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The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction wave for bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in [21, 23], we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as the applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction wave, are proved for the 1D bipolar Vlasov-Poisson-Boltzmann system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of rarefaction wave fan to compressible Euler equations is proved to 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in [18] sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces., to appear in Archive for Rational Mechanics and Analysis

High-resolution bathymetry, seismic reflection, and piston core data from a submarine channel on the western Niger Delta slope demonstrate that thick, coarse-grained, amalgamated sands in the channel thalweg/axis transition to thin, fine-grained, bedded sands and muds in the channel margin. Radiocarbon ages indicate that axis and margin deposits are coeval. Core data show that bed thickness, grain size, and deposition rate strongly decrease with increasing height above channel thalweg and/or distance from channel centerline. A 5 times decrease in bed thickness and 1–2 ψ decrease in grain size are evident over a 20 m elevation change (approximately the elevation difference between axis and margin). A simplified in-channel sedimentation model that solves vertical concentration and velocity profiles of turbidity currents accurately reproduces the vertical trends in grain size and bed thickness shown in the core data set. The close match between data and model suggests that the vertical distribution of grain size and bed thickness shown in this study is widely applicable and can be used to predict grain size and facies variation in data-poor areas (e.g., subsurface cores). This study emphasizes that facies models for submarine channel deposits should recognize that grain-size and thickness trends within contemporaneous axis-margin packages require a change in elevation above the thalweg. The transition from thick-bedded, amalgamated, coarser-grained sands to thin-bedded, nonamalgamated, finer-grained successions is primarily a reflection of a change in elevation. Even a relatively small elevation change (e.g., 1 m) is enough to result in a significant change in grain size, bed thickness, and facies.

Computer models are essential tools in the earth system sciences. They underpin our search for understanding of earth system functioning and support decision- and policy-making across spatial and temporal scales. To understand the implications of uncertainty and environmental variability on the identification of such earth system models and their predictions, we can rely on increasingly powerful Global Sensitivity Analysis (GSA) methods. Previous reviews have characterised the variability of GSA methods available and their usability for different tasks. In our paper we rather focus on reviewing what has been learned so far by applying GSA to models across the earth system sciences, independently of the specific algorithm that was applied. We identify and discuss 10 key findings with general applicability and relevance for the earth sciences. We further provide an A-B-C-D of best practise in applying GSA methods, which we have derived from analysing why some GSA applications provided more insight than others.

We prove that if $T$ is a complete theory with weak elimination of imaginaries, then there is an explicit bijection between strict independence relations for $T$ and strict independence relations for $T^{\text{eq}}$. We use this observation to show that if $T$ is the theory of the Fra\"{i}ss\'{e} limit of finite metric spaces with integer distances, then $T^{\text{eq}}$ has more than one strict independence relation. This answers a question of Adler [1, Question 1.7]., Comment: 9 pages, to appear in Archive for Mathematical Logic

A compactness framework is established for approximate solutions to subsonic-sonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop applies for both non-homentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles., Comment: 17 pages, 2 figures, Archive for Rational Mechanics and Analysis, 2015

A generic class of metamaterials is introduced and is shown to exhibit frequency dependent double negative effective properties. We develop a rigorous method for calculating the frequency intervals where either double negative or double positive effective properties appear and show how these intervals imply the existence of propagating Bloch waves inside sub-wavelength structures. The branches of the dispersion relation associated with Bloch modes are shown to be explicitly determined by the Dirichlet spectrum of the high dielectric phase and the generalized electrostatic spectra of the complement. Comment: This is the final version of the paper accepted for publication in Archive for Rational Mechanics and Analysis March 4, 2013

We introduce the notion of $R$-analytic functions. These are definable in an o-minimal expansion of a real closed field $R$ and are locally the restriction of a $K$-differentiable function (defined by Peterzil and Starchenko) where $K=R[\sqrt{-1}]$ is the algebraic closure of $R$. The class of these functions in this general setting exhibits the nice properties of real analytic functions. We also define strongly $R$-analytic functions. These are globally the restriction of a $K$-differentiable function. We show that in arbitrary models of important o-minimal theories strongly $R$-analytic functions abound and that the concept of analytic cell decomposition can be transferred to non-standard models., Comment: Final version; to appear at Archive for Mathematical Logic

We consider singularities in the ElectroHydroDynamic equations. In a regime where we are allowed to neglect surface tension, and assuming that the free surface is given by an injective curve and that either the fluid velocity or the electric field satisfies a certain non-degeneracy condition, we prove that either the fluid region or the gas region is asymptotically a cusp. Our proofs depend on a combination of monotonicity formulas and a non-vanishing result by Caffarelli and Friedman. As a by-product of our analysis we also obtain a special solution with convex conical air-phase which we believe to be new., Comment: To appear in Archive for Rational Mechanics and Analysis

We provide a complete and rigorous description of phase transitions for kinetic models of self-propelled particles interacting through alignment. These models exhibit a competition between alignment and noise. Both the alignment frequency and noise intensity depend on a measure of the local alignment. We show that, in the spatially homogeneous case, the phase transition features (number and nature of equilibria, stability, convergence rate, phase diagram, hysteresis) are totally encoded in how the ratio between the alignment and noise intensities depend on the local alignment. In the spatially inhomogeneous case, we derive the macroscopic models associated to the stable equilibria and classify their hyperbolicity according to the same function., Archive for Rational Mechanics and Analysis (2014) accepted

We introduce what we call the second-order Boltzmann-Gibbs principle, which allows to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann-Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture., Fundação para a Ciência e a Tecnologia (FCT)

We show that the local equivalence problem for second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also show that, modulo Cartan duality and point transformations, the Painlev\'e-I equation can be characterized as the simplest second-order ODE belonging to the class of equations requiring 10th order jets for their classification., Comment: revised introduction, minor correctoins, added mathematica notebook, resubmitting using a tgz archive to give direct access to ancillary file

In view of applications to conformal field theory or to other branches of theoretical physics and mathematics, new examples of character tables for Drinfeld doubles of finite groups (modular data) are made available on a website., Comment: 7 pages, 1 figure, 7th International Conference on Mathematical Methods in Physics, Rio de Janeiro, Brazil, April 2012. Version 2: a misleading sentence was removed from section 2. http://pos.sissa.it/archive/conferences/175/024/ICMP%202012_024.pdf

Poroelasticity theory models the dynamics of porous, fluid-saturated media. It was pioneered by Maurice Biot in the 1930s through 1960s, and has applications in several fields, including geophysics and modeling of in vivo bone. A wide variety of methods have been used to model poroelasticity, including finite difference, finite element, pseudospectral, and discontinuous Galerkin methods. In this work we use a Cartesian-grid high-resolution finite volume method to numerically solve Biot's equations in the time domain for orthotropic materials, with the stiff relaxation source term in the equations incorporated using operator splitting. This class of finite volume method has several useful properties, including the ability to use wave limiters to reduce numerical artifacts in the solution, ease of incorporating material inhomogeneities, low memory overhead, and an explicit time-stepping approach. To the authors' knowledge, this is the first use of high-resolution finite volume methods to model poroelasticity. The solution code uses the CLAWPACK finite volume method software, which also includes block-structured adaptive mesh refinement in its AMRCLAW variant. We present convergence results for known analytic plane wave solutions, achieving second-order convergence rates outside of the stiff regime of the system. Our convergence rates are degraded in the stiff regime, but we still achieve similar levels of error on the finest grids examined. We also demonstrate good agreement against other numerical results from the literature. To aid in reproducibility, we provide all of the code used to generate the results of this paper, at https://bitbucket.org/grady_lemoine/poro-2d-cartesian-archive ., Comment: 26 pages, 7 figures, to appear in SIAM Journal on Scientific Computing Section 2, code provided at https://bitbucket.org/grady_lemoine/poro-2d-cartesian-archive

In a recent article (Canc\`es, Deleurence and Lewin, Commun. Math. Phys., 281 (2008), pp. 129-177), we have rigorously derived, by means of bulk limit arguments, a new variational model to describe the electronic ground state of insulating or semiconducting crystals in the presence of local defects. In this so-called reduced Hartree-Fock model, the ground state electronic density matrix is decomposed as $\gamma = \gamma^0_{\rm per} + Q_{\nu,\epsilon_{\rm F}}$, where $\gamma^0_{\rm per}$ is the ground state density matrix of the host crystal and $Q_{\nu,\epsilon_{\rm F}}$ the modification of the electronic density matrix generated by a modification $\nu$ of the nuclear charge of the host crystal, the Fermi level $\epsilon_{\rm F}$ being kept fixed. The purpose of the present article is twofold. First, we study more in details the mathematical properties of the density matrix $Q_{\nu,\epsilon_{\rm F}}$ (which is known to be a self-adjoint Hilbert-Schmidt operator on $L^2(\R^3)$). We show in particular that if $\int_{\RR^3} \nu \neq 0$, $Q_{\nu,\epsilon_{\rm F}}$ is not trace-class. Moreover, the associated density of charge is not in $L^1(\R^3)$ if the crystal exhibits anisotropic dielectric properties. These results are obtained by analyzing, for a small defect $\nu$, the linear and nonlinear terms of the resolvent expansion of $Q_{\nu,\epsilon_{\rm F}}$. Second, we show that, after an appropriate rescaling, the potential generated by the microscopic total charge (nuclear plus electronic contributions) of the crystal in the presence of the defect, converges to a homogenized electrostatic potential solution to a Poisson equation involving the macroscopic dielectric permittivity of the crystal. This provides an alternative (and rigorous) derivation of the Adler-Wiser formula., Comment: Archive for Rational Mechanics and Analysis (2009) in press

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain ${\mathcal D} \subset {\mathbb R}^d$ until it hits the boundary and bounces randomly inside according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord "picked at random" in ${\mathcal D}$, and we study the angle of intersection of the process with a $(d-1)$-dimensional manifold contained in ${\mathcal D}$., Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains)

We study the real Monge-Amp\`ere equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine differential geometry, where solutions yield parabolic affine sphere hypersurfaces. We find explicit examples, connect the holomorphic function representation to Hitchin's description of special Lagrangian moduli space, and construct the developing map explicitly for a singularity corresponding to the type $I_n$ elliptic fiber (after hyper-K\"ahler rotation). Following Baues and Cort\'es, we show that various types of metric cones over two-dimensional elliptic affine spheres generate solutions of the Monge-Amp\`ere equation in three dimensions. We then prove a local and global existence theorem for an elliptic affine two-sphere metric with prescribed singularities. The metric cone over the two-sphere minus three points yields a parabolic affine sphere with singularities along a "Y"-shaped locus. This gives a semi-flat Calabi-Yau metric in a neighborhood of the "Y" vertex. In the erratum, we correct a gap in the solution of the elliptic affine sphere equation (with the small caveat that a parameter in the equation, the cubic differential, must be smaller than we assumed before). The new proof still allows us to construct semi-flat Calabi-Yau metrics in a neighborhood of the "Y" vertex by appealing to the result of Baues-Cort\'es. We also give an alternate construction of such semi-flat Calabi-Yau metrics by using a result on hyperbolic affine spheres due to the first author., Comment: 28 pages, plus 7-page erratum; original archive version of paper replaced by published version, and erratum added. J. Differential Geom. 71 (2005), no. 1, 129--158 (erratum to appear later)

We study rational curves on algebraic varieties, especially on normal affine varieties endowed with a $\C^*$-action. For varieties with an isolated singularity, we show that the presence of sufficiently many rational curves outside the singular point strongly affects the character of the singularity. This provides an explanation of classical results due to H.A. Schwartz and G.H. Halphen on polynomial solutions of the generalized Fermat equation., Comment: 24 pages; uses Paul Taylor's diagrams.tex, see e.g. ftp://ftp.dante.de/tex-archive/macros/generic/diagrams/taylor/V3,88.tex

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [9]. We describe our results in terms of a sequence of parameters $(s_n)_{n\geq 1}$ that quantifies the extreme-value behavior of small weights, and that describes different universality classes for first passage percolation on the complete graph. We consider both $n$-independent as well as $n$-dependent edge weights. The simplest example consists of edge weights of the form $E^{s_n}$, where $E$ is an exponential random variable with mean 1. In this paper, we investigate the case where $s_n\rightarrow \infty$, and focus on the local neighborhood of a vertex. We establish that the smallest-weight tree of a vertex locally converges to the invasion percolation cluster on the Poisson weighted infinite tree. In addition, we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices., This is the first paper in a series of two papers on "Long paths in first passage percolation on the complete graph". We have update the archive reference to the second paper

The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations along a moving interface. Unlike our approach for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in srong norms of our sequence of regularized problems., 43 pages, to appear in Archive for Rational Mechanics and Analysis

The function $\epsilon(x)=\mbox{li}(x)-\pi(x)$ is known to be positive up to the (very large) Skewes' number. Besides, according to Robin's work, the functions $\epsilon_{\theta}(x)=\mbox{li}[\theta(x)]-\pi(x)$ and $\epsilon_{\psi}(x)=\mbox{li}[\psi(x)]-\pi(x)$ are positive if and only if Riemann hypothesis (RH) holds (the first and the second Chebyshev function are $\theta(x)=\sum_{p \le x} \log p$ and $\psi(x)=\sum_{n=1}^x \Lambda(n)$, respectively, $\mbox{li}(x)$ is the logarithmic integral, $\mu(n)$ and $\Lambda(n)$ are the M\"obius and the Von Mangoldt functions). Negative jumps in the above functions $\epsilon$, $\epsilon_{\theta}$ and $\epsilon_{\psi}$ may potentially occur only at $x+1 \in \mathcal{P}$ (the set of primes). One denotes $j_p=\mbox{li}(p)-\mbox{li}(p-1)$ and one investigates the jumps $j_p$, $j_{\theta(p)}$ and $j_{\psi(p)}$. In particular, $j_p1$ for $p, Comment: 15 pages section 2.2 added, new sequences added, Fig. 2 and 3 are new

In this paper, we propose a novel variable selection approach in the framework of multivariate linear models taking into account the dependence that may exist between the responses. It consists in estimating beforehand the covariance matrix Σ of the responses and to plug this estimator in a Lasso criterion, in order to obtain a sparse estimator of the coefficient matrix. The properties of our approach are investigated both from a theoretical and a numerical point of view. More precisely, we give general conditions that the estimators of the covariance matrix and its inverse have to satisfy in order to recover the positions of the null and non null entries of the coefficient matrix when the size of Σ is not fixed and can tend to infinity. We prove that these conditions are satisfied in the particular case of some Toeplitz matrices. Our approach is implemented in the R package MultiVarSel available from the Comprehensive R Archive Network (CRAN) and is very attractive since it benefits from a low computational load. We also assess the performance of our methodology using synthetic data and compare it with alternative approaches. Our numerical experiments show that including the estimation of the covariance matrix in the Lasso criterion dramatically improves the variable selection performance in many cases.

We show that the Newton polytope of a polynomial has a strong impact on the distribution of its mass and zeros. The basic theme is that Newton polytopes determine allowed and forbidden regions for these distributions. We equip the space of (holomorphic) polynomials of degree $\leq p$ in $m$ complex variables with its usual $SU(m + 1)$-invariant Gaussian probability measure and then consider the conditional measure induced on the subspace of polynomials with fixed Newton polytope $P$. We then determine the asymptotics of the conditional expectation $E_{|N P}(Z^k)$ of simultaneous zeros of $k$ polynomials with Newton polytope $NP$ as $N \to \infty$. When $P = \Sigma$, the unit simplex, then it is obvious and well-known that the expected zero distribution $E_{|N\Sigma}(Z^k)$ is uniform relative to the Fubini-Study form. For a convex polytope $P\subset p\Sigma$, we show that there is an allowed region on which $N^{-k}E_{|N P}(Z^k)$ is asymptotically uniform as the scaling factor $N\to\infty$. However, the zeros have an exotic distribution in the complementary forbidden region, and when $k=m$, the expected percentage of simultaneous zeros in the forbidden region approaches 0 as $N\to\infty$, yielding a quantitative version of the Kouchnirenko-Bernstein theorem., Comment: Revised exposition: terminology and facts concerning toric varieties have been moved to an appendix. This posting supersedes our article of similar title posted on the archive as math.AG/0203074. By using Euler-Maclaurin sum formulae for polytopes instead of Toeplitz analysis on toric varieties, we significantly simplied the proofs and generalized the results

The conditions that can lead to the exploitative depletion of a shared resource, i.e, the tragedy of the commons, can be reformulated as a game of prisoner's dilemma: while preserving the common resource is in the best interest of the group, over-consumption is in the interest of each particular individual at any given point in time. One way to try and prevent the tragedy of the commons is through infliction of punishment for over-consumption and/or encouraging under-consumption, thus selecting against over-consumers. Here, the effectiveness of various punishment functions in an evolving consumer-resource system is evaluated within a framework of a parametrically heterogeneous system of ordinary differential equations (ODEs). Conditions leading to the possibility of sustainable coexistence with the common resource for a subset of cases are identified analytically using adaptive dynamics; the effects of punishment on heterogeneous populations with different initial composition are evaluated using the Reduction theorem for replicator equations. Obtained results suggest that one cannot prevent the tragedy of the commons through rewarding of under-consumers alone - there must also be an implementation of some degree of punishment that increases in a non-linear fashion with respect to over-consumption and which may vary depending on the initial distribution of clones in the population.

I provide an explicit construction of spectral curves for the affine $\mathrm{E}_8$ relativistic Toda chain. Their closed form expression is obtained by determining the full set of character relations in the representation ring of $\mathrm{E}_8$ for the exterior algebra of the adjoint representation; this is in turn employed to provide an explicit construction of both integrals of motion and the action-angle map for the resulting integrable system. I consider two main areas of applications of these constructions. On the one hand, I consider the resulting family of spectral curves in the context of the correspondences between Toda systems, 5d Seiberg-Witten theory, Gromov-Witten theory of orbifolds of the resolved conifold, and Chern-Simons theory to establish a version of the B-model Gopakumar-Vafa correspondence for the $\mathrm{sl}_N$ L\^e-Murakami-Ohtsuki invariant of the Poincar\'e integral homology sphere to all orders in $1/N$. On the other, I consider a degenerate version of the spectral curves and prove a 1-dimensional Landau-Ginzburg mirror theorem for the Frobenius manifold structure on the space of orbits of the extended affine Weyl group of type $\mathrm{E}_8$ introduced by Dubrovin-Zhang (equivalently, the orbifold quantum cohomology of the type-$\mathrm{E}_8$ polynomial $\mathbb{C} P^1$ orbifold). This leads to closed-form expressions for the flat co-ordinates of the Saito metric, the prepotential, and a higher genus mirror theorem based on the Chekhov-Eynard-Orantin recursion. I will also show how the constructions of the paper lead to a generalisation of a conjecture of Norbury-Scott to ADE $\mathbb{P}^1$-orbifolds, and a mirror of the Dubrovin-Zhang construction for all Weyl groups and choices of marked roots., Comment: 87 pages, 5 figures. Raw binaries containing spectral curve data available with an accompanying Mathematica notebook at https://tiny.cc/E8SpecCurve (180Mb ZIP archive; beware this currently unpacks to 912Mb). v2: a few corrections in section 5, typos fixed. v3: minor changes in the introduction, references added, version accepted for publication on Proc. Lond. Math. Soc

This paper focuses on the stability analysis of WKB approximate solutions in geometric optics with the absence of strong transparency conditions. We introduce a compatible condition and a singular localization method which allows us to prove the stability of WKB solutions over long time intervals. This compatible condition is weaker than the strong transparency condition. The singular localization method allows us to do delicate analysis near resonances. As an application, we show the long time approximation of Klein-Gordon equations by Schr\"odinger equations in the non-relativistic limit regime., Comment: 41 pages, final version, to appear in Archive for Rational Mechanics and Analysis

Consider the Cauchy problem for a strictly hyperbolic, $N\times N$ quasilinear system in one space dimension $$ u_t+A(u) u_x=0,\qquad u(0,x)=\bar u(x), \eqno (1) $$ where $u \mapsto A(u)$ is a smooth matrix-valued map, and the initial data $\overline u$ is assumed to have small total variation. We investigate the rate of convergence of approximate solutions of (1) constructed by the Glimm scheme, under the assumption that, letting $\lambda_k(u)$, $r_k(u)$ denote the $k$-th eigenvalue and a corresponding eigenvector of $A(u)$, respectively, for each $k$-th characteristic family the linearly degenerate manifold $$ \mathcal{M}_k \doteq \big\{u\in\Omega : \nabla\lambda_k(u)\cdot r_k(u)=0\big\} $$ is either the whole space, or it is empty, or it consists of a finite number of smooth, $N-1$-dimensional, connected, manifolds that are transversal to the characteristic vector field $r_k$. We introduce a Glimm type functional which is the sum of the cubic interaction potential defined in \cite{sie}, and of a quadratic term that takes into account interactions of waves of the same family with strength smaller than some fixed threshold parameter. Relying on an adapted wave tracing method, and on the decrease amount of such a functional, we obtain the same type of error estimates valid for Glimm approximate solutions of hyperbolic systems satisfying the classical Lax assumptions of genuine nonlinearity or linear degeneracy of the characteristic families., Comment: To appear on Archive for Rational Mechanics and Analysis