De Prony Research Articles

Theory-Inspired Deep Network for Instantaneous-Frequency Extraction and Subsignals Recovery From Discrete Blind-Source Data.

In the mathematical and engineering literature on signal processing and time-series analysis, there are two opposite points of view concerning the extraction of time-varying frequencies (commonly called instantaneous frequencies, IFs). One is to consider the given signal as a composite signal consisting of a finite number of subsignals that are oscillating, and the goal is to decompose the signal into the sum of the (unknown) subsignals, followed by extracting the IF from each subsignal; the other is first to extract from the given signal, the IFs of the (unknown) subsignals, from which the subsignals that constitute the given signal are recovered. Let us call the first the ``signal decomposition approach'' and the second the ``signal resolution approach.'' For the ``signal decomposition approach,'' rigorous mathematical theories on function decomposition have been well developed in the mathematical literature, with the most relevant one, called ``atomic decomposition'' initiated by R. Coifman, with various extensions by others, notably by D. Donoho, with the goal of extracting the signal building blocks, but without concern of which building blocks constitute any of the subsignals, and consequently, the subsignals along with their IFs cannot be recovered. On the other hand, the most popular of the decomposition approach is the ``empirical mode decomposition (EMD),'' proposed by N. Huang et al., with many variations by others. In contrast to atomic decomposition, all variations of EMD are ad hoc algorithms, without any rigorous mathematical theory. Unfortunately, all existing versions of EMD fail to resolve the inverse problem on the recovery of the subsignals that constitute the given composite signal, and consequently, extracting the IFs is not satisfactory. For example, EMD fails to extract even two IFs that are not far apart from each other. In contrast to the signal decomposition approach, the ``signal resolution approach'' has a very long history dated back to the Prony method, introduced by G. de Prony in 1795, for solving the inverse problem of time-invariant linear systems. On the other hand, for nonstationary signals, the synchrosqueezed wavelet transform (SST), proposed by I. Daubechies over a decade ago, with various extensions and variations by others, was introduced to resolving the inverse problem, by first extracting the IFs from some reference frequency, followed by recovering the subsignals. Unfortunately, the SST approximate IFs could not be separated when the target IFs are close to one another at certain time instants, and even if they could be separated, the approximation is usually not sufficiently accurate. For these reasons, some signal components could not be recovered, and those that could be recovered are usually inexact. More recently, we introduced and developed a more direct method, called signal separation operation (SSO), published in 2016, to accurately compute the IFs and to accurately recover all signal components even if some of the target IFs are close to each other. The main contributions of this article are twofold. First, the SSO method is extended from uniformly sampled data to arbitrarily sampled data. This method is localized as illustrated by a number of numerical examples, including components with different subsignal arrival and departure times. It also yields a short-term prediction of the digital components along with their IFs. Second, we present a novel theory-inspired implementation of our method as a deep neural network (DNN). We have proved that a major advantage of DNN over shallow networks is that DNN can take advantage of any inherent compositional structure in the target function, while shallow networks are necessarily blind to such structure. Therefore, DNN can avoid the so-called curse of dimensionality using what we have called the blessing of compositionality. However, the compositional structure of the target function is not uniquely defined, and the constituent functions are typically not known so that the networks still need to be trained end-to-end. In contrast, the DNN introduced in this article implements a mathematical procedure so that no training is required at all, and the compositional structure is evident from the procedure. We will disclose the extension of the SSO method in Sections II and III and explain the construction of the deep network in Section IV.

Validated analysis of modulated signals: From de Prony to Padé and beyond

The spectral analysis of modulated signals has attracted quite some research, mainly because of the fact that Fourier methods are not particularly suitable. Among the challenges, we mention the separation of close components that differ significantly in magnitude, the limitation of the sampling duration, the probable ill-conditioning of certain structured matrices.We show how a validated exponential analysis add-on, for use with any standard exponential analysis method, offers a lot of advantages in the context of these challenges. The add-on uses an alias-free decimation technique and essentially combines the basics of de Prony’s method for exponential fitting with the theory of Padé approximation theory.

Kanaky New Caledonia

Kanaky New Caledonia Mathias Chauchat (bio) The year 2020 left Kanaky New Caledonia in a worrying situation, with an economic crisis linked to the pandemic, rising debt, and the country's 2022 budget not being approved and a yes/no referendum that placed two camps in an almost equal face-off. Turnout in the 2020 referendum was overwhelming at 85.69 percent, and out of 180,799 registered voters, 53.26 percent voted no to independence and 46.74 percent voted yes (High Commission of the French Republic in New Caledonia 2020). Adding to these troubles, as 2021 began, the tension around the sale of Vale's Southern nickel processing plant became increasingly violent. However, the country showed resilience. In March, an agreement was reached on the Southern plant, ending roadblocks and demonstrations across New Caledonia. Despite the covid-19-linked bereavements, the country's health-care system held up well against the disease, and good progress was made with vaccination. Further, all institutions (Congress, customary Senate, and government) were renewed to the advantage of the pro-independence parties. Nevertheless, a dark shadow hung over the picture: the French State and the nonindependentists demanded a rapid third referendum to remove uncertainties—as early as December 2021 instead of October 2022, as provided for in the Nouméa Agreement—resulting in massive nonparticipation of the independentists. The Nouméa Accord ended in a huge mess. In 2021, the multinational company Vale pushed hard to sell its 95 percent share in the Southern hydrometallurgical complex, the remaining 5 percent belonging to the Société de Participation Minière du Sud Calédonien (spmsc, South New Caledonian Mining Holding Company), a holding company run by the three New Caledonian provinces. Above all, Vale demanded a radical change in strategy: in addition to [End Page 466] the export of a less processed product (nickel hydroxide cake), it demanded access to the mineral reserves of Goro saprolites, richer in content than laterites, for export. Vale required this access to the mine to balance its finances and to find a buyer. This would be a very serious blow to the "nickel doctrine" supported by the pro-independence parties, which stipulates that the ore must be transformed rather than sold raw. Political conflict was inevitable. The situation became violent at the end of 2020 and reached its climax in early 2021. Vale drew up a report on the impacts of this violence on 1 February: more than two billion cfp francs in direct destruction onsite and at least five times that in indirect damage (us$1 = about 107.23 cfp francs). Heavy mining machinery, vehicles, tires, oil stocks, and buildings were set on fire. Forty-eight people came under judicial investigation (La Première 2020). However, on 4 March, a "political agreement" on the proposed sale of Vale NC was signed between the Southern Province and the Usine du Sud = Usine Pays (Southern Plant = Country's Plant) collective, which unites customary chiefs, environmental groups, trade unionists, and members of the independence coalition Front de Libération Nationale Kanak et Socialiste (flnks), ending six months of violent mobilization. This agreement was the first of its kind, leading to the emergence of a new mining model involving the rental rather than sale of mining titles to the operator. Vale NC sold its titles to a new structure representing New Caledonian interests, which then leased them to the new operator, Prony Resources (associated with the Swiss multinational company Trafigura). In other words, these mining titles, which are worth more than 50 billion cfp francs, belong to a local company, Sud Nickel, a subsidiary of PromoSud (which is owned by the Southern Province). Prony Resources is governed by four shareholding entities: the spmsc (30 percent), which represents the three provinces; Fidal fiducie (21 percent), which represents Prony Resources employees and the local population; Trafigura (19 percent); and Compagnie Financière de Prony (30 percent), which represents factory executives and investors (Prony Resources 2022). Together, spmsc and Fidal fiducie hold a 51 percent share, while Trafigura and Compagnie Financière de Prony (ie, the private sector) hold only a 49 percent share. Vale Canada, owner of 95 percent of Vale Nouvelle-Cal...

19th-century real analysis, forward and backward

19th century real analysis received a major impetus from Cauchy's work. Cauchy mentions variable quantities, limits, and infinitesimals, but the meaning he attached to these terms is not identical to their modern meaning. Some Cauchy historians work in a conceptual scheme dominated by an assumption of a teleological nature of the evolution of real analysis toward a preordained outcome. Thus, Gilain and Siegmund-Schultze assume that references to limite in Cauchy's work necessarily imply that Cauchy was working with an Archi-medean continuum, whereas infinitesimals were merely a convenient figure of speech, for which Cauchy had in mind a complete justification in terms of Archimedean limits. However, there is another formalisation of Cauchy's procedures exploiting his limite, more consistent with Cauchy's ubiquitous use of infinitesimals, in terms of the standard part principle of modern infinitesimal analysis. We challenge a misconception according to which Cauchy was allegedly forced to teach infinitesimals at the Ecole Polytechnique. We show that the debate there concerned mainly the issue of rigor, a separate one from infinitesimals. A critique of Cauchy's approach by his contemporary de Prony sheds light on the meaning of rigor to Cauchy and his contemporaries. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis, and indicates that he was a pioneer of infinitesimal techniques as much as a harbinger of the Epsilontik.

Cauchy's Work on Integral Geometry, Centers of Curvature, and Other Applications of Infinitesimals

Like his colleagues de Prony, Petit, and Poisson at the Ecole Polytechnique, Cauchy used infinitesimals in the Leibniz-Euler tradition both in his research and teaching. Cauchy applied infinitesimals in an 1826 work in differential geometry where infinitesimals are used neither as variable quantities nor as sequences but rather as numbers. He also applied infinitesimals in an 1832 article on integral geometry, similarly as numbers. We explore these and other applications of Cauchy's infinitesimals as used in his textbooks and research articles. An attentive reading of Cauchy's work challenges received views on Cauchy's role in the history of analysis and geometry. We demonstrate the viability of Cauchy's infinitesimal techniques in fields as diverse as geometric probability, differential geometry, elasticity, Dirac delta functions, continuity and convergence. Keywords: Cauchy--Crofton formula; center of curvature; continuity; infinitesimals; integral geometry; limite; standard part; de Prony; Poisson

A scale and shift paradigm for sparse interpolation in one and more dimensions

Sparse interpolation from at least 2 n uniformly spaced interpolation points t j can be traced back to the exponential fitting method [MATH HERE] of de Prony from the 18-th century [5]. Almost 200 years later this basic problem is also reformulated as a generalized eigenvalue problem [8]. We generalize (1) to sparse interpolation problems of the form [MATH HERE] and some multivariate formulations thereof, from corresponding regular interpolation point patterns. Concurrently we introduce the wavelet inspired paradigm of dilation and translation for the analysis (2) of these complex-valued structured univariate or multivariate samples. The new method is the result of a search on how to solve ambiguity problems in exponential analysis, such as aliasing which arises from too coarsely sampled data, or collisions which may occur when handling projected data.

Superresolution without separation

Abstract This article provides a theoretical analysis of diffraction-limited superresolution, demonstrating that arbitrarily close point sources can be resolved in ideal situations. Precisely, we assume that the incoming signal is a linear combination of $M$ shifted copies of a known waveform with unknown shifts and amplitudes, and one only observes a finite collection of evaluations of this signal. We characterize properties of the base waveform such that the exact translations and amplitudes can be recovered from $2M+1$ observations. This recovery can be achieved by solving a weighted version of basis pursuit over a continuous dictionary. Our analysis shows that $\ell_1$-based methods enjoy the same separation-free recovery guarantees as polynomial root finding techniques, such as de Prony’s method or Vetterli’s method for signals of finite rate of innovation. Our proof techniques combine classical polynomial interpolation techniques with contemporary tools from compressed sensing.

Animal Welfare: Focusing on the Future - Edited by DJ Mellor and ACD Bayvel (2014). Published by the World Organisation for Animal Health, 12 Rue de Prony, 75017 Paris, France. 360 pages Paperback (ISBN 978-92-9044-929-4). Scientific and Technical Review. Volume 33(1), April 2014. Price €65.00 www.oie.int.

Animal Welfare: Focusing on the Future - Edited by DJ Mellor and ACD Bayvel (2014). Published by the World Organisation for Animal Health, 12 Rue de Prony, 75017 Paris, France. 360 pages Paperback (ISBN 978-92-9044-929-4). Scientific and Technical Review. Volume 33(1), April 2014. Price €65.00 www.oie.int. - Volume 25 Issue 1

Fluid chemistry of the low temperature hyperalkaline hydrothermal system of Prony Bay (New Caledonia)

Abstract. The terrestrial hyperalkaline springs of Prony Bay (southern lagoon, New Caledonia) have been known since the nineteenth century, but a recent high-resolution bathymetric survey of the seafloor has revealed the existence of numerous submarine structures similar to the well-known Aiguille de Prony, which are also the location of high-pH fluid discharge into the lagoon. During the HYDROPRONY cruise (28 October to 13 November 2011), samples of waters, gases and concretions were collected by scuba divers at underwater vents. Four of these sampling sites are located in Prony Bay at depths up to 50 m. One (Bain des Japonais spring) is also in Prony Bay but uncovered at low tide and another (Rivière des Kaoris spring) is on land slightly above the seawater level at high tide. We report the chemical composition (Na, K, Ca, Mg, Cl, SO4, dissolved inorganic carbon, SiO2(aq)) of 45 water samples collected at six sites of high-pH water discharge, as well as the composition of gases. Temperatures reach 37 °C at the Bain des Japonais and 32 °C at the spring of the Kaoris. Gas bubbling was observed only at these two springs. The emitted gases contain between 12 and 30% of hydrogen in volume of dry gas, 6 to 14% of methane, and 56 to 72% of nitrogen, with trace amounts of carbon dioxide, ethane and propane. pH values and salinities of all the 45 collected water samples range from the seawater values (8.2 and 35 g L−1) to hyperalkaline freshwaters of the Ca-OH type (pH 11 and salinities as low as 0.3 g L−1) showing that the collected samples are always a mixture of a hyperalkaline fluid of meteoric origin and ambient seawater. Cl-normalized concentrations of dissolved major elements first show that the Bain des Japonais is distinct from the other sites. Water collected at this site are three component mixtures involving the high-pH fluid, the lagoon seawater and the river water from the nearby Rivière du Carénage. The chemical compositions of the hyperalkaline endmembers (at pH 11) are not significantly different from one site to the other although the sites are several kilometres away from each other and are located on different ultramafic substrata. The very low salinity of the hyperalkaline endmembers shows that seawater does not percolate through the ultramafic formation. Mixing of the hyperalkaline hydrothermal endmember with local seawater produces large ranges and very sharp gradients of pH, salinity and dissolved element concentrations. There is a major change in the composition of the water samples at a pH around 10, which delimitates the marine environment from the hyperalkaline environment. The redox potential evolves toward negative values at high pH indicative of the reducing conditions due to bubbling of the H2-rich gas. The calculation of the mineral saturation states carried out for the Na-K-Ca-Mg-Cl-SO4-DIC-SiO2-H2O system shows that this change is due to the onset of brucite formation. While the saturation state of the Ca carbonates over the whole pH range is typical of that found in a normal marine environment, Mg- and Mg-Ca carbonates (magnesite, hydromagnesite, huntite, dolomite) exhibit very large supersaturations with maximum values at a pH of around 10, very well marked for the Bain des Japonais, emphasizing the role of water mixing in mineral formation. The discharge of high-pH waters of meteoric origin into the lagoon marine environment makes the hydrothermal system of Prony Bay unique compared to other low temperature serpentinizing environments such as Oman (fully continental) or Lost City (fully marine).

スマホの中の計算工場

スマホの中の計算工場

Scientific Assessment and Management of Animal Pain: Volume 10, Technical Series Co-ordinated by DJ Mellor, PM Thornber, AC David Bayvel and S Kahn (2008). Published by the World Organisation for Animal Health (OIE), 12 rue de Prony, 75017 Paris, France. 218 pp Paperback (ISBN 978-92-9044-720-7). Price £41.00, €25.00.

Scientific Assessment and Management of Animal Pain: Volume 10, Technical Series Co-ordinated by DJ Mellor, PM Thornber, AC David Bayvel and S Kahn (2008). Published by the World Organisation for Animal Health (OIE), 12 rue de Prony, 75017 Paris, France. 218 pp Paperback (ISBN 978-92-9044-720-7). Price £41.00, €25.00. - Volume 19 Issue 1

Animal Welfare: global issues, trends and challenges ACD Bayvel, SA Rahman and A Gavinelli (2005). Published by the World Organisation for Animal Health, 12 rue de Prony, 75017 Paris, France. 300 pp Paperback (ISBN 92-9044-658-7) Scientific and Technical Review 24 (2), August 2005. Price €50. www.oie.int.

Animal Welfare: global issues, trends and challenges ACD Bayvel, SA Rahman and A Gavinelli (2005). Published by the World Organisation for Animal Health, 12 rue de Prony, 75017 Paris, France. 300 pp Paperback (ISBN 92-9044-658-7) Scientific and Technical Review 24 (2), August 2005. Price €50. www.oie.int. - Volume 16 Issue 4

Biological Disasters of Animal Origin. The Role and Preparedness of Veterinary and Public Health Services M Hugh-Jones (2006). Published by the World Organisation for Animal Health, 12 rue de Prony, 75017 Paris, France. 461 pp Paperback (ISBN 92-9044-661-7): Scientific and Technical Review 25 (1), April 2006. www.oie.int. Price €50.00

Biological Disasters of Animal Origin. The Role and Preparedness of Veterinary and Public Health Services M Hugh-Jones (2006). Published by the World Organisation for Animal Health, 12 rue de Prony, 75017 Paris, France. 461 pp Paperback (ISBN 92-9044-661-7): Scientific and Technical Review 25 (1), April 2006. www.oie.int. Price €50.00 - Volume 16 Issue 3

The History of mathematical tables: from Sumer to spreadsheets

Introduction Table and tabular formatting in Sumer, Babylonia and Assyria, 2500 BCE - 50 CE The making of logarithm tables The computation factory: de Prony's project for making tables in the 1790's Difference engines: from Muller to Comrie The 'unerring certainty of mechanical agency': machines and table making in the nineteenth century Table making in astronomy The General Registry Office and the tabulation of data, 1837 - 1939 Table making by committee British table maker 1871 - 1965 Table making for the relief of labour The making of astronomical tables in H.M. Nautical Almanac Office The rise and rise of the spreadsheet Biographical Notes

The logarithmic tables of Edward Sang and his daughters

Edward Sang (1805–1890), aided only by his daughters Flora and Jane, compiled vast logarithmic and other mathematical tables. These exceed in accuracy and extent the tables of the French Bureau du Cadastre, produced by Gaspard de Prony and a multitude of assistants during 1794–1801. Like Prony's, only a small part of Sang's tables was published: his 7-place logarithmic tables of 1871. The contents and fate of Sang's manuscript volumes, the abortive attempts to publish them, and some of Sang's methods are described. A brief biography of Sang outlines his many other contributions to science and technology in both Scotland and Turkey. Remarkably, the tables were mostly compiled in his spare time.

Machine Condition Monitoring Based on Transient Vibration Signal Analysis

Over the past several decades a significant amount of research activity has been devoted to Machine Signature Analysis (MSA). The major tool used for MSA has been spectral analysis. Classical spectral estimation techniques include Periodogram, Averaged Periodogram and BlackmanTukey spectral estimation. Newer approaches to spectral analysis include a variety of Parametric modeling techniques. Within this category are the Rational Transfer Function Modelling Method, Autoregressive (AR) Power Spectral Density (PSD) Estimation, Moving Average (MA) PSD Estimation, Autoregressive Moving Average (ARMA) PSD Estimation, Prony Spectral Density Estimation and Maximum Likelihood Method (MLM). The majority of the research and development work carried out to date with regard to signal processing strategies for machine condition monitoring and diagnostics applications has focused on signals generated from stationary processes. Non-stationary processes have been left relatively unstudied. Examples of methods that are applicable to non-stationary processes are the Wavelet and the Short Time Fast Fourier Transforms. The Prony method, originated by the French scientist Baron de Prony in 1795, is also capable of analysing non-stationary processes and is inherently suitable for the study of exponentially decaying dynamic signals, such as those that develop as a result of many different types of machinery condition deterioration, for instance, induction motor current waveforms, power transformer tank vibrations and fluorescence decay behaviour for optical temperature sensing. This paper explores the use of the Prony method to study such transient signals. The Prony method was originally used for interpolation between available data points. The method and its modified versions are techniques used for modelling data of equally spaced data samples. The Prony model is similar to the wellknown system identification techniques such as the AR and ARMA models. The Prony model seeks to fit an exponential model, which is a linear combination of a series of exponential functions, to sampled data. The Prony method first determines the linear prediction parameters that fit the sampled data. Such linear prediction parameters are then used as coefficients to form a polynomial. The roots of this polynomial are finally employed to estimate the damping coefficients, the sinusoidal frequencies, the exponential amplitude and the sinusoidal initial phase of each of the exponential terms.

The historical development of the linear minimax absolute residual estimation procedure 1786–1960

This paper is concerned with the historical development of a traditional procedure for determining appropriate values for the parameters defining a linear relationship. This traditional procedure is variously known as the minimax absolute residual, Chebyshev, or L ∞-norm procedure. Besides being of considerable interest in its own right as one of the earliest objective methods for estimating the parameters of such relationships, this procedure is also closely related to Rousseeuw's least median of squared residuals and to the least sum of absolute residuals or L 1-norm procedures. The minimax absolute residual procedure was first proposed by Laplace in 1786 and developed over the next 40 years by de Prony, Cauchy, Fourier, and Laplace himself. More recent contributions to this traditional literature include those of de la Vallée Poussin and Stiefel. Nowadays, the minimax absolute residual procedure is usually implemented as the solution of a primal or dual linear programming problem. It therefore comes as no surprise to discover that some of the more prominent features of such problems, including early variants of the simplex algorithm are to be found in these contributions. In this paper we re-examine some of the conclusions reached by Grattan-Guinness (1970), Franksen (1985) and Grattan-Guinness (1994) and suggest several amendments to their findings. In particular, we establish the nature of de Prony's geometrical fitting procedure and trace the origins of Fourier's prototype of the simplex algorithm.

Prony the Bridge-builder The Life and Times of Gaspard de Prony, Educator and Scientist

CentaurusVolume 37, Issue 3 p. 230-268 Prony the Bridge-builder The Life and Times of Gaspard de Prony, Educator and Scientist Margaret Bradley, Margaret Bradley School of International Studies and Law. Coventry University, Priory Street, Coventry CV1 5FB EnglandSearch for more papers by this author Margaret Bradley, Margaret Bradley School of International Studies and Law. Coventry University, Priory Street, Coventry CV1 5FB EnglandSearch for more papers by this author First published: July 1994 https://doi.org/10.1111/j.1600-0498.1994.tb00010.xCitations: 1 AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume37, Issue3July 1994Pages 230-268 RelatedInformation

Work for the Hairdressers: The Production of de Prony's Logarithmic and Trigonometric Tables

A summary account is given of the large set of logarithmic and trigonometric tables produced at the end of the 18th century under the direction of Gaspard Riche de Prony. Although they were completed in 1801, their size made publication a costly task; and it was never done, despite the fact that printing was started more than once and various efforts were made over the years to find finance. Eventually a reduced edition of some tables appeared in 1891. In addition to the history some details are given of the mathematical methods used in the compilation of the tables. An appendix considers the possible influence of this project upon Babbage's ideas on computers and tables.

Science News a Century Ago

Baron de Prony, 1755–1839 ON July 29, 1839, the distinguished French mathematician and engineer, Gaspard-Claire-Francois-Marie-Riche, Baron de Prony, died at Asnières, near Paris, at eighty-four years of age. For many years he had been the senior member of the Corps des Ponts et Chaussées and was famous both as a writer and a constructor. Born near Lyons on July 11, 1755, he had been trained under Perronet, “the Telford of France”, at the École des Ponts et Chaussées, and afterwards became his assistant. With Perronet, too, he visited England. During the Revolutionary era, Prony formed one of the commission entrusted with the reformation of the French weights and measures and with the great survey of France. In connexion with this he organized and trained a body of computers for forming the logarithmic tables required, the result being seventeen large volumes of tables, afterwards deposited in the Paris Observatory. In 1798 he was raised to the rank of inspector-general in his corps and soon afterwards succeeded Chezy as director of his old school.