A correspondence between rooted planar maps and normal planar lambda terms

A rooted planar map is a connected graph embedded in the 2-sphere, with one edge marked and assigned an orientation. A term of the pure lambda calculus is said to be linear if every variable is used exactly once, normal if it contains no beta-redexes, and planar if it is linear and the use of variables moreover follows a deterministic stack discipline. We begin by showing that the sequence counting normal planar lambda terms by a natural notion of size coincides with the sequence (originally computed by Tutte) counting rooted planar maps by number of edges. Next, we explain how to apply the machinery of string diagrams to derive a graphical language for normal planar lambda terms, extracted from the semantics of linear lambda calculus in symmetric monoidal closed categories equipped with a linear reflexive object or a linear reflexive pair. Finally, our main result is a size-preserving bijection between rooted planar maps and normal planar lambda terms, which we establish by explaining how Tutte decomposition of rooted planar maps (into vertex maps, maps with an isthmic root, and maps with a non-isthmic root) may be naturally replayed in linear lambda calculus, as certain surgeries on the string diagrams of normal planar lambda terms.Comment: Corrected title field in metadat

Realization of rigid C∗^*-tensor categories via Tomita bimodules

Starting from a (small) rigid C$^*$-tensor category $\mathscr{C}$ with simple unit, we construct von Neumann algebras associated to each of its objects. These algebras are factors and can be either semifinite (of type II$_1$ or II$_\infty$, depending on whether the spectrum of the category is finite or infinite) or they can be of type III$_\lambda$, $\lambda\in (0,1]$. The choice of type is tuned by the choice of Tomita structure (defined in the paper) on certain bimodules we use in the construction. Moreover, if the spectrum is infinite we realize the whole tensor category directly as endomorphisms of these algebras, with finite Jones index, by exhibiting a fully faithful unitary tensor functor $F:\mathscr{C} \hookrightarrow End_0(\Phi)$ where $\Phi$ is a factor (of type II or III). The construction relies on methods from free probability (full Fock space, amalgamated free products), it does not depend on amenability assumptions, and it can be applied to categories with uncountable spectrum (hence it provides an alternative answer to a conjecture of Yamagami \cite{Y3}). Even in the case of uncountably generated categories, we can refine the previous equivalence to obtain realizations on $\sigma$-finite factors as endomorphisms (in the type III case) and as bimodules (in the type II case). In the case of trivial Tomita structure, we recover the same algebra obtained in \cite{PopaS} and \cite{AMD}, namely the (countably generated) free group factor $L(F_\infty)$ if the given category has denumerable spectrum, while we get the free group factor with uncountably many generators if the spectrum is infinite and non-denumerable.Comment: 39 page

The looping probability of random heteropolymers helps to understand the scaling properties of biopolymers

Random heteropolymers are a minimal description of biopolymers and can provide a theoretical framework to the investigate the formation of loops in biophysical experiments. A two--state model provides a consistent and robust way to study the scaling properties of loop formation in polymers of the size of typical biological systems. Combining it with self--adjusting simulated--tempering simulations, we can calculate numerically the looping properties of several realizations of the random interactions within the chain. Differently from homopolymers, random heteropolymers display at different temperatures a continuous set of scaling exponents. The necessity of using self--averaging quantities makes finite--size effects dominant at low temperatures even for long polymers, shadowing the length--independent character of looping probability expected in analogy with homopolymeric globules. This could provide a simple explanation for the small scaling exponents found in experiments, for example in chromosome folding

Enumeration of Hypermaps of a Given Genus

This paper addresses the enumeration of rooted and unrooted hypermaps of a given genus. For rooted hypermaps the enumeration method consists of considering the more general family of multirooted hypermaps, in which darts other than the root dart are distinguished. We give functional equations for the generating series counting multirooted hypermaps of a given genus by number of darts, vertices, edges, faces and the degrees of the vertices containing the distinguished darts. We solve these equations to get parametric expressions of the generating functions of rooted hypermaps of low genus. We also count unrooted hypermaps of given genus by number of darts, vertices, hyperedges and faces.Comment: 42 page

A Symbolic Transformation Language and its Application to a Multiscale Method

The context of this work is the design of a software, called MEMSALab, dedicated to the automatic derivation of multiscale models of arrays of micro- and nanosystems. In this domain a model is a partial differential equation. Multiscale methods approximate it by another partial differential equation which can be numerically simulated in a reasonable time. The challenge consists in taking into account a wide range of geometries combining thin and periodic structures with the possibility of multiple nested scales. In this paper we present a transformation language that will make the development of MEMSALab more feasible. It is proposed as a Maple package for rule-based programming, rewriting strategies and their combination with standard Maple code. We illustrate the practical interest of this language by using it to encode two examples of multiscale derivations, namely the two-scale limit of the derivative operator and the two-scale model of the stationary heat equation.Comment: 36 page

Limits on Sparse Data Acquisition: RIC Analysis of Finite Gaussian Matrices

One of the key issues in the acquisition of sparse data by means of compressed sensing (CS) is the design of the measurement matrix. Gaussian matrices have been proven to be information-theoretically optimal in terms of minimizing the required number of measurements for sparse recovery. In this paper we provide a new approach for the analysis of the restricted isometry constant (RIC) of finite dimensional Gaussian measurement matrices. The proposed method relies on the exact distributions of the extreme eigenvalues for Wishart matrices. First, we derive the probability that the restricted isometry property is satisfied for a given sufficient recovery condition on the RIC, and propose a probabilistic framework to study both the symmetric and asymmetric RICs. Then, we analyze the recovery of compressible signals in noise through the statistical characterization of stability and robustness. The presented framework determines limits on various sparse recovery algorithms for finite size problems. In particular, it provides a tight lower bound on the maximum sparsity order of the acquired data allowing signal recovery with a given target probability. Also, we derive simple approximations for the RICs based on the Tracy-Widom distribution.Comment: 11 pages, 6 figures, accepted for publication in IEEE transactions on information theor

Frictional behavior of talc-calcite mixtures

Faults involving phyllosilicates appear weak when compared to the laboratory-derived strength of most crustal rocks. Among phyllosilicates, talc, with very low friction, is one of the weakest minerals involved in various tectonic settings. As the presence of talc has been recently documented in carbonate faults, we performed laboratory friction experiments to better constrain how various amounts of talc could alter these fault’s frictional properties. We used a biaxial apparatus to systematically shear different mixtures of talc and calcite as powdered gouge at room temperature, normal stresses up to 50 MPa and under different pore fluid saturated conditions, i.e., CaCO3-equilibrated water and silicone oil. We performed slide-hold-slide tests, 1–3000 s, to measure the amount of frictional healing and velocity-stepping tests, 0.1–1000 μm/s, to evaluate frictional stability. We then analyzed microstructures developed during our experiments. Our results show that with the addition of 20% talc the calcite gouge undergoes a 70% reduction in steady state frictional strength, a complete reduction of frictional healing and a transition from velocity-weakening to velocity-strengthening behavior. Microstructural analysis shows that with increasing talc content, deformation mechanisms evolve from distributed cataclastic flow of the granular calcite to localized sliding along talc-rich shear planes, resulting in a fully interconnected network of talc lamellae from 20% talc onward. Our observations indicate that in faults where talc and calcite are present, a low concentration of talc is enough to strongly modify the gouge’s frictional properties and specifically to weaken the fault, reduce its ability to sustain future stress drops, and stabilize slip

Graph Based Reduction of Program Verification Conditions

Increasing the automaticity of proofs in deductive verification of C programs is a challenging task. When applied to industrial C programs known heuristics to generate simpler verification conditions are not efficient enough. This is mainly due to their size and a high number of irrelevant hypotheses. This work presents a strategy to reduce program verification conditions by selecting their relevant hypotheses. The relevance of a hypothesis is determined by the combination of a syntactic analysis and two graph traversals. The first graph is labeled by constants and the second one by the predicates in the axioms. The approach is applied on a benchmark arising in industrial program verification