) the possibility to switch from an ORIF to a reconstruction arthrodesis with the same approach and instrumentation in case of severely damaged posterior facet

Strichartz estimates for the Schroedinger equation on non-rectangular two-dimensional tori

We propose a conjecture for long time Strichartz estimates on generic (non-rectangular) at tori. We proceed to partially prove it in dimension 2. Our arguments involve on the one hand Weyl bounds; and on the other hands bounds on the number of solutions of Diophantine problems

The behaviour of political parties and MPs in the parliaments of the Weimar Republic

Copyright @ 2012 The Authors. This is the author's accepted manuscript. The final published article is available from the link below.Analysing the roll-call votes of the MPs of the Weimar Republic we find: (1) that party competition in the Weimar parliaments can be structured along two dimensions: an economic left–right and a pro-/anti-democratic. Remarkably, this is stable throughout the entire lifespan of the Republic and not just in the later years and despite the varying content of votes across the lifespan of the Republic, and (2) that nearly all parties were troubled by intra-party divisions, though, in particular, the national socialists and communists became homogeneous in the final years of the Republic.Zukunftskolleg, University of Konstan

Bounded Rationality and the Emergence of Simplicity Amidst Complexity

The purpose of this essay is to explore the relationship between the simple and the complex in economics by anchoring our analysis on bounded rationality. Much of the conventional literature focuses on ‘un-bounded rationality’ of the rationality-as-consistency variety. Theorizing of bounded rationality tends to assume that the problem to be solved is independent of the nature of bounded rationality. Following the insights from the works of Herbert Simon and contributions from outside economics, both bounded rationality and the environment are inextricably linked. The boundaries between bounded rationality and its environment can shift. The form in which bounded rationality is found depends on the complexity of the environment. Furthermore, if local interactions between bounded rational agents result in the formation of hierarchies – the complexity of the collective system will change. Whether this will occur depends on the nature of bounded rationality at the individual level