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Hector Giacomini

Publications and source records attributed to Hector Giacomini.

At least 19 recordsLinked to original sources
arXiv2026

Conrad Habicht 1914 Manuscript on Special Relativity and Einstein 1907 Reframing of the 1905 Theory

This note examines an apparently unpublished manuscript on special relativity written by Conrad Habicht in 1914 and made available online by the ETH-Bibliothek Z\"urich in December 2024. To the best of my knowledge, no study of its content has yet been published. Habicht was one of Einstein's closest companions during the Bern years. Between February 1902 and mid-1904 he shared with Einstein many occasions for discussion and companionship in Bern. After leaving the city, he remained in close contact with Einstein through visits, reciprocal stays, and a substantial correspondence extending from the years immediately following 1905 to the eve of the First World War. The manuscript offers a clear and pedagogical presentation of special relativity. Its historical interest lies in the structure of the exposition and in the memory of the theory that the text preserves. Habicht does not present special relativity as an isolated creation beginning from Einstein's 1905 paper alone. He devotes considerable space to the pre-Einsteinian problem situation: the classical principle of relativity, the ether, Fizeau's experiment, Michelson--Morley, Lorentz's theory, the contraction hypothesis, local time, and the privileged system of the stationary ether. Lorentz is treated as the central figure who brought the electrodynamics of moving bodies to its most acute form before Einstein's intervention. This note provides a qualitative description of the manuscript, with particular attention to its structure, its treatment of the relation between classical mechanics and electrodynamics, and the respective roles assigned to Lorentz, Michelson--Morley, Einstein, and Minkowski.

physics.hist-ph↗
arXiv2026

Henri Poincare Saint Louis Lecture of 1904: Publication, Dissemination, and Historiographical Implications

Henri Poincare Saint Louis lecture of 1904 occupies an important place in the prehistory of relativity. In it, Poincare formulated the principle of relativity in general terms and presented it as one of the guiding principles of mathematical physics, together with the principles of least action and energy conservation. This article reconstructs the early publication and international dissemination of the lecture before the end of June 1905, through La Revue des idees, the Bulletin des sciences math\'ematiques, The Monist, and La valeur de la science. Drawing on library records, accession data, booksellers' advertisements, press notices, and correspondence, it shows that Poincare text circulated rapidly through scholarly, commercial, and institutional channels in Europe and North America. The significance of this circulation is not merely bibliographical. It bears directly on the documentary landscape within which Einstein 1905 relativity paper should be historically situated. The early availability of Poincare lecture and of La valeur de la science shifts attention toward the concrete conditions under which texts, concepts, and problems circulated in the months and weeks preceding Einstein's June 1905 paper. Evidence from Einstein Bern milieu further supports this view. Joseph Sauter later testimony and the possibility of redating the Habicht letter to 1 June 1905, on the basis of several converging chronological and documentary arguments, suggest that the intellectual environment of 1905 was denser than simplified narratives of solitary discovery imply. Availability is not influence; but without reconstructing availability, the historical problem of Einstein relation to Poincar\'e and Lorentz remains ill posed.

physics.hist-ph↗
arXiv2025

Lorentz, Poincare, Einstein, and the Genesis of the Theory of Special Relativity

.This article reexamines the genesis of special relativity by situating the contributions of Lorentz, Poincare, and Einstein within the scientific, documentary, and editorial context of the years 1895--1913. It emphasizes the rapid circulation of Lorentz 1904 work, in particular through German-speaking channels such as the Beiblatter zu den Annalen der Physik, and reassesses the significance of Richard Gans 1905 review as a concise access point to Lorentz results. The article also discusses Poincar\'e role in formulating the principle of relativity, interpreting local time, establishing the group property of the Lorentz transformations, and developing an invariant formulation of electrodynamics. Against this background, Einstein 1905 paper appears not as an isolated creation, but as a powerful reformulation of problems already posed by Maxwellian electrodynamics and by the failure to detect motion through the ether. The article finally examines the subsequent construction of the Lorentz--Einstein--Minkowski canon and the relative exclusion of Poincare from that narrative. Its central claim is that special relativity should be understood as the crystallization of a broader electrodynamic transformation of physics, rather than as a sudden break detached from its immediate scientific context.

physics.hist-ph↗
arXiv2025

The Rapid Arrival of Josiah Willard Gibbs's Elementary Principles in Statistical Mechanics in European University Libraries

This note offers an overview of how Josiah Willard Gibbs's Elementary Principles in Statistical Mechanics, published simultaneously in London and New York in 1902, spread through European university libraries. Contrary to the received idea that the circulation of this text was slow, information gathered through direct contacts with numerous academic libraries, together with an examination of Yale University's archives, reveals an unexpectedly rapid material diffusion beginning on 15 March 1902. This early propagation can be explained by several channels: presentation copies sent by Yale University to leading universities, personal mailings by Gibbs himself to prominent scientists, and the distribution of copies by the American publisher to major scientific journals.

physics.hist-ph↗
arXiv2022

Number of limit cycles for planar systems with invariant algebraic curves

For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Lienard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.

math.CA↗
arXiv2021

Effectiveness of the Bendixson-Dulac theorem

We illustrate with several new applications the power and elegance of the Bendixson Dulac theorem to obtain upper bounds of the number of limit cycles for several families of planar vector fields. In some cases we propose to use a function related with the curvature of the orbits of the vector field as a Dulac function. We get some general results for Lienard type equations and for rigid planar systems. We also present a remarkable phenomenon: for each integer m greater than one, we provide a simple one parametric differential system for which we prove that it has limit cycles only for the values of the parameter in a subset of an interval that decreases exponentially when m grows. One of the strengths of the results presented in this work is that although they are obtained with simple calculations, that can be easily checked by hand, they improve and extend previous studies. Another one is that, for certain systems, it is possible to reduce the question of the number of limit cycles to the study of the shape of a planar curve and the sign of an associated function in one or two variables.

math.CA↗
arXiv2020

Solving polynomials with ordinary differential equations

In this work we consider a given root of a family of n-degree polynomials as a one-variable function that depends only on the independent term. Then we prove that this function satisfies several ordinary differential equations (ODE). More concretely, it satisfies several simple separated variables ODE, a first order generalized Abel ODE of degree n-1 and an (n-1)-th order linear ODE. Although some of our results are not new, our approach is simple and self-contained. For n=2, 3 and 4 we recover, from these ODE, the classical formulas for solving these polynomials.

math.CA↗
arXiv2017

When is the growth index constant?

The growth index $γ$ is an interesting tool to assess the phenomenology of dark energy (DE) models, in particular of those beyond general relativity (GR). We investigate the possibility for DE models to allow for a constant $γ$ during the entire matter and DE dominated stages. It is shown that if DE is described by quintessence (a scalar field minimally coupled to gravity), this behaviour of $γ$ is excluded either because it would require a transition to a phantom behaviour at some finite moment of time, or, in the case of tracking DE at the matter dominated stage, because the relative matter density $Ω_m$ appears to be too small. An infinite number of solutions, with $Ω_m$ and $γ$ both constant, are found with $w_{DE}=0$ corresponding to Einstein-de Sitter universes. For all modified gravity DE models satisfying $G_{\rm eff}\ge G$, among them the $f(R)$ DE models suggested in the literature, the condition to have a constant $w_{DE}$ is strongly violated at the present epoch. In contrast, DE tracking dust-like matter deep in the matter era, but with $Ω_m <1$, requires $G_{\rm eff} > G$ and an example is given using scalar-tensor gravity for a range of admissible values of $γ$. For constant $w_{DE}$ inside GR, departure from a quasi-constant value is limited until today. Even a large variation of $w_{DE}$ may not result in a clear signature in the change of $γ$. The change however is substantial in the future and the asymptotic value of $γ$ is found while its slope with respect to $Ω_m$ (and with respect to $z$) diverges and tends to $-\infty$.

astro-ph.CO↗
arXiv2013

Some Applications of the Extended Bendixson-Dulac Theorem

During the last years the authors have studied the number of limit cycles of several families of planar vector fields. The common tool has been the use of an extended version of the celebrated Bendixson-Dulac Theorem. The aim of this work is to present an unified approach of some of these results, together with their corresponding proofs. We also provide several applications.

math.CA↗
arXiv2013

Explicit Traveling Waves and Invariant Algebraic Curves

In this paper we introduce a precise definition of algebraic traveling wave solution for general n-th order partial differential equations. All examples of explicit traveling waves known by the authors fall in this category. Our main result proves that algebraic traveling waves exist if and only if an associated n- dimensional first order ordinary differential system has some invariant algebraic curve. As a paradigmatic application we prove that, for the celebrated Fisher- Kolmogorov equation, the only algebraic traveling waves solutions are the ones found in 1979 by Ablowitz and Zeppetella. To the best of our knowledge, this is the first time that this type of results have been obtained.

math.AP↗
arXiv2013

Bifurcation diagram and stability for a one-parameter family of planar vector fields

We consider the 1-parameter family of planar quintic systems, $\dot x= y^3-x^3$, $\dot y= -x+my^5$, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter $m$ is in $(0.36,0.6)$. In this paper, using the Bendixon-Dulac theorem, we give a new unified proof of all the previous results, we shrink this to $(0.547,0.6)$, and we prove the hyperbolicity of the limit cycle. We also consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally we answer an open question about the change of stability of the origin for an extension of the above systems.

math.DS↗
arXiv2013

A proof of Perko's conjectures for the Bogdanov-Takens system

The Bogdanov-Takens system has at most one limit cycle and, in the parameter space, it exists between a Hopf and a saddle-loop bifurcation curves. The aim of this paper is to prove the Perko's conjectures about some analytic properties of the saddle-loop bifurcation curve. Moreover, we provide sharp piecewise algebraic upper and lower bounds for this curve.

math.DS↗
arXiv2011

Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations

It is well-known that the existence of traveling wave solutions for reaction-diffusion partial differential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar differential equations. We introduce a method for finding explicit upper and lower bounds of these heteroclinic orbits. In particular, for the classical Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to locate these solutions analytically and with very high accuracy.

math.DS↗
arXiv2010

A new dynamical approach of Emden-Fowler equations and systems

We give a new approach on general systems of the form \[(G){[c]{c}% -Δ_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=ε_{1}|x| ^{a}u^{s}v^δ, -Δ_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla u)=ε_{2}|x|^{b}u^μv^{m},\] where $Q,p,q,δ,μ,s,m,$ $a,b$ are real parameters, $Q,p,q\neq1,$ and $ε_{1}=\pm1,$ $ε_{2}=\pm1.$ In the radial case we reduce the problem to a quadratic system of order 4, of Kolmogorov type. Then we obtain new local and global existence or nonexistence results. In the case $ε_{1}=ε_{2}=1,$ we also describe the behaviour of the ground states in two cases where the system is variational. We give an important result on existence of ground states for a nonvariational system with $p=q=2$ and $s=m>0.$ In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$ and $δ=m+1$ and $μ=s+1.$

math.AP↗
arXiv2009

Some results on homoclinic and heteroclinic connections in planar systems

Consider a family of planar systems depending on two parameters $(n,b)$ and having at most one limit cycle. Assume that the limit cycle disappears at some homoclinic (or heteroclinic) connection when $Φ(n,b)=0.$ We present a method that allows to obtain a sequence of explicit algebraic lower and upper bounds for the bifurcation set ${Φ(n,b)=0}.$ The method is applied to two quadratic families, one of them is the well-known Bogdanov-Takens system. One of the results that we obtain for this system is the bifurcation curve for small values of $n$, given by $b=\frac5 7 n^{1/2}+{72/2401}n- {30024/45294865}n^{3/2}- {2352961656/11108339166925} n^2+O(n^{5/2})$. We obtain the new three terms from purely algebraic calculations, without evaluating Melnikov functions.

math.DS↗
arXiv2009

Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, θ)$, to an equation over a cylinder in which the singular point $p_0$ corresponds to a limit cycle $γ_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $γ_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $γ_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.

math.DS↗
arXiv2008

Upper bounds for the number of limit cycles of some planar polynomial differential systems

We give an effective method for controlling the maximum number of limit cycles of some planar polynomial systems. It is based on a suitable choice of a Dulac function and the application of the well-known Bendixson-Dulac Criterion for multiple connected regions. The key point is a new approach to control the sign of the functions involved in the criterion. The method is applied to several examples.

math.DS↗