Ensuring the boundedness of the core of games with restricted cooperation

The core of a cooperative game on a set of players N is one of the most popular concept of solution. When cooperation is restricted (feasible coalitions form a subcollection F of 2N), the core may become unbounded, which makes it usage questionable in practice. Our proposal is to make the core bounded by turning some of the inequalities defining the core into equalities (additional efficiency constraints). We address the following mathematical problem : can we find a minimal set of inequalities in the core such that, if turned into equalities, the core becomes bounded ? The new core obtained is called the restricted core. We completely solve the question when F is a distributive lattice, introducing also the notion of restricted Weber set. We show that the case of regular set systems amounts more or less to the case of distributive lattices. We also study the case of weakly union-closed systems and give some results for the general case.Cooperative game, core, restricted cooperation, bounded core, Weber set.

Genesis of indifference thresholds and infinitely many indifference points in discrete time infinite horizon optimisation problems

This article investigates the discrete time indifference-attractor bifurcation of infinite horizon continous-time optimal control problems with a single state variable. We show that these bifurcations are linked to a heteroclinic bifurcation scenario of the state-costate equations, and we analyse the consequences for the optimal solutions. In particular, we can characterise the bifurcation value at which indifference thresholds appear by a geometric condition, and we find that for certain parameter values, there are countably infinitely many indifference points. We apply our results to a modified version of the shallow lake pollution management problem.

Conceptual Analysis of Black Hole Entropy in String Theory

The microscopic state counting of the extremal Reissner-Nordstr\"om black hole performed by Andrew Strominger and Cumrun Vafa in 1996 has proven to be a central result in string theory. Here, with a philosophical readership in mind, the argument is presented in its contemporary context and its rather complex conceptual structure is analysed. In particular, we will identify the various inter-theoretic relations, such as duality and linkage relations, on which it depends. We further aim to make clear why the argument was immediately recognised as a successful accounting for the entropy of this black hole and how it engendered subsequent work that intended to strengthen the string theoretic analysis of black holes. Its relation to the formulation of the AdS/CFT conjecture will be briefly discussed, and the familiar reinterpretation of the entropy calculation in the context of the AdS/CFT correspondence is given. Finally, we discuss the heuristic role that Strominger and Vafa's microscopic account of black hole entropy played for the black hole information paradox. A companion paper analyses the ontology of the Strominger-Vafa black hole states, the question of emergence of the black hole from a collection of D-branes, and the role of the correspondence principle in the context of string theory black holes.Comment: 70 page

Mixing-like properties for some generic and robust dynamics

We show that the set of Bernoulli measures of an isolated topologically mixing homoclinic class of a generic diffeomorphism is a dense subset of the set of invariant measures supported on the class. For this, we introduce the large periods property and show that this is a robust property for these classes. We also show that the whole manifold is a homoclinic class for an open and dense subset of the set of robustly transitive diffeomorphisms far away from homoclinic tangencies. In particular, using results from Abdenur and Crovisier, we obtain that every diffeomorphism in this subset is robustly topologically mixing

Almost-Fisher families

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $\lambda$ can have at most $n$ non-empty sets. One may weaken the condition by requiring that for every set in $\mathcal F$, all but at most $k$ of its pairwise intersections have size $\lambda$. We call such families $k$-almost $\lambda$-Fisher. Vu was the first to study the maximum size of such families, proving that for $k=1$ the largest family has $2n-2$ sets, and characterising when equality is attained. We substantially refine his result, showing how the size of the maximum family depends on $\lambda$. In particular we prove that for small $\lambda$ one essentially recovers Fisher's bound. We also solve the next open case of $k=2$ and obtain the first non-trivial upper bound for general $k$.Comment: 27 pages (incluiding one appendix

Polynomials that Sign Represent Parity and Descartes' Rule of Signs

A real polynomial $P(X_1,..., X_n)$ sign represents $f: A^n \to \{0,1\}$ if for every $(a_1, ..., a_n) \in A^n$, the sign of $P(a_1,...,a_n)$ equals $(-1)^{f(a_1,...,a_n)}$. Such sign representations are well-studied in computer science and have applications to computational complexity and computational learning theory. In this work, we present a systematic study of tradeoffs between degree and sparsity of sign representations through the lens of the parity function. We attempt to prove bounds that hold for any choice of set $A$. We show that sign representing parity over $\{0,...,m-1\}^n$ with the degree in each variable at most $m-1$ requires sparsity at least $m^n$. We show that a tradeoff exists between sparsity and degree, by exhibiting a sign representation that has higher degree but lower sparsity. We show a lower bound of $n(m -2) + 1$ on the sparsity of polynomials of any degree representing parity over $\{0,..., m-1\}^n$. We prove exact bounds on the sparsity of such polynomials for any two element subset $A$. The main tool used is Descartes' Rule of Signs, a classical result in algebra, relating the sparsity of a polynomial to its number of real roots. As an application, we use bounds on sparsity to derive circuit lower bounds for depth-two AND-OR-NOT circuits with a Threshold Gate at the top. We use this to give a simple proof that such circuits need size $1.5^n$ to compute parity, which improves the previous bound of ${4/3}^{n/2}$ due to Goldmann (1997). We show a tight lower bound of $2^n$ for the inner product function over $\{0,1\}^n \times \{0, 1\}^n$.Comment: To appear in Computational Complexit

Unlabeled sample compression schemes and corner peelings for ample and maximum classes

We examine connections between combinatorial notions that arise in machine learning and topological notions in cubical/simplicial geometry. These connections enable to export results from geometry to machine learning. Our first main result is based on a geometric construction by Tracy Hall (2004) of a partial shelling of the cross-polytope which can not be extended. We use it to derive a maximum class of VC dimension 3 that has no corners. This refutes several previous works in machine learning from the past 11 years. In particular, it implies that all previous constructions of optimal unlabeled sample compression schemes for maximum classes are erroneous. On the positive side we present a new construction of an unlabeled sample compression scheme for maximum classes. We leave as open whether our unlabeled sample compression scheme extends to ample (a.k.a. lopsided or extremal) classes, which represent a natural and far-reaching generalization of maximum classes. Towards resolving this question, we provide a geometric characterization in terms of unique sink orientations of the 1-skeletons of associated cubical complexes

Small doubling, atomic structure and ℓ\ell-divisible set families

Let $\mathcal{F}\subset 2^{[n]}$ be a set family such that the intersection of any two members of $\mathcal{F}$ has size divisible by $\ell$. The famous Eventown theorem states that if $\ell=2$ then $|\mathcal{F}|\leq 2^{\lfloor n/2\rfloor}$, and this bound can be achieved by, e.g., an `atomic' construction, i.e. splitting the ground set into disjoint pairs and taking their arbitrary unions. Similarly, splitting the ground set into disjoint sets of size $\ell$ gives a family with pairwise intersections divisible by $\ell$ and size $2^{\lfloor n/\ell\rfloor}$. Yet, as was shown by Frankl and Odlyzko, these families are far from maximal. For infinitely many $\ell$, they constructed families $\mathcal{F}$ as above of size $2^{\Omega(n\log \ell/\ell)}$. On the other hand, if the intersection of {\em any number} of sets in $\mathcal{F}\subset 2^{[n]}$ has size divisible by $\ell$, then it is easy to show that $|\mathcal{F}|\leq 2^{\lfloor n/\ell\rfloor}$. In 1983 Frankl and Odlyzko conjectured that $|\mathcal{F}|\leq 2^{(1+o(1)) n/\ell}$ holds already if one only requires that for some $k=k(\ell)$ any $k$ distinct members of $\mathcal{F}$ have an intersection of size divisible by $\ell$. We completely resolve this old conjecture in a strong form, showing that $|\mathcal{F}|\leq 2^{\lfloor n/\ell\rfloor}+O(1)$ if $k$ is chosen appropriately, and the $O(1)$ error term is not needed if (and only if) $\ell \, | \, n$, and $n$ is sufficiently large. Moreover the only extremal configurations have `atomic' structure as above. Our main tool, which might be of independent interest, is a structure theorem for set systems with small 'doubling'