The t-stability number of a random graph

Given a graph G = (V,E), a vertex subset S is called t-stable (or t-dependent) if the subgraph G[S] induced on S has maximum degree at most t. The t-stability number of G is the maximum order of a t-stable set in G. We investigate the typical values that this parameter takes on a random graph on n vertices and edge probability equal to p. For any fixed 0 < p < 1 and fixed non-negative integer t, we show that, with probability tending to 1 as n grows, the t-stability number takes on at most two values which we identify as functions of t, p and n. The main tool we use is an asymptotic expression for the expected number of t-stable sets of order k. We derive this expression by performing a precise count of the number of graphs on k vertices that have maximum degree at most k. Using the above results, we also obtain asymptotic bounds on the t-improper chromatic number of a random graph (this is the generalisation of the chromatic number, where we partition of the vertex set of the graph into t-stable sets).Comment: 25 pages; v2 has 30 pages and is identical to the journal version apart from formatting and a minor amendment to Lemma 8 (and its proof on p. 21

The Dirichlet Problem for Harmonic Functions on Compact Sets

For any compact set $K\subset \mathbb{R}^n$ we develop the theory of Jensen measures and subharmonic peak points, which form the set $\mathcal{O}_K$, to study the Dirichlet problem on $K$. Initially we consider the space $h(K)$ of functions on $K$ which can be uniformly approximated by functions harmonic in a neighborhood of $K$ as possible solutions. As in the classical theory, our Theorem 8.1 shows $C(\mathcal{O}_K)\cong h(K)$ for compact sets with $\mathcal{O}_K$ closed. However, in general a continuous solution cannot be expected even for continuous data on \rO_K as illustrated by Theorem 8.1. Consequently, we show that the solution can be found in a class of finely harmonic functions. Moreover by Theorem 8.7, in complete analogy with the classical situation, this class is isometrically isomorphic to $C_b(\mathcal{O}_K)$ for all compact sets $K$.Comment: There have been a large number of changes made from the first version. They mostly consists of shortening the article and supplying additional reference

Fast evaluation of union-intersection expressions

We show how to represent sets in a linear space data structure such that expressions involving unions and intersections of sets can be computed in a worst-case efficient way. This problem has applications in e.g. information retrieval and database systems. We mainly consider the RAM model of computation, and sets of machine words, but also state our results in the I/O model. On a RAM with word size $w$, a special case of our result is that the intersection of $m$ (preprocessed) sets, containing $n$ elements in total, can be computed in expected time $O(n (\log w)^2 / w + km)$, where $k$ is the number of elements in the intersection. If the first of the two terms dominates, this is a factor $w^{1-o(1)}$ faster than the standard solution of merging sorted lists. We show a cell probe lower bound of time $\Omega(n/(w m \log m)+ (1-\tfrac{\log k}{w}) k)$, meaning that our upper bound is nearly optimal for small $m$. Our algorithm uses a novel combination of approximate set representations and word-level parallelism

Rare Kaon Decay Experiments at CERN

The NA62 experiment at the CERN SPS aims to measure the branching ratio of the rare decay $K^+\to\pi^+\nu\bar\nu$ with a relative precision of $\sim10$. To achieve that goal, it is designed to be exposed to $1.2\times 10^{13}$ $K^+$ decays in its fiducial volume. The unprecedented $K^+$ flux will lead to record sensitivities to rare and forbidden decays of $K^+$ and $\pi^0$, including those that violate lepton flavour or lepton number conservation. The expected NA62 performances for lepton flavour conservation and lepton universality tests are discussed. Relevant on-going or recently completed measurements from the $K^\pm$ decay data sets collected by earlier kaon experiments at CERN (NA48/2 and NA62-$R_K$) are also presented.Comment: Talk given at the 1st Charged Lepton Flavour Conference, Lecce, May 201

Comparing Different Information Levels

Given a sequence of random variables ${\bf X}=X_1,X_2,\ldots$ suppose the aim is to maximize one's return by picking a `favorable' $X_i$. Obviously, the expected payoff crucially depends on the information at hand. An optimally informed person knows all the values $X_i=x_i$ and thus receives $E (\sup X_i)$. We will compare this return to the expected payoffs of a number of observers having less information, in particular $\sup_i (EX_i)$, the value of the sequence to a person who only knows the first moments of the random variables. In general, there is a stochastic environment (i.e. a class of random variables $\cal C$), and several levels of information. Given some ${\bf X} \in {\cal C}$, an observer possessing information $j$ obtains $r_j({\bf X})$. We are going to study `information sets' of the form $$R_{\cal C}^{j,k} = \{ (x,y) | x = r_j({\bf X}), y=r_k({\bf X}), {\bf X} \in {\cal C} \},$$ characterizing the advantage of $k$ relative to $j$. Since such a set measures the additional payoff by virtue of increased information, its analysis yields a number of interesting results, in particular `prophet-type' inequalities.Comment: 14 pages, 3 figure

Answering Spatial Multiple-Set Intersection Queries Using 2-3 Cuckoo Hash-Filters

We show how to answer spatial multiple-set intersection queries in O(n(log w)/w + kt) expected time, where n is the total size of the t sets involved in the query, w is the number of bits in a memory word, k is the output size, and c is any fixed constant. This improves the asymptotic performance over previous solutions and is based on an interesting data structure, known as 2-3 cuckoo hash-filters. Our results apply in the word-RAM model (or practical RAM model), which allows for constant-time bit-parallel operations, such as bitwise AND, OR, NOT, and MSB (most-significant 1-bit), as exist in modern CPUs and GPUs. Our solutions apply to any multiple-set intersection queries in spatial data sets that can be reduced to one-dimensional range queries, such as spatial join queries for one-dimensional points or sets of points stored along space-filling curves, which are used in GIS applications.Comment: Full version of paper from 2017 ACM SIGSPATIAL International Conference on Advances in Geographic Information System

Ignorance is Almost Bliss: Near-Optimal Stochastic Matching With Few Queries

The stochastic matching problem deals with finding a maximum matching in a graph whose edges are unknown but can be accessed via queries. This is a special case of stochastic $k$-set packing, where the problem is to find a maximum packing of sets, each of which exists with some probability. In this paper, we provide edge and set query algorithms for these two problems, respectively, that provably achieve some fraction of the omniscient optimal solution. Our main theoretical result for the stochastic matching (i.e., $2$-set packing) problem is the design of an \emph{adaptive} algorithm that queries only a constant number of edges per vertex and achieves a $(1-\epsilon)$ fraction of the omniscient optimal solution, for an arbitrarily small $\epsilon>0$. Moreover, this adaptive algorithm performs the queries in only a constant number of rounds. We complement this result with a \emph{non-adaptive} (i.e., one round of queries) algorithm that achieves a $(0.5 - \epsilon)$ fraction of the omniscient optimum. We also extend both our results to stochastic $k$-set packing by designing an adaptive algorithm that achieves a $(\frac{2}{k} - \epsilon)$ fraction of the omniscient optimal solution, again with only $O(1)$ queries per element. This guarantee is close to the best known polynomial-time approximation ratio of $\frac{3}{k+1} -\epsilon$ for the \emph{deterministic} $k$-set packing problem [Furer and Yu, 2013] We empirically explore the application of (adaptations of) these algorithms to the kidney exchange problem, where patients with end-stage renal failure swap willing but incompatible donors. We show on both generated data and on real data from the first 169 match runs of the UNOS nationwide kidney exchange that even a very small number of non-adaptive edge queries per vertex results in large gains in expected successful matches

Extremal and typical results in Real Algebraic Geometry

In the first part of the dissertation we show that $2((d-1)^n-1)/(d-2)$ is the maximum possible number of critical points that a generic $(n-1)$-dimensional spherical harmonic of degree $d$ can have. Our result in particular shows that there exist generic real symmetric tensors whose all eigenvectors are real. The results of this part are contained in Chapter \ref{ch:harmonics}. In the second part of the thesis we are interested in expected outcomes in three different problems of probabilistic real algebraic and differential geometry. First, in Chapter \ref{ch:discriminant} we compute the volume of the projective variety \Delta\subset \txt{P}\txt{Sym}(n,\K{R}) of real symmetric matrices with repeated eigenvalues. Our computation implies that the expected number of real symmetric matrices with repeated eigenvalues in a uniformly distributed projective $2$-plane L\subset \txt{P}\txt{Sym}(n,\K{R}) equals \mean\#(\Delta\cap L) = {n\choose 2}. The sharp upper bound on the number of matrices in the intersection $\Delta\cap L$ of $\Delta$ with a generic projective $2$-plane $L$ is ${n+1 \choose 3}$. Second, in Chapter \ref{ch:pevp} we provide explicit formulas for the expected condition number for the polynomial eigenvalue problem defined by matrices drawn from various Gaussian matrix ensembles. Finally, in Chapter \ref{ch:tangents} we are interested in the expected number of lines that are simultaneously tangent to the boundaries of several convex sets randomly positioned in the sphere. We express this number in terms of the integral mean curvatures of the boundaries of the convex sets