Abstract A two-dimensional version of the Pennsylvania State University mesoscale model has been applied to Winter Monsoon Experiment data in order to simulate the diurnally occurring convection observed over the South China Sea. The domain includes a representation of part of Borneo as well as the sea so that the model can simulate the initiation of convection. Also included in the model are parameterizations of mesoscale ice phase and moisture processes and longwave and shortwave radiation with a diurnal cycle. This allows use of the model to test the relative importance of various heating mechanisms to the stratiform cloud deck, which typically occupies several hundred kilometers of the domain. Frank and Cohen's cumulus parameterization scheme is employed to represent vital unresolved vertical transports in the convective area. The major conclusions are: Ice phase processes are important in determining the level of maximum large-scale heating and vertical motion because there is a strong anvil componen...

Numerical study of convection observed during the Winter Monsoon Experiment using a mesoscale two-dimensional model [presentation]

Groupes et algébres de lie

Boosting is a general method for improving the accuracy of learning algorithms. We use boosting to construct improved {\em privacy-preserving synopses} of an input database. These are data structures that yield, for a given set $\Q$ of queries over an input database, reasonably accurate estimates of the responses to every query in~$\Q$, even when the number of queries is much larger than the number of rows in the database. Given a {\em base synopsis generator} that takes a distribution on $\Q$ and produces a ``weak'' synopsis that yields ``good'' answers for a majority of the weight in $\Q$, our {\em Boosting for Queries} algorithm obtains a synopsis that is good for all of~$\Q$. We ensure privacy for the rows of the database, but the boosting is performed on the {\em queries}. We also provide the first synopsis generators for arbitrary sets of arbitrary low-sensitivity queries, {\it i.e.}, queries whose answers do not vary much under the addition or deletion of a single row. In the execution of our algorithm certain tasks, each incurring some privacy loss, are performed many times. To analyze the cumulative privacy loss, we obtain an $O(\eps^2)$ bound on the {\em expected} privacy loss from a single $\eps$-\dfp{} mechanism. Combining this with evolution of confidence arguments from the literature, we get stronger bounds on the expected cumulative privacy loss due to multiple mechanisms, each of which provides $\eps$-differential privacy or one of its relaxations, and each of which operates on (potentially) different, adaptively chosen, databases.

/pdf/boosting-and-differential-privacy-yjmdnv70yn.pdf

Boosting and Differential Privacy

Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as "property testing" in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization). This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future. --Persi Diaconis, Stanford University This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor. --Noga Alon, Tel Aviv University, Israel Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory. --Terence Tao, University of California, Los Angeles, CA Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike. --Bela Bollobas, Cambridge University, UK

Large Networks and Graph Limits

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of positive density contains progressions of arbitrary length. The second, which is the main new ingredient of this paper, is a certain transference principle. This allows us to deduce from Szemeredi�s theorem that any subset of a suficiently pseudorandom set (or measure) of positive relative density contains progressions of arbitrary length. The third ingredient is a recent result of Goldston and Yildirim, which we reproduce here. Using this, one may place (a large fraction of) the primes inside a pseudorandom set of �almost primes� (or more precisely, a pseudorandom measure concentrated on almost primes) with positive relative density.

https://annals.math.princeton.edu/wp-content/uploads/annals-v167-n2-p03.pdf

The primes contain arbitrarily long arithmetic progressions

Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation of a famous and difficult open conjecture of Hardy and Littlewood predicts an asymptotic, as N →∞, for the number of integer points n ∈ ℤ d ∩ K for which the integers ψ 1 (n), ... , ψ t (n) are simultaneously prime. This implies many other well-known conjectures, such as the twin prime conjecture and the (weak) Goldbach conjecture. It also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations, in which all unknowns are required to be prime. In this paper we (conditionally) verify this asymptotic under the assumption that no two of the affine-linear forms ψ 1 , ... , ψ t are affinely related; this excludes the important "binary" cases such as the twin prime or Goldbach conjectures, but does allow one to count "nondegenerate" configurations such as arithmetic progressions. Our result assumes two families of conjectures, which we term the inverse Gowers-norm conjecture (GI(s)) and the Mobius and nilsequences conjecture (MN(s)), where s ∈ {1,2,...} is the complexity of the system and measures the extent to which the forms ψ i depend on each other. The case s = 0 is somewhat degenerate, and follows from the prime number theorem in APs. Roughly speaking, the inverse Gowers-norm conjecture GI(s) asserts the Gowers U s+1 -norm of a function f: [N] → [-1, 1] is large if and only if f correlates with an s-step nilsequence, while the Mobius and nilsequences conjecture MN(s) asserts that the Mobius function μ is strongly asymptotically orthogonal to s-step nilsequences of a fixed complexity. These conjectures have long been known to be true for s = 1 (essentially by work of Hardy-Littlewood and Vinogradov), and were established for s = 2 in two papers of the authors. Thus our results in the case of complexity s ≤ 2 are unconditional. In particular we can obtain the expected asymptotics for the number of 4-term progressions p 1 < p 2 < p 3 < p 4 ≤ N of primes, and more generally for any (nondegenerate) problem involving two linear equations in four prime unknowns.

https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n3-p08-p.pdf

Linear equations in primes

We show that the Mobius function (n) is strongly asymptotically or- thogonal to any polynomial nilsequence (F (g(n))) n2N. Here, G is a sim- ply-connected nilpotent Lie group with a discrete and cocompact subgroup (so G= is a nilmanifold), g : Z ! G is a polynomial sequence, and F : G= ! R is a Lipschitz function. More precisely, we show that j 1 N P N n=1 (n)F (g(n)) j F;G; ;A log A N for all A > 0. In particu- lar, this implies the Mobius and Nilsequence conjecture MN(s) from our earlier paper for every positive integer s. This is one of two major in- gredients in our programme to establish a large number of cases of the generalised Hardy-Littlewood conjecture, which predicts how often a collec- tion 1;:::; t : Z d ! Z of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper. We give some applications of our main theorem. We show, for exam- ple, that the Mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c. We also obtain a result about the distribution of nilse- quences (a n x) n2N as n ranges only over the primes.

http://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p03-p.pdf

The Mobius function is strongly orthogonal to nilsequences

A theorem of Leibman asserts that a polynomial orbit (g(n)G) n?Z on a nilmanifold G/G is always equidistributed in a union of closed sub-nilmanifolds of G/G . In this paper we give a quantitative version of Leibman�s result, describing the uniform distribution properties of a finite polynomial orbit (g(n)G) n?[N] in a nilmanifold. More specifically we show that there is a factorisation g=eg ' ? , where e(n) is �smooth,� (?(n)G) n?Z is periodic and �rational,� and (g ' (n)G) n?P is uniformly distributed (up to a specified error d ) inside some subnilmanifold G ' /G ' of G/G for all sufficiently dense arithmetic progressions P?[N] .
 
Our bounds are uniform in N and are polynomial in the error tolerance d . In a companion paper we shall use this theorem to establish the Mobius and Nilsequences conjecture from an earlier paper of ours.

http://annals.math.princeton.edu/wp-content/uploads/annals-v175-n2-p02-p.pdf

The quantitative behaviour of polynomial orbits on nilmanifolds

Let K⩾1 be a parameter. A K-approximate group is a finite set A in a (local) group which contains the identity, is symmetric, and such that A⋅A is covered by K left translates of A. The main result of this paper is a qualitative description of approximate groups as being essentially finite-by-nilpotent, answering a conjecture of H. Helfgott and E. Lindenstrauss. This may be viewed as a generalisation of the Freiman-Ruzsa theorem on sets of small doubling in the integers to arbitrary groups. We begin by establishing a correspondence principle between approximate groups and locally compact (local) groups that allows us to recover many results recently established in a fundamental paper of Hrushovski. In particular we establish that approximate groups can be approximately modeled by Lie groups. To prove our main theorem we apply some additional arguments essentially due to Gleason. These arose in the solution of Hilbert’s fifth problem in the 1950s. Applications of our main theorem include a finitary refinement of Gromov’s theorem, as well as a generalized Margulis lemma conjectured by Gromov and a result on the virtual nilpotence of the fundamental group of Ricci almost nonnegatively curved manifolds.

Benjamin W. Green

Papers

The primes contain arbitrarily long arithmetic progressions

Linear equations in primes

The Mobius function is strongly orthogonal to nilsequences

The quantitative behaviour of polynomial orbits on nilmanifolds

The structure of approximate groups