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A finite classical polar space of rank $n$ consists of the totally isotropic subspaces of a finite vector space equipped with a nondegenerate form such that $n$ is the maximal dimension of such a subspace. A $t$-Steiner system in a finite classical polar space of rank $n$ is a collection $Y$ of totally isotropic $n$-spaces such that each totally isotropic $t$-space is contained in exactly one member of $Y$. Nontrivial examples are known only for $t=1$ and $t=n-1$. We give an almost complete classification of such $t$-Steiner systems, showing that such objects can only exist in some corner cases. This classification result arises from a more general result on packings in polar spaces., Comment: 25 pages; this revision contains a strengthened version of Cor. 3.4, and also small changes taking into account referee comments

A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erd\H{o}s, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\mid p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\mid p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p(z)$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials., Comment: 20 pages

Three measures of pseudorandomness of finite binary sequences were introduced by Mauduit and S\'ark\"ozy in 1997 and have been studied extensively since then: the normality measure, the well-distribution measure, and the correlation measure of order r. Our main result is that the correlation measure of order r for random binary sequences converges strongly, and so has a limiting distribution. This solves a problem due to Alon, Kohayakawa, Mauduit, Moreira, and R\"odl. We also show that the best known lower bounds for the minimum values of the correlation measures are simple consequences of a celebrated result due to Welch, concerning the maximum nontrivial scalar products over a set of vectors., Comment: 19 pages, this version contains small changes taking into account referee comments

Littlewood asked how small the ratio $||f||_4/||f||_2$ (where $||.||_\alpha$ denotes the $L^\alpha$ norm on the unit circle) can be for polynomials $f$ having all coefficients in $\{1,-1\}$, as the degree tends to infinity. Since 1988, the least known asymptotic value of this ratio has been $\sqrt[4]{7/6}$, which was conjectured to be minimum. We disprove this conjecture by showing that there is a sequence of such polynomials, derived from the Fekete polynomials, for which the limit of this ratio is less than $\sqrt[4]{22/19}$., Comment: minor revisions

The Kyoto Protocol to the United Nations Framework Convention on Climate Change (UNFCCC) provides for three ‘market-oriented mechanisms’: joint implementation (JI); emissions trading (ET); and the clean development mechanism (CDM). The main purpose of the three mechanisms is to maximize the cost-effectiveness of climate change mitigation by providing an opportunity to reduce greenhouse gas (GHG) emissions inexpensively abroad, rather than at home. This chapter provides an overview of the CDM actors and the steps in the CDM project cycle with an analysis of the role of the Secretariat in supporting these stages.

Concern about the increasingly high-probability, high-impact risks posed by global warming is driving the exploration of new techniques to artificially cool the planet through an approach known as solar radiation modification (SRM). Would the world be better off with or without such techniques? Would there be winners and losers? And how can we sufficiently compare the relative risks presented in a future with SRM against the risks faced in a future without it? Such ‘risk-risk’ assessment poses particular challenges given uncertainties around the techniques and the extent of human-induced changes to the climate system that might be expected in the future. These uncertainties are further compounded by differences in stakeholders’ framing and risk tolerance, as well as the level of complexity and the intertemporal nature of such assessment. To help address these questions, mindful of such challenges, we present here a basic risk-risk assessment framework providing a structure to help strengthen risk-management considerations in the design of future climate response strategies.