Vito Volterra Research Articles

The Volterra Integrable case. Novel analytical and numerical results

In the present paper we reconsider the integrable case of the Hamiltonian $N$-species Volterra system, as it has been introduced by Vito Volterra in 1937 and significantly enrich the results already published in the ArXiv in 2019 by two of the present authors (M. Scalia and O. Ragnisco). In fact, we present a new approach to the construction of conserved quantities and comment about the solutions of the equations of motion; we display mostly new analytical and numerical results, starting from the classical predator-prey model and arriving at the general $N$-species model

Regularization of the Solution of a Nonlinear Integral Equation of the First Kind

Integral equations, the main branch of mathematics, are widely used in physics, engineering, mechanics, control theory and other fields. Related to the application of integral equations, new fields are developing, such as economics, some sections of biology, etc. The theory of integral equations mainly developed in the late nineteenth-early twentieth century, starting with Vito Volterra (1982, 1986), Eric Ivar Fredholm (2010), David Hilbert, Erhard Schmidt, etc. scientists began to study it. Nevertheless, within the framework of mathematical concepts that existed before the first half of the twentieth century, such problems were considered incorrect because a small change in the given functions led to a greater change in the desired functions. The Volterra equation of the first kind is an integral equation that has an exact solution only in some cases. The limit of integration has been carried out in very small quantities on non-classical linear and nonlinear integral equations with variable limits and the construction of solutions in these works is based on numerical methods. Therefore, for the so-called non-classical Volterra integral equations, it is relevant to determine the conditions that ensure the uniqueness and regularization of their solutions. In this paper, the uniqueness of the solution of the non-classical nonlinear integral Volterra equation of the first kind is resolved. The aim of the study is to solve the non-classical Volterra integral equation of the first kind, that is, to determine the conditions that ensure the uniqueness of the solution of the nonlinear non-classical Volterra integral equation of the first kind. The proposed methods can be used for the study of integral, integral-differential equations such as the Volterra integral equation of the first kind, as well as for the qualitative study of some applied processes in physics, ecology, medicine, geophysics, and the theory of control of complex systems.

Application of the Bilateral Hybrid Methods to Solving Initial -Value Problems for the Volterra Integro-Differential Equations

The many problems of natural sciences are reduced to solving integro-differential equations with variable boundaries. It is known that Vito Volterra, for the study of the memory of Earth, has constructed the integro-differential equations. As is known, there is a class of analytical and numerical methods for solving the Volterra integro-differential equation. Among them, the numerical methods are the most popular. For solving this equation Volterra himself used the quadrature methods. How known in solving the initial-value problem for the Volterra integro-differential equations, increases the volume of calculations, when moving from one point to another, which is the main disadvantage of the quadrature methods. Here the method is exempt from the specified shortcomings and has found the maximum value for the order of accuracy and also the necessary conditions imposed on the coefficients of the constructed methods. The results received here are the development of Dahlquist’s results. Using Dahlquist’s theory in solving initial-value problem for the Volterra integro-differential equation engaged the known scientists as P.Linz, J.R.Sobka, A.Feldstein, A.A.Makroglou, V.R.Ibrahimov, M.N.Imanova, O.S.Budnikova, M.V.Bulatova, I.G.Buova and ets. The scientists taking into account the direct connection between the initial value problem for both ODEs and the Volterra integrodifferential equations, the scientists tried to modify methods, that are used in solving ODEs and applied them to solve Integro-differential equations. Here, proved that some modifications of the methods, which are usually applied to solve initial-value problems for ODEs, can be adapted for solving the Volterra integro-differential equations. Here, for this aim, it is suggested to use a multistep method with the new properties. In this case, a question arises, how one can determine the validity of calculated values. For this purpose, it is proposed here to use bilateral methods. As is known for the calculation of the validity values of the solution of investigated problems, usually have used the predictor-corrector method or to use some bounders for the step-size. And to define the value of the boundaries, one can use the stability region using numerical methods. As was noted above, for this aim proposed to use bilateral methods. For the illustration advantage of bilateral methods is the use of very simple methods, which are called Euler’s explicit and implicit methods. In the construction of the bilateral methods it often becomes necessary to define the sign for some coefficients. By taking this into account, here have defined the sign for some coefficients.

Vito Volterra by Angelo Guerraggio and Giovanni Paoloni Reviewed by Stefano Gattei; Franco Muzzio, 2008, 243 pp. €18. English translation by Kim Williams, Springer, 2012, 187 pp, €31.19

Vito Volterra by Angelo Guerraggio and Giovanni Paoloni Reviewed by Stefano Gattei; Franco Muzzio, 2008, 243 pp. €18. English translation by Kim Williams, Springer, 2012, 187 pp, €31.19

Series representations of the Volterra function and the Fransén–Robinson constant

The Volterra function μ(t,β,α) was introduced by Vito Volterra in 1916 as the solution to certain integral equations with a logarithmic kernel. Despite the large number of applications of the Volterra function, the only known analytic representations of this function are given in terms of integrals. In this paper we derive several convergent expansion of μ(t,β,α) in terms of incomplete gamma functions. These expansions may be used to implement numerical evaluation techniques for this function. As a particular application, we derive a numerical series representation of the Fransén–Robinson constant F≔μ(1,1,0)=∫0∞1Γ(x)dx. Some numerical examples illustrate the accuracy of the approximations.

GauseR: Simple methods for fitting Lotka-Volterra models describing Gause's "Struggle for Existence".

Point 1: The ecological models of Alfred J. Lotka and Vito Volterra have had an enormous impact on ecology over the past century. Some of the earliest—and clearest—experimental tests of these models were famously conducted by Georgy Gause in the 1930s. Although well known, the data from these experiments are not widely available and are often difficult to analyze using standard statistical and computational tools.Point 2: Here, we introduce the gauseR package, a collection of tools for fitting Lotka‐Volterra models to time series data of one or more species. The package includes several methods for parameter estimation and optimization, and includes 42 datasets from Gause's species interaction experiments and related work. Additionally, we include with this paper a short blog post discussing the historical importance of these data and models, and an R vignette with a walk‐through introducing the package methods. The package is available for download at github.com/adamtclark/gauseR.Point 3: To demonstrate the package, we apply it to several classic experimental studies from Gause, as well as two other well‐known datasets on multi‐trophic dynamics on Isle Royale, and in spatially structured mite populations. In almost all cases, models fit observations closely and fitted parameter values make ecological sense.Point 4: Taken together, we hope that the methods, data, and analyses that we present here provide a simple and user‐friendly way to interact with complex ecological data. We are optimistic that these methods will be especially useful to students and educators who are studying ecological dynamics, as well as researchers who would like a fast tool for basic analyses.

Masters and students in Italian Physics between the 19th and 20th centuries: the Felici-Bartoli-Stracciati-Corbino case

In the second half of the 19th century, a special practice of research and training in physics took shape in Pisa, characterized by a particular attention to theoretical studies and to combining experimental activity with a profound mastery of mathematical tools. This peculiar approach, started by Carlo Matteucci and Ottaviano Mossotti, continued and spread by Riccardo Felici, Enrico Betti, Adolfo Bartoli and Vito Volterra, was quite an exception in the framework generally marked by strict experimentalism and positivist empiricism of the Italian physics cabinets of the time. The present paper highlights a special path connecting this tradition of the Pisan school to the scientific environment that was formed in the early years of the 20th century at the Royal Physical Institute in Via Panisperna in Rome, through the interaction of Orso Mario Corbino with Volterra on one side, and the imprinting left on Corbino by Adolfo Bartoli and his student and collaborator Enrico Stracciati.

A remark on a dynamical stochastic model in G. V. Schiaparelli's evolution theory.

It is well known that Giovanni Virginio Schiaparelli (1834-1910) wrote in 1898 an essay, Forme organiche naturali e forme geometriche pure. Studio comparativo, which was also appreciated by Vito Volterra (1860-1940). Recent studies have revised Schiaparelli's evolutionary ideas with new proposals and from this it is possible to obtain a dynamical (deterministic) model. In this paper I propose a related stochastic model (as toy model): thanks to this tool, we can describe a speciation phenomenology.

Economic and Mathematical Model for Size and Structure Optimisation of Predator and Prey Populations

The paper proposes an original economic and mathematical model for size and structure optimisation of Predator and Prey populations. The most well-known mathematical model in biology for periodical dynamics of antagonistic animal species was developed independently by Alfred Lotka and Vito Volterra. This classical mathematical Predator-Prey model is known as the Lotka-Volterra model.

MATHEMATICS IS THE LANTERN: VITO VOLTERRA, LÉON WALRAS, AND IRVING FISHER ON THE MATHEMATIZATION OF ECONOMICS

The interdisciplinary project to unite the field of mathematics with the social and biological sciences marks the work of Vito Volterra, one of Italy’s most prominent mathematicians of the twentieth century. This paper explores the connections between Volterra’s 1901 inaugural address at the Accademia dei Lincei in Rome and the work of two of his contemporaries, Léon Walras and Irving Fisher. All three thinkers were ardent advocates of the mathematical turn in economic thinking. This paper argues, however, that it is the previously unexplored relationship between Volterra and Fisher that sheds the most light on the way in which mechanical physics contributed to the project of mathematization within economics more generally. Furthermore, it explores the way in which mathematical inquiry postulated a new and coherent abstraction of the economy, at the same time that it gave epistemological authority to the economist.

The paradox of Vito Volterra’s predator-prey model

This article is dedicated to the late Giorgio Israel. R{\'e}sum{\'e}. The aim of this article is to propose on the one hand a brief history of modeling starting from the works of Fibonacci, Robert Malthus, Pierre Francis Verhulst and then Vito Volterra and, on the other hand, to present the main hypotheses of the very famous but very little known predator-prey model elaborated in the 1920s by Volterra in order to solve a problem posed by his son-in-law, Umberto D'Ancona. It is thus shown that, contrary to a widely-held notion, Volterra's model is realistic and his seminal work laid the groundwork for modern population dynamics and mathematical ecology, including seasonality, migration, pollution and more. 1. A short history of modeling 1.1. The Malthusian model. If the rst scientic view of population growth seems to be that of Leonardo Fibonacci [2], also called Leonardo of Pisa, whose famous sequence of numbers was presented in his Liber abaci (1202) as a solution to a population growth problem, the modern foundations of population dynamics clearly date from Thomas Robert Malthus [20]. Considering an ideal population consisting of a single homogeneous animal species, that is, neglecting the variations in age, size and any periodicity for birth or mortality, and which lives alone in an invariable environment or coexists with other species without any direct or indirect inuence, he founded in 1798, with his celebrated claim Population, when unchecked, increases in a geometrical ratio, the paradigm of exponential growth. This consists in assuming that the increase of the number N (t) of individuals of this population, during a short interval of time, is proportional to N (t). This translates to the following dierential equation : (1) dN (t) dt = $\epsilon$N (t) where $\epsilon$ is a constant factor of proportionality that represents the growth coe-cient or growth rate. By integrating (1) we obtain the law of exponential growth or law of Malthusian growth (see Fig. 1). This law, which does not take into account the limits imposed by the environment on growth and which is in disagreement with the actual facts, had a profound inuence on Charles Darwin's work on natural selection. Indeed, Darwin [1] founded the idea of survival of the ttest on the 1. According to Frontier and Pichod-Viale [3] the correct terminology should be population kinetics, since the interaction between species cannot be represented by forces. 2. A population is dened as the set of individuals of the same species living on the same territory and able to reproduce among themselves.

Y for y = f(x)

This paper surveys the history of the notion of function, back to its original identification with curves and (only) elementary functions. Making the definitions of the derivative and of limit rigorous leads to making calculus for functions of one and n variables rigorous as well, and to the subsequent extension to the functional calculus due to Vito Volterra.

The Oceanographic Achievements of Vito Volterra in Italy and Abroad

The Oceanographic Achievements of Vito Volterra in Italy and Abroad

Characterizing nonlinear systems with memory while combating and reducing the curse of dimensionality using new volterra expansion technique

A generalized model for characterizing nonlinear systems was originally proposed by Italian mathematician and physicist Vito Volterra (1860-1940). A further development by American mathematician and MIT Professor Norbert Wiener (1894-1964) was published in 1958. After direct involvement with Norbert Wiener publication, Albert H. Nuttall has recently made new inroads along with his coauthors in applying the Wiener-Volterra model. A general description of a nonlinear system to the third order is termed the Nuttall-Wiener-Volterra model (NWV) after its co-founders. In this formulation, two measurement waveforms on the system are required in order to characterize a specified nonlinear system under consideration: an excitation input, x(t) (the transmitted signal) and a response output, z(t) (the received signal). Given these two measurement waveforms for a given system, a kernel response, h= [h0,h1,h2,h3] between the two measurement points, is computed via a least squares approach that optimizes modeled kernel values by performing a best fit between measured response z(t) and a modeled response y(t). New procedures developed by A. Nuttall are invoked to significantly diminish the exponential growth of the computed number of kernel coefficients with respect to third order and higher orders to combat and reasonably reduce the curse of dimensionality.

Characterizing nonlinear systems with memory while combating and reducing the curse of dimensionality using new Volterra Expansion Technique

A generalized model for characterizing nonlinear systems was originally proposed by Italian mathematician and physicist Vito Volterra (1860-1940). A further development by American mathematician and MIT Professor Norbert Wiener (1894-1964) was published in 1958. After direct involvement with Norbert Wiener publication, Albert H. Nuttall has recently made new inroads along with his coauthors in applying the Wiener-Volterra model. A general description of a nonlinear system to the third order is termed the Nuttall-Wiener-Volterra model (NWV) after its co-founders. In this formulation, two measurement waveforms on the system are required in order to characterize a specified nonlinear system under consideration: an excitation input, x(t) (the transmitted signal) and a response output, z(t) (the received signal). Given these two measurement waveforms for a given system, a kernel response, h= [h0,h1,h2,h3] between the two measurement points, is computed via a least squares approach that optimizes modeled kernel...

Allee effects and resilience in stochastic populations

Allee effects, or positive functional relationships between a population’s density (or size) and its per unit abundance growth rate, are now considered to be a widespread if not common influence on the growth of ecological populations. Here we analyze how stochasticity and Allee effects combine to impact population persistence. We compare the deterministic and stochastic properties of four models: a logistic model (without Allee effects), and three versions of the original model of Allee effects proposed by Vito Volterra representing a weak Allee effect, a strong Allee effect, and a strong Allee effect with immigration. We employ the diffusion process approach for modeling single-species populations, and we focus on the properties of stationary distributions and of the mean first passage times. We show that stochasticity amplifies the risks arising from Allee effects, mainly by prolonging the amount of time a population spends at low abundance levels. Even weak Allee effects become consequential when the ubiquitous stochastic forces affecting natural populations are accounted for in population models. Although current concepts of ecological resilience are bound up in the properties of deterministic basins of attraction, a complete understanding of alternative stable states in ecological systems must include stochasticity.

“Une jeune fille fort sérieuse”: Suzanne Lambin (1902-2008), A Pioneer in Biomedicine and Biomathematics

In 1951 Suzanne Lambin (1902-2008) became the first woman to occupy a chair (microbiology) at the Faculty of Pharmacy of Paris University. Her scientific career offers an example of the rise to visibility of a young woman scientist from an educated, middle class family thanks to the opening up of public education towards women in France during the Third Republic. She had the opportunity to collaborate with four outstanding figures in the golden age of theoretical ecology and to participate in this intellectual enterprise: as a student at Nantes, she worked with Stéphane Leduc (1853-1939), and in the ‘Thirties, in Paris, together with her mentor, Jean Régnier (1892-1946), she collaborated with two mathematicians, the Italian Vito Volterra (1860-1940) and the Russian Vladimir Kostizin (1883-1963). Thus, Lambin’s scientific career was also helped by the encounters with men who were open-minded both regarding the collaboration with women in research, and regarding the exploration of new paths in science. This essay is based on unpublished sources from Lambin’s personal archive, and on an interview with her in her Paris home. Régnier’s and Lambin’s technical ability and theoretical, dynamical approach to the evolution of microbiological populations made it possible to set up a series of experiments in the Régnier microbiological laboratory at the Ambroise Paré Hospital to provide biological feedback for Volterra’s equations of population dynamics. Their research was a seminal contribution to the evolution of biomathematics, and the underlying scientific project can be better grasped thanks to the unpublished introduction to a projected book on the numerical approach in microbiology, or - in the authors’ words - the “cyphered knowledge of the rhythms of life”.

Phonotelemetry: sound-ranging techniques in World War I

During the World War I, new devices and new methods for waging war were developed and applied. Among these, the Italian phonotelemetry, sound-ranging techniques for the determination of the position of enemy artillery by the sound emitted by its fire. This paper considers how research in sound detection, which involved mathematicians and their knowledge, became part of the Office of Inventions directed by Vito Volterra. The aim is to situate the scientific instrument in its working context and to show, in particular, how a preconceived concept of precision makes no sense when considered according to the logic of military use. In the case of the Italian device, which relied on human rather than automatic listeners, I will show in particular how the instrument, though rudimentary when in comparison to those of the Allies, especially the French, was actually more efficient in the context in which it was used.

Vito Volterra at the front: from the Office of Inventions to the Italian National Research Council

Vito Volterra fervently served his country, first in a military capacity during World War I, founding the Office of Inventions in 1917, and then in an institutional capacity, founding the Italian National Research Council in 1923. The key to understanding these two entities lies first in the mobilisation of scientific forces during the Great war, and then in the context of post-war international scientific collaboration, as well as the institutions being set up in other countries to organise scientific research and experimentation.

Vito Volterra's construction of a nonconstant function with a bounded, non-Riemann integrable derivative

In the 1880s the research on the theory of integration was focused mainly on the properties of infinite sets. The development of nowhere dense sets with positive outer content, known nowadays as nowhere dense sets with positive measure, allowed the construction of general functions with the purpose of extending Riemann's definition of integral. Vito Volterra provided, in 1881, an example of a differentiable function whose derivative is bounded but not Riemann integrable. In this article we present and discuss Volterra's example.