Papers by Enrico De Micheli
Applied Mathematics and Computation, May 1, 2017
We present a new algorithm for the computation of the inverse Abel transform, a problem which eme... more We present a new algorithm for the computation of the inverse Abel transform, a problem which emerges in many areas of physics and engineering. We prove that the Legendre coefficients of a given function coincide with the Fourier coefficients of a suitable periodic function associated with its Abel transform. This allows us to compute the Legendre coefficients of the inverse Abel transform in an easy, fast and accurate way by means of a single Fast Fourier Transform. The algorithm is thus appropriate also for the inversion of Abel integrals given in terms of samples representing noisy measurements. Rigorous stability estimates are proved and the accuracy of the algorithm is illustrated also by some numerical experiments.
Journal of the Optical Society of America, May 21, 2009
The inverse problem in optics, which is closely related to the classical question of the resolvin... more 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 Mε of ε-distinguishable messages (ε 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 ε-capacity, we obtain: Mε ∼ 2 S log(1/ε) ---→ ε→0 ∞, where S is the Shannon number. Moreover, we show that the ε-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 ε-entropy and ε-capacity, which are concepts related to the evaluation of the massiveness of compact sets.
Nuclear Physics, May 1, 2004
We propose a theory of the resonance-antiresonance scattering process which differs considerably ... more 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.
The European Physical Journal A, Apr 1, 2001
Sequences of rotational resonances (rotational bands) and corresponding antiresonances are observ... more 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 manybody 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.
arXiv (Cornell University), May 27, 2019
In this paper, we present a mixed-type integral-sum representation of the cylinder functions Cµ(z... more In this paper, we present a mixed-type integral-sum representation of the cylinder functions Cµ(z), which holds for unrestricted complex values of the order µ and for any complex value of the variable z. Particular cases of these representations and some applications, which include the discussion of limiting forms and representations of related functions, are also discussed.
medRxiv (Cold Spring Harbor Laboratory), May 5, 2020
In this paper we present a new approach to deterministic modelling of COVID-19 epidemic. Our mode... more In this paper we present a new approach to deterministic modelling of COVID-19 epidemic. Our model dynamics is expressed by a single prognostic variable which satisfies an integro-differential equation. All unknown parameters are described with a single, time-dependent variable R(t). We show that our model has similarities to classic compartmental models, such as SIR, and that the variable R(t) can be interpreted as a generalized effective reproduction number. The advantages of our approach are the simplicity of having only one equation, the numerical stability due to an integral formulation and the reliability since the model is formulated in terms of the most trustable statistical data variable: the number of cumulative diagnosed positive cases of COVID-19. Once this dynamic variable is calculated, other non-dynamic variables, such as the number of heavy cases (hospital beds), the number of intensive-care cases (ICUs) and the fatalities, can be derived from it using a similarly stable, integral approach. The formulation with a single equation allows us to calculate from real data the values of the sample effective reproduction number, which can then be fitted. Extrapolated values of R(t) can be used in the model to make reliable forecasts, though under the assumption that measures for reducing infections are maintained. We have applied our model to more than 15 countries and the ongoing results are available on a web-based platform [1]. In this paper, we focus on the data for two exemplary countries, Italy and Germany, and show that the model is capable of reproducing the course of the epidemic in the past and forecasting its course for a period of four to five weeks with a reasonable numerical stability.
Communications in Mathematical Physics, 2001
By exploiting the analyticity and boundary value properties of the thermal Green functions that r... more By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recovering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalism. The key property on which we rely is the fact that the Fourier transforms of the retarded and advanced functions in the energy variable have to be the "unique Carlsonian analytic interpolations" of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete Matsubara imaginary energies, respectively in the upper and lower half-planes. Starting from the Fourier coefficients regarded as "data set", we then develop a method based on the Pollaczek polynomials for constructing explicitly their analytic interpolations.
Physical Review A, Mar 1, 2002
A family of orbiting resonances in molecular scattering is globally described by using a single p... more 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 π/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.
Applicable Analysis, 2006
In this article we describe the generation of the evanescent waves which are present in the rarer... more 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-Hänchen effect is proved.
Journal of the Optical Society of America, Nov 1, 2000
The problem of object restoration in the case of spatially incoherent illumination is considered.... more 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.
arXiv (Cornell University), Dec 4, 2020
In this paper, we prove a new integral formula for the Bessel function of the first kind J µ (z).... more In this paper, we prove a new integral formula for the Bessel function of the first kind J µ (z). This formula generalizes to any µ, z ∈ C the classical representations of Bessel and Poisson.
arXiv (Cornell University), Dec 13, 2005
The problem of evaluating the information associated with Fredholm integral equations of the firs... more The problem of evaluating the information associated with Fredholm integral equations of the first kind, when the integral operator is selfadjoint 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 ε-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.
arXiv (Cornell University), Jan 13, 2006
In this paper we study a class of Hausdorff-transformed power series whose convergence is extreme... more 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.
arXiv (Cornell University), Dec 1, 2005
In a previous paper we have presented a new method for solving a class of Cauchy integral equatio... more 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.
arXiv (Cornell University), Sep 30, 2005
In this paper the problem of recovering a regularized solution of the Fredholm integral equations... more 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.
arXiv (Cornell University), Nov 24, 2005
In this paper we consider power and trigonometric series whose coefficients are supposed to satis... more 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.
Forum Mathematicum, 2012
We study the holomorphic extension associated with power series, i.e., the analytic continuation ... more We study the holomorphic extension associated with power series, i.e., the analytic continuation from the unit disk to the cut-plane C \ [1, +∞). 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. is denoted by H 2 (S). They prove that G ∈ H 2 (S) if and only if g k = g(λ)| λ=k
Journal of Mathematical Analysis and Applications, Jun 1, 2000
In a previous paper we have presented a new method for solving a class of Cauchy integral equatio... more 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.
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Papers by Enrico De Micheli