Hyper-polynomial hierarchies and the polynomial jump

AbstractAssuming that the polynomial hierarchy (PH) does not collapse, we show the existence of ascending sequences of ptime Turing degrees of length ω1CK in PSPACE such that successors are polynomial jumps of their predecessors. Moreover these ptime degrees are all uniformly hard for PH. This is analogous to the hyperarithmetic hierarchy, which is defined similarly but with the (computable) Turing degrees. The lack of uniform least upper bounds for ascending sequences of ptime degrees causes the limit levels of our hyper-polynomial hierarchy to be inherently non-canonical. This problem is investigated in depth, and various possible structures for hyper-polynomial hierarchies are explicated, as are properties of the polynomial jump operator on the languages which are in PSPACE but not in PH

On Integrability and Exact Solvability in Deterministic and Stochastic Laplacian Growth

We review applications of theory of classical and quantum integrable systems to the free-boundary problems of fluid mechanics as well as to corresponding problems of statistical mechanics. We also review important exact results obtained in the theory of multi-fractal spectra of the stochastic models related to the Laplacian growth: Schramm-Loewner and Levy-Loewner evolutions

Discrete scale invariance and complex dimensions

We discuss the concept of discrete scale invariance and how it leads to complex critical exponents (or dimensions), i.e. to the log-periodic corrections to scaling. After their initial suggestion as formal solutions of renormalization group equations in the seventies, complex exponents have been studied in the eighties in relation to various problems of physics embedded in hierarchical systems. Only recently has it been realized that discrete scale invariance and its associated complex exponents may appear ``spontaneously'' in euclidean systems, i.e. without the need for a pre-existing hierarchy. Examples are diffusion-limited-aggregation clusters, rupture in heterogeneous systems, earthquakes, animals (a generalization of percolation) among many other systems. We review the known mechanisms for the spontaneous generation of discrete scale invariance and provide an extensive list of situations where complex exponents have been found. This is done in order to provide a basis for a better fundamental understanding of discrete scale invariance. The main motivation to study discrete scale invariance and its signatures is that it provides new insights in the underlying mechanisms of scale invariance. It may also be very interesting for prediction purposes.Comment: significantly extended version (Oct. 27, 1998) with new examples in several domains of the review paper with the same title published in Physics Reports 297, 239-270 (1998

Symmetry reduction in convex optimization with applications in combinatorics

This dissertation explores different approaches to and applications of symmetry reduction in convex optimization. Using tools from semidefinite programming, representation theory and algebraic combinatorics, hard combinatorial problems are solved or bounded. The first chapters consider the Jordan reduction method, extend the method to optimization over the doubly nonnegative cone, and apply it to quadratic assignment problems and energy minimization on a discrete torus. The following chapter uses symmetry reduction as a proving tool, to approach a problem from queuing theory with redundancy scheduling. The final chapters propose generalizations and reductions of flag algebras, a powerful tool for problems coming from extremal combinatorics

Activity representation with motion hierarchies

International audienceComplex activities, e.g., pole vaulting, are composed of a variable number of sub-events connected by complex spatio-temporal relations, whereas simple actions can be represented as sequences of short temporal parts. In this paper, we learn hierarchical representations of activity videos in an unsupervised manner. These hierarchies of mid-level motion components are data-driven decompositions specific to each video. We introduce a spectral divisive clustering algorithm to efficiently extract a hierarchy over a large number of tracklets (i.e., local trajectories). We use this structure to represent a video as an unordered binary tree. We model this tree using nested histograms of local motion features. We provide an efficient positive definite kernel that computes the structural and visual similarity of two hierarchical decompositions by relying on models of their parent-child relations. We present experimental results on four recent challenging benchmarks: the High Five dataset [Patron-Perez et al, 2010], the Olympics Sports dataset [Niebles et al, 2010], the Hollywood 2 dataset [Marszalek et al, 2009], and the HMDB dataset [Kuehne et al, 2011]. We show that pervideo hierarchies provide additional information for activity recognition. Our approach improves over unstructured activity models, baselines using other motion decomposition algorithms, and the state of the art

The complexity of coverability in ν-Petri nets

We show that the coverability problem in ν-Petri nets is complete for ‘double Ackermann’ time, thus closing an open complexity gap between an Ackermann lower bound and a hyper-Ackermann upper bound. The coverability problem captures the verification of safety properties in this nominal extension of Petri nets with name management and fresh name creation. Our completeness result establishes ν-Petri nets as a model of intermediate power among the formalisms of nets enriched with data, and relies on new algorithmic insights brought by the use of well-quasi-order ideals

Polynomial optimization: matrix factorization ranks, portfolio selection, and queueing theory

Inspired by Leonhard Euler’s belief that every event in the world can be understood in terms of maximizing or minimizing a specific quantity, this thesis delves into the realm of mathematical optimization. The thesis is divided into four parts, with optimization acting as the unifying thread. Part 1 introduces a particular class of optimization problems called generalized moment problems (GMPs) and explores the moment method, a powerful tool used to solve GMPs. We introduce the new concept of ideal sparsity, a technique that aids in solving GMPs by improving the bounds of their associated hierarchy of semidefinite programs. Part 2 focuses on matrix factorization ranks, in particular, the nonnegative rank, the completely positive rank, and the separable rank. These ranks are extensively studied using the moment method, and ideal sparsity is applied (whenever possible) to enhance the bounds on these ranks and speed-up their computation. Part 3 centers around portfolio optimization and the mean-variance-skewness kurtosis (MVSK) problem. Multi-objective optimization techniques are employed to uncover Pareto optimal solutions to the MVSK problem. We show that most linear scalarizations of the MVSK problem result in specific convex polynomial optimization problems which can be solved efficiently. Part 4 explores hypergraph-based polynomials emerging from queueing theory in the setting of parallel-server systems with job redundancy policies. By exploiting the symmetry inherent in the polynomials and some classical results on matrix algebras, the convexity of these polynomials is demonstrated, thereby allowing us to prove that the polynomials attain their optima at the barycenter of the simplex.<br/