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We present a new algorithm for imitation learning in infinite horizon linear MDPs dubbed ILARL which greatly improves the bound on the number of trajectories that the learner needs to sample from the environment. In particular, we remove exploration assumptions required in previous works and we improve the dependence on the desired accuracy $\epsilon$ from $\mathcal{O}(\epsilon^{-5})$ to $\mathcal{O}(\epsilon^{-4})$. Our result relies on a connection between imitation learning and online learning in MDPs with adversarial losses. For the latter setting, we present the first result for infinite horizon linear MDP which may be of independent interest. Moreover, we are able to provide a strengthen result for the finite horizon case where we achieve $\mathcal{O}(\epsilon^{-2})$. Numerical experiments with linear function approximation shows that ILARL outperforms other commonly used algorithms., Comment: Accepted at ICML 2024

In this work, we introduce a new variant of online gradient descent, which provably converges to Nash Equilibria and simultaneously attains sublinear regret for the class of congestion games in the semi-bandit feedback setting. Our proposed method admits convergence rates depending only polynomially on the number of players and the number of facilities, but not on the size of the action set, which can be exponentially large in terms of the number of facilities. Moreover, the running time of our method has polynomial-time dependence on the implicit description of the game. As a result, our work answers an open question from (Du et. al, 2022)., Comment: ICML 2023

In online reinforcement learning (RL), instead of employing standard structural assumptions on Markov decision processes (MDPs), using a certain coverage condition (original from offline RL) is enough to ensure sample-efficient guarantees (Xie et al. 2023). In this work, we focus on this new direction by digging more possible and general coverage conditions, and study the potential and the utility of them in efficient online RL. We identify more concepts, including the $L^p$ variant of concentrability, the density ratio realizability, and trade-off on the partial/rest coverage condition, that can be also beneficial to sample-efficient online RL, achieving improved regret bound. Furthermore, if exploratory offline data are used, under our coverage conditions, both statistically and computationally efficient guarantees can be achieved for online RL. Besides, even though the MDP structure is given, e.g., linear MDP, we elucidate that, good coverage conditions are still beneficial to obtain faster regret bound beyond $\widetilde{O}(\sqrt{T})$ and even a logarithmic order regret. These results provide a good justification for the usage of general coverage conditions in efficient online RL., Comment: Accepted by ICML 2023

While Reinforcement Learning (RL) aims to train an agent from a reward function in a given environment, Inverse Reinforcement Learning (IRL) seeks to recover the reward function from observing an expert's behavior. It is well known that, in general, various reward functions can lead to the same optimal policy, and hence, IRL is ill-defined. However, (Cao et al., 2021) showed that, if we observe two or more experts with different discount factors or acting in different environments, the reward function can under certain conditions be identified up to a constant. This work starts by showing an equivalent identifiability statement from multiple experts in tabular MDPs based on a rank condition, which is easily verifiable and is shown to be also necessary. We then extend our result to various different scenarios, i.e., we characterize reward identifiability in the case where the reward function can be represented as a linear combination of given features, making it more interpretable, or when we have access to approximate transition matrices. Even when the reward is not identifiable, we provide conditions characterizing when data on multiple experts in a given environment allows to generalize and train an optimal agent in a new environment. Our theoretical results on reward identifiability and generalizability are validated in various numerical experiments.

This work develops new algorithms with rigorous efficiency guarantees for infinite horizon imitation learning (IL) with linear function approximation without restrictive coherence assumptions. We begin with the minimax formulation of the problem and then outline how to leverage classical tools from optimization, in particular, the proximal-point method (PPM) and dual smoothing, for online and offline IL, respectively. Thanks to PPM, we avoid nested policy evaluation and cost updates for online IL appearing in the prior literature. In particular, we do away with the conventional alternating updates by the optimization of a single convex and smooth objective over both cost and Q-functions. When solved inexactly, we relate the optimization errors to the suboptimality of the recovered policy. As an added bonus, by re-interpreting PPM as dual smoothing with the expert policy as a center point, we also obtain an offline IL algorithm enjoying theoretical guarantees in terms of required expert trajectories. Finally, we achieve convincing empirical performance for both linear and neural network function approximation.

This paper provides a theoretical study of deep neural function approximation in reinforcement learning (RL) with the $\epsilon$-greedy exploration under the online setting. This problem setting is motivated by the successful deep Q-networks (DQN) framework that falls in this regime. In this work, we provide an initial attempt on theoretical understanding deep RL from the perspective of function class and neural networks architectures (e.g., width and depth) beyond the ``linear'' regime. To be specific, we focus on the value based algorithm with the $\epsilon$-greedy exploration via deep (and two-layer) neural networks endowed by Besov (and Barron) function spaces, respectively, which aims at approximating an $\alpha$-smooth Q-function in a $d$-dimensional feature space. We prove that, with $T$ episodes, scaling the width $m = \widetilde{\mathcal{O}}(T^{\frac{d}{2\alpha + d}})$ and the depth $L=\mathcal{O}(\log T)$ of the neural network for deep RL is sufficient for learning with sublinear regret in Besov spaces. Moreover, for a two layer neural network endowed by the Barron space, scaling the width $\Omega(\sqrt{T})$ is sufficient. To achieve this, the key issue in our analysis is how to estimate the temporal difference error under deep neural function approximation as the $\epsilon$-greedy exploration is not enough to ensure ``optimism''. Our analysis reformulates the temporal difference error in an $L^2(\mathrm{d}\mu)$-integrable space over a certain averaged measure $\mu$, and transforms it to a generalization problem under the non-iid setting. This might have its own interest in RL theory for better understanding $\epsilon$-greedy exploration in deep RL., Comment: Accepted by NeurIPS22

Abstract object properties and their relations are deeply rooted in human common sense, allowing people to predict the dynamics of the world even in situations that are novel but governed by familiar laws of physics. Standard machine learning models in model-based reinforcement learning are inadequate to generalize in this way. Inspired by the classic framework of noisy indeterministic deictic (NID) rules, we introduce here Neural NID, a method that learns abstract object properties and relations between objects with a suitably regularized graph neural network. We validate the greater generalization capability of Neural NID on simple benchmarks specifically designed to assess the transition dynamics learned by the model., Comment: Physical Reasoning and Inductive Biases for the Real World at NeurIPS 2021

Imitation learning (IL) is a popular paradigm for training policies in robotic systems when specifying the reward function is difficult. However, despite the success of IL algorithms, they impose the somewhat unrealistic requirement that the expert demonstrations must come from the same domain in which a new imitator policy is to be learned. We consider a practical setting, where (i) state-only expert demonstrations from the real (deployment) environment are given to the learner, (ii) the imitation learner policy is trained in a simulation (training) environment whose transition dynamics is slightly different from the real environment, and (iii) the learner does not have any access to the real environment during the training phase beyond the batch of demonstrations given. Most of the current IL methods, such as generative adversarial imitation learning and its state-only variants, fail to imitate the optimal expert behavior under the above setting. By leveraging insights from the Robust reinforcement learning (RL) literature and building on recent adversarial imitation approaches, we propose a robust IL algorithm to learn policies that can effectively transfer to the real environment without fine-tuning. Furthermore, we empirically demonstrate on continuous-control benchmarks that our method outperforms the state-of-the-art state-only IL method in terms of the zero-shot transfer performance in the real environment and robust performance under different testing conditions., Comment: accepted to AAMAS 2022 (camera-ready version)

We study the inverse reinforcement learning (IRL) problem under a transition dynamics mismatch between the expert and the learner. Specifically, we consider the Maximum Causal Entropy (MCE) IRL learner model and provide a tight upper bound on the learner's performance degradation based on the $\ell_1$-distance between the transition dynamics of the expert and the learner. Leveraging insights from the Robust RL literature, we propose a robust MCE IRL algorithm, which is a principled approach to help with this mismatch. Finally, we empirically demonstrate the stable performance of our algorithm compared to the standard MCE IRL algorithm under transition dynamics mismatches in both finite and continuous MDP problems.

The Fredholm integral equations of the first kind are a classical example of ill-posed problem in the sense of Hadamard. If the integral operator is self-adjoint and admits a set of eigenfunctions, then a formal solution can be written in terms of eigenfunction expansions. One of the possible methods of regularization consists in truncating this formal expansion after restricting the class of admissible solutions through a-priori global bounds. In this paper we reconsider various possible methods of truncation from the viewpoint of the $\varepsilon$-coverings of compact sets., Comment: 17 pages

We study the holomorphic extension associated with power series, i.e., the analytic continuation from the unit disk to the cut-plane $\mathbb{C} \setminus [1,+\infty)$. Analogous results are obtained also in the study of trigonometric series: we establish conditions on the series coefficients which are sufficient to guarantee the series to have a KMS analytic structure. In the case of power series we show the connection between the unique (Carlsonian) interpolation of the coefficients of the series and the Laplace transform of a probability distribution. Finally, we outline a procedure which allows us to obtain a numerical approximation of the jump function across the cut starting from a finite number of power series coefficients. By using the same methodology, the thermal Green functions at real time can be numerically approximated from the knowledge of a finite number of noisy Fourier coefficients in the expansion of the thermal Green functions along the imaginary axis of the complex time plane., Comment: 38 pages, 4 figures

We present a method able to recover location and residue of poles of functions meromorphic in a half--plane from samples of the function on the real positive semi-axis. The function is assumed to satisfy appropriate asymptotic conditions including, in particular, that required by Carlson's theorem. The peculiar features of the present procedure are: (i) it does not make use of the approximation of meromorphic functions by rational functions; (ii) it does not use the standard methods of regularization of ill-posed problems. The data required for the determination of the pole parameters (i.e., location and residue) are the approximate values of the meromorphic function on a finite set of equidistant points on the real positive semi-axis. We show that this method is numerically stable by proving that the algorithm is convergent as the number of data points tends to infinity and the noise on the input data goes to zero. Moreover, we can also evaluate the degree of approximation of the estimates of pole location and residue which we obtain from the knowledge of a finite number of noisy samples., Comment: 33 pages, 8 figures

The inverse problem in optics, which is closely related to the classical question of the resolving power, is reconsidered as a communication channel problem. The main result is the evaluation of the maximum number $M_\epsilon$ of $\epsilon$-distinguishable messages ($\epsilon$ being a bound on the noise of the image) which can be conveyed back from the image to reconstruct the object. We study the case of coherent illumination. By using the concept of Kolmogorov's $\epsilon$-capacity, we obtain: $M_\epsilon ~ 2^{S \log(1/\epsilon)} \to \infty$ as $\epsilon \to 0$, where S is the Shannon number. Moreover, we show that the $\epsilon$-capacity in inverse optical imaging is nearly equal to the amount of information on the object which is contained in the image. We thus compare the results obtained through the classical information theory, which is based on the probability theory, with those derived from a form of topological information theory, based on Kolmogorov's $\epsilon$-entropy and $\epsilon$-capacity, which are concepts related to the evaluation of the massiveness of compact sets., Comment: 18 pages, 2 figures

In this article we study resonances and surface waves in $\pi^+$--p scattering. We focus on the sequence whose spin-parity values are given by $J^p = {3/2}^+,{7/2}^+, {11/2}^+, {15/2}^+,{19/2}^+$. A widely-held belief takes for granted that this sequence can be connected by a moving pole in the complex angular momentum (CAM) plane, which gives rise to a linear trajectory of the form $J = \alpha_0+\alpha' m^2$, $\alpha'\sim 1/(\mathrm{GeV})^2$, which is the standard expression of the Regge pole trajectory. But the phenomenology shows that only the first few resonances lie on a trajectory of this type. For higher $J^p$ this rule is violated and is substituted by the relation $J\sim kR$, where $k$ is the pion--nucleon c.m.s.-momentum, and $R\sim 1$ fm. In this article we prove: (a) Starting from a non-relativistic model of the proton, regarded as composed by three quarks confined by harmonic potentials, we prove that the first three members of this $\pi^+$-p resonance sequence can be associated with a vibrational spectrum of the proton generated by an algebra $Sp(3,R)$. Accordingly, these first three members of the sequence can be described by Regge poles and lie on a standard linear trajectory. (b) At higher energies the amplitudes are dominated by diffractive scattering, and the creeping waves play a dominant role. They can be described by a second class of poles, which can be called Sommerfeld's poles, and lie on a line nearly parallel to the imaginary axis of the CAM-plane. (c) The Sommerfeld pole which is closest to the real axis of the CAM-plane is dominant at large angles, and describes in a proper way the backward diffractive peak in both the following cases: at fixed $k$, as a function of the scattering angle, and at fixed scattering angle $\theta=\pi$, as a function of $k$. (d) The evolution of this pole, as a function of $k$, is given in first approximation by $J\simeq kR$., Comment: 35 pages, 8 figures

In this article, the ray tracing method is studied beyond the classical geometrical theory. The trajectories are here regarded as geodesics in a Riemannian manifold, whose metric and topological properties are those induced by the refractive index (or, equivalently, by the potential). First, we derive the geometrical quantization rule, which is relevant to describe the orbiting bound-states observed in molecular physics. Next, we derive properties of the diffracted rays, regarded here as geodesics in a Riemannian manifold with boundary. A particular attention is devoted to the following problems: (i) modification of the classical stationary phase method suited to a neighborhood of a caustic; (ii) derivation of the connection formulae which enable one to obtain the uniformization of the classical eikonal approximation by patching up geodesic segments crossing the axial caustic; (iii) extension of the eikonal equation to mixed hyperbolic-elliptic systems, and generation of complex-valued rays in the shadow of the caustic. By these methods, we can study the creeping waves in diffractive scattering, describe the orbiting resonances present in molecular scattering beside the orbiting bound-states, and, finally, describe the generation of the evanescent waves, which are relevant in the nuclear rainbow., Comment: 30 pages, 2 figures

We present a simple and fast algorithm for the computation of the coefficients of the expansion of a function f(cos u) in ultraspherical (Gegenbauer) polynomials. We prove that these coefficients coincide with the Fourier coefficients of an Abel-type transform of the function f(cos u). This allows us to fully exploit the computational efficiency of the Fast Fourier Transform, computing the first N ultraspherical coefficients in just O (N log_2 N) operations., Comment: 7 pages, 1 figure, references added, typos corrected

An efficient procedure for the computation of the coefficients of Legendre expansions is here presented. We prove that the Legendre coefficients associated with a function f(x) can be represented as the Fourier coefficients of an Abel-type transform of f(x). The computation of N Legendre coefficients can then be performed in O(N log N) operations with a single Fast Fourier Transform of the Abel-type transform of f(x)., Comment: 5 pages

This paper presents a backfitting-type method for estimating and forecasting a periodically correlated partially linear model with exogeneous variables and heteroskedastic input noise. A rate of convergence of the estimator is given. The results are valid even if the period is unknown.

The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(lambda,k) associated with physically relevant classes N_{w,alpha}^gamma of nonlocal potentials in corresponding domains D_{gamma,alpha}^delta of the space C^2 of the complex angular momentum lambda and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Theta(lambda,k)/sigma(lambda,k) of two holomorphic functions in D_{gamma,alpha}^delta is obtained by using the Fredholm-Smithies theory for complex k, at first for lambda=l integer, and in a second step for lambda complex (Real(lambda)>-1/2). Finally, we justify the "Watson resummation" of the partial wave amplitudes in an angular sector of the lambda-plane in terms of the various components of the polar manifold of T with equation sigma(lambda,k)=0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of lambda, this unified representation of the singularities of T also provides an attractive possible description of antiresonances in the lower half-plane of lambda. Such a possibility, which is forbidden in the usual theory of local potentials, represents an enriching alternative to the standard Breit-Wigner hard-sphere picture of antiresonances., Comment: 85 pages, 2 figures

We study the geometrical optics generated by a refractive index of the form $n(x,y)=1/y$ $(y>0)$, where $y$ is the coordinate of the vertical axis in an orthogonal reference frame in $\R^2$. We thus obtain what we call "hyperbolic geometrical optics" since the ray trajectories are geodesics in the Poincar\'e-Lobachevsky half--plane $\HH^2$. Then we prove that the constant phase surface are horocycles and obtain the \emph{horocyclic waves}, which are closely related to the classical Poisson kernel and are the analogs of the Euclidean plane waves. By studying the transport equation in the Beltrami pseudosphere, we prove(i) the conservation of the flow in the entire strip $0

This paper investigates the second order properties of a stationary process after random sampling. While a short memory process gives always rise to a short memory one, we prove that long-memory can disappear when the sampling law has heavy enough tails. We prove that under rather general conditions the existence of the spectral density is preserved by random sampling. We also investigate the effects of deterministic sampling on seasonal long-memory.

In this article we describe the generation of the evanescent waves which are present in the rarer medium at total reflection by using a mixed-type system, the Ludwig system, which leads naturally to consider a complex-valued phase. The Ludwig system is derived from the Helmholtz equation by using an appropriate modification of the stationary phase procedure: the Chester, Friedman and Ursell's method. The passage from the illuminated to the shadow region is described by means of the ray switching mechanism based on the Stokes phenomenon applied to the Airy function. Finally, the transport system connected to the Ludwig eikonal system is studied in the case of linear wavefronts and the existence of the Goos-Hanchen effect is proved., Comment: 30 pages, 3 figures

We propose a theory of the resonance-antiresonance scattering process which differs considerably from the classical one (the Breit-Wigner theory), which is commonly used in the phenomenological analysis. Here both resonances and antiresonances are described in terms of poles of the scattering amplitude: the resonances by poles in the first quadrant while the antiresonances by poles in the fourth quadrant of the complex angular momentum plane. The latter poles are produced by non-local potentials, which derive from the Pauli exchange forces acting among the nucleons or the quarks composing the colliding particles., Comment: 30 pages, 7 figures

The problem of object restoration in the case of spatially incoherent illumination is considered. A regularized solution to the inverse problem is obtained through a probabilistic approach, and a numerical algorithm based on the statistical analysis of the noisy data is presented. Particular emphasis is placed on the question of the positivity constraint, which is incorporated into the probabilistically regularized solution by means of a quadratic programming technique. Numerical examples illustrating the main steps of the algorithm are also given., Comment: 21 pages, 4 figures

In this paper we study a class of Hausdorff--transformed power series whose convergence is extremely slow for large values of the argument. We perform a Watson-type resummation of these expansions, and obtain, by the use of the Pollaczek polynomials, a new representation whose convergence is much faster. We can thus propose a new algorithm for the numerical evaluation of these expansions, which include series playing a relevant role in the computation of the partition function in statistical mechanics. By the same procedure we obtain also a solution of the classical Hausdorff moment problem., Comment: 18 pages, 4 Postscript figures

The problem of evaluating the information associated with Fredholm integral equations of the first kind, when the integral operator is self-adjoint and compact, is considered here. The data function is assumed to be perturbed gently by an additive noise so that it still belongs to the range of the operator. First we estimate upper and lower bounds for the epsilon-capacity (and then for the metric information), and explicit computations in some specific cases are given; then the problem is reformulated from a probabilistic viewpoint and use is made of the probabilistic information theory. The results obtained by these two approaches are then compared., Comment: 19 pages

In this article we prove that if the coefficients of a Fourier-Legendre expansion satisfy a suitable Hausdorff-type condition, then the series converges to a function which admits a holomorphic extension to a cut-plane. Furthermore, we prove that a Laplace-type (Laplace composed with Radon) transform of the function describing the jump across the cut is the unique Carlsonian interpolation of the Fourier coefficients of the expansion. We can thus reconstruct the discontinuity function from the coefficients of the Fourier-Legendre series by the use of the Pollaczek polynomials., Comment: 19 pages, 2 Postiscript figures

In a previous paper we have presented a new method for solving a class of Cauchy integral equations. In this work we discuss in detail how to manage this method numerically, when only a finite and noisy data set is available: particular attention is focused on the question of the numerical stability., Comment: 19 pages, 5 Postscript figures

In this paper we consider power and trigonometric series whose coefficients are supposed to satisfy the Hausdorff conditions, which play a relevant role in the moment problem theory. We prove that these series converge to functions analytic in cut domains. We are then able to reconstruct the jump functions across the cuts from the coefficients of the series expansions by the use of the Pollaczek polynomials. We can thus furnish a solution for a class of Cauchy integral equations., Comment: 17 pages, 2 Postscript figures

In this paper the problem of recovering a regularized solution of the Fredholm integral equations of the first kind with Hermitian and square-integrable kernels, and with data corrupted by additive noise, is considered. Instead of using a variational regularization of Tikhonov type, based on a priori global bounds, we propose a method of truncation of eigenfunction expansions that can be proved to converge asymptotically, in the sense of the $L^2$--norm, in the limit of noise vanishing. Here we extend the probabilistic counterpart of this procedure by constructing a probabilistically regularized solution without assuming any structure of order on the sequence of the Fourier coefficients of the data. This probabilistic approach allows us to use the statistical tools proper of time-series analysis, and in this way we attain a new regularizing algorithm, which is illustrated by some numerical examples. Finally, a comparison with solutions obtained by the means of the variational regularization exhibits how some intrinsic limits of the variational-based techniques can be overcome., Comment: 23 pages, 3 Postscript figures

In this paper we show how rotational bands of resonances can be described by using trajectories of poles of the scattering amplitude in the complex angular momentum plane: each band of resonances is represented by the evolution of a single pole lying in the first quadrant of the plane. The main result of the paper consists in showing that also the antiresonances (or echoes) can be described by trajectories of the scattering amplitude poles, instead of using the hard-sphere potential scattering as prescribed by the classical Breit-Wigner theory. The antiresonance poles lie in the fourth quadrant of the complex angular momentum plane, and are associated with non-local potentials which take into account the exchange forces; it derives a clear-cut separation between resonance and antiresonance poles. The evolution of these latter poles describes the passage from quantum to semi-classical physics. The theory is tested on the rotational band produced by $\alpha$-$\alpha$ elastic scattering and on the hadronic rotational bands in $\pi^+$-p elastic scattering., Comment: 26 pages, 4 figures, to be published in Phys. Rev. C

The geometrical diffraction theory, in the sense of Keller,is here reconsidered as an obstacle problem in the Riemannian geometry. The first result is the proof of the existence and the analysis of the main properties of the diffracted rays, which follow from the non-uniqueness of the Cauchy problem for geodesics in a Riemannian manifold with boundary. Then,the axial caustic is here regarded as a conjugate locus, in the sense of the Riemannian geometry, and the results of the Morse theory can be applied.The methods of the algebraic topology allow us to introduce the homotopy classes of diffracted rays. These geometrical results are related to the asymptotic approximations of a solution of a boundary value problem for the reduced wave equation. In particular, we connect the results of the Morse theory to the Maslov construction, which is used to obtain the uniformization of the asymptotic approximations. Then, the border of the diffracting body is the envelope of the diffracted rays and, instead of the standard saddle point method, use is made of the procedure of Chester, Friedman and Ursell to derive the damping factors associated with the rays which propagate along the boundary. Finally, the amplitude of the diffracted rays when the diffracting body is an opaque sphere is explicitly calculated., Comment: 26 pages, 3 figures

Sequences of rotational resonances (rotational bands) and corresponding antiresonances are observed in ion collisions. In this paper we propose a description which combines collective and single-particle features of cluster collisions. It is shown how rotational bands emerge in many-body dynamics, when the degeneracies proper of the harmonic oscillator spectra are removed by adding interactions depending on the angular momentum. These interactions can be properly introduced in connection with the exchange-forces and the antisymmetrization, and give rise to a class of non-local potentials whose spectral properties are analyzed in detail. In particular, we give a classification of the singularities of the resolvent, which are associated with bound states and resonances. The latter are then studied using an appropriate type of collective coordinates, and a hydrodynamical model of the trapping, responsible for the resonances, is then proposed. Accordingly, we derive, from the uncertainty principle, a spin-width of the unstable states which can be related to their angular lifetime., Comment: 25 pages, 2 figures

A family of orbiting resonances in molecular scattering is globally described by using a single pole moving in the complex angular momentum plane. The extrapolation of this pole at negative energies gives the location of the bound states. Then a single pole trajectory, that connects a rotational band of bound states and orbiting resonances, is obtained. These complex angular momentum singularities are derived through a geometrical theory of the orbiting. The downward crossing of the phase-shifts through pi/2, due to the repulsive region of the molecular potential, is estimated by using a simple hard-core model. Some remarks about the difference between diffracted rays and orbiting are also given., Comment: 18 pages, 3 figures, to appear in Physical Review A

Research on accelerator driven systems (ADS), related new fuels and their ability for nuclear waste incineration has led to a revival of interest in nuclear cross sections of many nuclides over a large energy range. Discrepancies observed between nuclear data bases require new measurements in several cases. A complete measurement of such cross sections including resonance resolution consists of an extensive beam time experiment associated to a long process of analysis ans validation. With a slowing down spectrometer associated to a pulsed neutron source, it is possible to determine good cross section profile in an energy range from 0.1 eV to 40 keV. These measurements performed at ISN (Grenoble) with neutron source GENEPI requires only small quantities of matter (as small as 0.1 g) and about one day of beam per target., Comment: 0 pages, 6 tables, to submit in NIM-A