The classical theory of elementary charges as originally proposed by Abraham and Lorentz included implicitly a stability problem, which was first discussed by Poincaré in a very artificial way. It is shown that the consideration of nonlinear terms in the field Lagrangian densities provides the necessary cohesive forces in a natural way. Some examples of particle models are considered.

Dans son beau discours sur M. Henri Poincaré, Émile Picard nous a rappelé une expression de J.-B. Dumas sur le fameux physiologiste Claude Bernard, et a employé cette expression pour Poincaré lui-même : « On pourrait dire pareillement d’Henri Poincaré qu’il ne fut pas seulement un grand mathéma­ticien, maïs la mathématique elle-même. » Et comme Poincaré, d’après l’expression que nous venons de dire, a com­pris et développé la mathématique dans toutes ses profondeurs, il est clair qu’il devait..

An axiomatic framework for describing general space-time models is presented. Space-time models to which irreducible propositional systems belong as causal logics are quantum (q) theoretically interpretable and their event spaces are Hilbert spaces. Such aq space-time is proposed via a “canonical” quantization. As a basic assumption, the time t and the radial coordinate r of aq particle satisfy the canonical commutation relation [t,r]=±i $h =$ . The two cases will be considered simultaneously. In that case the event space is the (...) Hilbert space L2(ℝ3). Unitary symmetries consist of Poincaré-like symmetries (translations, rotations, and inversion) and of gauge-like symmetries. Space inversion implies time inversion. Thisq space-time reveals a confinement phenomenon: Theq particle is “confined” in an $h =$ size region of Minkowski space $\mathbb{M}^4$ at any time. One particle mechanics overq space-time provides mass eigenvalue equations for elementary particles. Prugovečki's stochasticq mechanics andq space-time offer a natural way for introducing and interpreting consistently such aq space-time andq particles existing in it. The mass eigenstates ofq particles generate Prugovečki's extended elementary particles. When $h =$ →0, these particles shrink to point particles and $\mathbb{M}^4$ is recovered as the classical (c) limit ofq space-time. Conceptual considerations favor the case [t,r]=+i $h =$ , and applications in hadron physics give the fit $h =$ ⋍2/5 fermi/GeV. (shrink)

In the late 19th century great changes in theories of light and electricity were in direct conflict with certitude, the view that scientific knowledge is infallible. What is, then, the epistemic status of scientific theory? To resolve this issue Duhem and Poincaré proposed images of fallible knowledge, Instrumentalism and Conventionalism, respectively. Only in 1919–1922, after Einstein's relativity was published, he offered arguments to support Fallibilism, the view that certainty cannot be achieved in science. Though Einstein did not consider Duhem's Instrumentalism, (...) he argued against Poincaré's Conventionalism. Hitherto, Einstein's Fallibilism, as presented at first in a rarely known essay of 1919, was left in the dark. Recently, Howard obscured its meaning. Einstein's essay was never translated into English. In my paper I provide its translation and attempt to shed light on Einstein's view and its context; I also direct attention to Einstein's images of philosophical opportunism in scientific practice. (shrink)

We propose a Lorentz-covariant deformed algebra describing a -dimensional quantized spacetime, which in the nonrelativistic limit leads to undeformed one. The deformed Poincaré transformations leaving the algebra invariant are identified. In the classical limit the Lorentz-covariant deformed algebra yields the deformed Lorentz-covariant Poisson brackets. Kepler problem with the deformed Lorentz-covariant Poisson brackets is studied. We obtain that the precession angle of an orbit of the relativistic particle in the gravitational field depends on the mass of the particle, i.e. equivalence principle (...) is violated. We propose a condition for the recovery of the equivalence principle in the space with the deformed Poisson brackets. Comparing our analytical result with the experimental data for the precession angle of Mercury’s orbit we provide an estimation of minimal length. (shrink)

In the context of a covariant mechanics with Poincaré-invariant evolution parameter τ, Sa'ad, Horwitz, and Arshansky have argued that for the electromagnetic interaction to be well posed, the local gauge function of the field should include dependence on τ, as well as on the spacetime coordinates. This requirement of full gauge covariance leads to a theory of five τ-dependent gauge compensation fields, which differs in significant aspects from conventional electrodynamics, but whose zero modes coincide with the Maxwell theory. The pre-Maxwell (...) fields may exchange mass with charged particles, permitting pair annihilation even at the classical level. The total mass-energy-momentum tensor of the fields and particles is conserved. The Green's functions for the fields provide spacelike and timelike support for correlations, as well as lightlike propagation. A τ-integration of the fields—singling out the massless photons—recovers the standard Maxwell theory, which then has the character of an equilibrium limit of the underlying microscopic dynamics. The pre-Maxwell theory also turns out to be the solution of the inverse problem in variational mechanics: it is shown to be the most general local gauge theory consistent with unconstrained commutation relations in four dimensions. Posed in this framework, the extension to n-dimensions, curved background space, and non-abelian gauge symmetry becomes straightforward. (shrink)

Press release. -/- The ebook entitled, Einstein’s Revolution: A Study of Theory-Unification, gives students of physics and philosophy, and general readers, an epistemological insight into the genesis of Einstein’s special relativity and its further unification with other theories, that ended well by the construction of general relativity. The book was developed by Rinat Nugayev who graduated from Kazan State University relativity department and got his M.Sci at Moscow State University department of philosophy of science and Ph.D at Moscow Institute of (...) Philosophy, Russian Academy of Science. He has forty years of philosophy of science and relativistic astrophysics teaching and research experience evincing in more than 200 papers in the scientific journals of Russia, Ukraine, Belorussia, USA, Great Britain, Germany, Spain, Italy, Sweden, Switzerland, Netherlands, Canada, Denmark, Poland, Romania, France, Greece, Japan and some other countries, and 8 monographs. Revolutions in physics all embody theoretical unification. Hence the overall aim of the present book is to unfold Einstein’s unificationist modus operandi, the hallmarks of actual Einstein’s methodology of unification that engendered his 1905 special relativity, as well as his 1915 general relativity. To achieve the object, a lucid epistemic model is exposed aimed at an analysis of the reasons for mature theory change in science (chapter1). According to the model, scientific revolutions were not due to fanciful creation of new ideas ‘ex nihilo’, but rather to the long-term processes of the reconciliation, interpenetration and intertwinement of ‘old’ research traditions preceding such breaks .Accordingly, origins of scientific revolutions lie not in a clash of fundamental theories with facts, but of “old” mature research traditions with each other, leading to contradictions that can only be attenuated in a more general theoretical approach. In chapter 2 it is contended that Einstein’s ingenious approach to special relativity creation, substantially distinguishing him from Lorentz’s and Poincaré’s invaluable impacts, turns to be a milestone of maxwellian electrodynamics, statistical mechanics and thermodynamics reconciliation design. Special relativity turns out to be grounded on Einstein’s breakthrough 1905 light quantum hypothesis. Eventually the author amends the received view on the general relativity genesis by stressing that the main reason for Einstein’s victory over the rival programmes of Abraham and Nordström was a unificationist character of Einstein’s research programme (chapter 3). Rinat M. Nugayev, Ph.D, professor of Volga Region Academy, Kazan, the Republic of Tatarstan, the Russian Federation. (shrink)

A [1983] review, 'Relativity before Einstein' made no mention of the work of Joseph Larmor, whose early derivation of the Lorentz transformation seems to be less well known than those of Lorentz and Poincare. In 1897, Larmor, starting from a first-order transformation similar to Lorentz's first order version, presented the correct form of what is now known as the Lorentz transformation. In his presentation of the theory in 1900 Larmor saw the time dilation effect as a consequence of Maxwell's electromagnetic (...) theory. It was Lorentz who, in 1895, introduced the notion of the relativity of simultaneity (local time), without the time dilation effect. Poincare in 1900 discussed how Lorentz's local time would arise from the procedure of synchronizing moving clocks by exchanging light signals assumed to travel at the same speed in either direction. Lorentz presented the correct version of the transformation in 1899, and discussed the variation of mass with velocity arising from it. In 1902 Lorentz was aware of Larmor's 1897 work but apparently missed its significance. Nevertheless, the credit for the first presentation of the Lorentz transformation including the crucial time dilation belongs to Larmor. (shrink)

A cornerstone of physics, Maxwell‘s theory of electromagnetism, apparently contains a fatal flaw. The standard expressions for the electromagnetic field energy and the self-mass of an electron of finite extension do not obey Einstein‘s famous equation, \, but instead fulfill this relation with a factor 4/3 on the left-hand side. Furthermore, the energy and momentum of the electromagnetic field associated with the charge fail to transform as a four-vector. Many famous physicists have contributed to the debate of this so-called 4/3-problem (...) without arriving at a complete solution. It has generally been assumed that, as originally suggested by Poincaré, the problems are connected to the question of stability of the charge distribution, and that relativistic equivalence between energy and self-mass can only be restored by inclusion of stabilizing forces. Alternative solutions to the problems have also been proposed. Nearly a century ago Fermi suggested a covariant definition of the electromagnetic energy and momentum, and sixty years later Kalckar et al. argued that the 4/3 problem is caused by omission of a relativistic correction in the standard evaluation of the self-force from Coulomb self-interaction. However, the relation between these suggestions has not been clear. We show that the relativistic correction implies that the mechanical momentum of an accelerated rigid body must be defined as the sum of the momenta of its parts for fixed time in the momentary rest frame of the body. For the total momentum of particles and field to be conserved, the total energy–momentum tensor must be divergence free, and this then requires that the momentum of the associated electromagnetic field be defined in the same way, consistent with the suggestion by Fermi. This comprehensive solution of the 4/3-problem demonstrates that there is no conflict of Maxwell‘s theory with special relativity and the questions of equivalence of electromagnetic energy and self-mass and of stability of a classical charge distribution are independent. In appendices we discuss the relations of our treatment with Fermi‘s seminal paper and with a classic paper by Dirac where he evaluated the damping self-force on a point electron from transport of energy and momentum in the electromagnetic field. (shrink)

In the framework of off-shell quantum electrodynamics—the quantum field theory of a covariant symplectic mechanics, in which events evolve according to a Poincaré-invariant parameter τ—we study the low-energy scattering of identical scalar particles. It is shown that exchange of mass is permitted in the formalism, and we calculate scattering cross-sections for this case. In these cross-sections, the usual forward pole of the standard scalar QED splits into two poles and a zero, slightly offset from the forward direction. As mass exchange (...) vanishes, a pole-zero pair cancel, the remaining pole moves to θ = 0, and the standard cross-section is recovered. (shrink)

Do numbers and the other objects of mathematics enjoy a timeless existence independent of human minds, or are they the products of cerebral invention? Do we discover them, as Plato supposed and many others have believed since, or do we construct them? Does mathematics constitute a universal language that in principle would permit human beings to communicate with extraterrestrial civilizations elsewhere in the universe, or is it merely an earthly language that owes its accidental existence to the peculiar evolution of (...) neuronal networks in our brains? Does the physical world actually obey mathematical laws, or does it seem to conform to them simply because physicists have increasingly been able to make mathematical sense of it? Jean-Pierre Changeux, an internationally renowned neurobiologist, and Alain Connes, one of the most eminent living mathematicians, find themselves deeply divided by these questions.The problematic status of mathematical objects leads Changeux and Connes to the organization and function of the brain, the ways in which its embryonic and post-natal development influences the unfolding of mathematical reasoning and other kinds of thinking, and whether human intelligence can be simulated, modeled,--or actually reproduced-- by mechanical means. The two men go on to pose ethical questions, inquiring into the natural foundations of morality and the possibility that it may have a neural basis underlying its social manifestations. This vivid record of profound disagreement and, at the same time, sincere search for mutual understanding, follows in the tradition of Poincaré, Hadamard, and von Neumann in probing the limits of human experience and intellectual possibility. Why order should exist in the world at all, and why it should be comprehensible to human beings, is the question that lies at the heart of these remarkable dialogues. (shrink)

Barut showed us how it is possible to get a Poincaré invariant n-body equation with a single time. Starting from the Barut equation for n-free particles, we show how to generalize it when they interact through Dirac oscillators with different frequencies. We then particularize the problem to n=2 and consider the particle-antiparticle system whose frequencies are respectively ω and −ω. We indicate how the resulting equation can be solved by perturbation theory, though the spectrum and its comparison with that of (...) the mesons will be given in another publication. (shrink)

A quantum theory of the photon is developed in a natural manner. Newton-Wigner and Wightman demonstrated that the photon could not be strictly localized according to natural criteria. These investigations involved the identification of an elementary system with a uirrep of the Poincare group. We identify a particle with the localized measurement of the states satisfying the uirrep. In the case of zero mass and unit spin, the photon is identified with those components of the state that can be localized. (...) A c-number four-vector potential and Lorentz condition are derived from the relativistic wave equation. The Wightman localization is demonstrated for the three independent space components of the vector potential, and the photon is identified with these components. A position operator and probability density follow immediately from the localization. A consequence of the subjective definition of a photon is that the transformations of the vector potential are unitary, and hence the unitary scalar product can be obtained for the four-vector potential. A Hilbert space is defined for the three space components of the vector potential. A position operator and probability density are derived from the scalar product, which compare directly with those obtained from the localization and the non-relativistic theory. As the longitudinal and scalar polarizations do not contribute to the measured transition probability, they are considered virtual. Lastly, a conserved four-vector current is derived from the scalar product. The possibility of observing a strict localization of the photon in the laboratory is suggested. (shrink)

A methodological model of origin and settlement of theory-choice situations (previously tried on the theories of Einstein and Lorentz in electrodynamics) is applied to modern Theory of Gravity. The process of origin and growth of empirically-equivalent relativistic theories of gravitation is theoretically reproduced. It is argued that all of them are proposed within the two rival research programmes – (1) metric (A. Einstein et al.) and (2) nonmetric (H. Poincare et al.). Each programme aims at elimination of the cross-contradiction between (...) the special theory of relativity and Newton’s theory of gravitation. New arguments in favor of Einstein’s programme are given. Nevertheless, this does not imply the necessity to rule out all the nonmetric theories, since Einstein’s and Poincare’s programmes are alternative only as different tools of the cross-contradiction elimination. In the other respects these programmes are complementary: description, explanation and prediction of gravitational experimental data entails the usage of the languages of nonmetric theories as well as of metric ones. The part of the present investigation elucidating the necessity of nonmetric theories is an implementation of the ideas of A.Z. Petrov, the founder of Kazan University Relativity Department. Late Alexei Zinovievich had frequently punctuated that the notion of Riemann space-time continuum common for all metric theories obfuscates all the gravitational notions considerably and hampers the analogies with other physical theories at hand. Since the ambiguity is a hallmark of all the general relativism notions, approach to their definitions “should be determined not by analogies and contingent facts, but by general considerations linked the physical measurements theory… No matter how far the events lie out of the frames of classical physical explanations, all the experimental data should be described by classical notions” (Petrov, 1965,pp. 59,66). Key words: Kip S. Thorne, A.P. Lightman, Stepin, theory of gravity . (shrink)

A translation of the 1913 volume Dernières Pensées, this collection of papers contains much material of interest to the logician and the philosopher of science. In "The Logic of Infinity" Poincaré clarifies the notion of "predicative set" and discusses Zermelo's and Russell's approaches to set theory. "The Evolution of Laws" attempts to formulate the question "do laws of nature evolve?" Two papers concern space and time, two others, the electrostatic and quantum theories of matter. The collection concludes with a pair (...) of papers on the ethical import of science.—P. J. M. (shrink)

To comprehend the special relativity genesis, one should unfold Einstein’s activities in quantum theory first . His victory upon Lorentz’s approach can only be understood in the wider context of a general programme of unification of classical mechanics and classical electrodynamics, with relativity and quantum theory being merely its subprogrammes. Because of the lack of quantum facets in Lorentz’s theory, Einstein’s programme, which seems to surpass the Lorentz’s one, was widely accepted as soon as quantum theory became a recognized part (...) of physics. A new approach to special relativity genesis enables to broaden the bothering “Trinity” group of its creators to include Gilbert N. Lewis. Notwithstanding that the links necessarily existing between all the 1905 papers were obscured by Einstein himself due to the reasons discussed below, Lewis revealed from the very beginning the connections between special relativity and quasi-corpuscular theory of light, as he punctuated: “The consequences which one of us obtained from a simple assumption as to the mass of a beam of light, and the fundamental conservation of mass, energy and momentum, Einstein has derived from the principle of relativity and the electromagnetic theory” (Lewis G.N.& Tolman R.C. “The Principle of Relativity and Non-Newtonian Mechanics”, Philosophical Magazine, 1908). (shrink)

Since the discovery of incommensurability in ancient Greece, arithmeticism and geometricism constantly switched roles. After ninetieth century arithmeticism Frege eventually returned to the view that mathematics is really entirely geometry. Yet Poincaré, Brouwer, Weyl and Bernays are mathematicians opposed to the explication of the continuum purely in terms of the discrete. At the beginning of the twenty-first century ‘continuum theorists’ in France (Longo, Thom and others) believe that the continuum precedes the discrete. In addition the last 50 years witnessed the (...) revival of infinitesimals (Laugwitz and Robinson—non-standard analysis) and—based upon category theory—the rise of smooth infinitesimal analysis and differential geometry. The spatial whole-parts relation is irreducible (Russell) and correlated with the spatial order of simultaneity. The human imaginative capacities are connected to the characterization of points and lines (Euclid) and to the views of Aristotle (the irreducibility of the continuity of a line to its points), which remained in force until the ninetieth century. Although Bolzano once more launched an attempt to arithmetize continuity, it appears as if Weierstrass, Cantor and Dedekind finally succeeded in bringing this ideal to its completion. Their views are assessed by analyzing the contradiction present in Grünbaum’s attempt to explain the continuum as an aggregate of unextended elements (degenerate intervals). Alternatively a line-stretch is characterized as a one-dimensional spatial subject, given at once in its totality (as a whole) and delimited by two points—but it is neither a breadthless length nor the (shortest) distance between two points. The overall aim of this analysis is to account for the uniqueness of discreteness and continuity by highlighting their mutual interconnections exemplified in the nature of a line as a one-dimensional spatial subject, while acknowledging that points are merely spatial objects which are always dependent upon an extended spatial subject. Instead of attempting to reduce continuity to discreteness or discreteness to continuity, a third alternative is explored: accept the irreducibility of number and space and then proceed by analyzing their unbreakable coherence. The argument may be seen as exploring some implications of the view of John Bell, namely that the “continuous is an autonomous notion, not explicable in terms of the discrete.” Bell points out that initially Brouwer, in his dissertation of 1907, “regards continuity and discreteness as complementary notions, neither of which is reducible to each other.”. (shrink)

The semigroup of trajectories in Minkowski space-time and its induced representations are constructed as a generalization of the Galilei case. They describe relativistic pointlike particles and yield the free propagator as a path integral in the space of trajectories parametrized by a fifth parameter. This non physical propagator in a five-dimensional space is integrated over the fifth parameter to yield the physical propagator in Minkowski space. Thereafter, this notion is applied to a model of extended particles with internal Poincaré symmetry (...) and moving in an external Minkowski space. The geometrical structure is of Hilbert bundles and the interaction is introduced as a connection. The propagator is a path integral with respect to either the internal and external trajectories and reduces to a product of an internal and an external propagator when the interaction is ignored, just as has been found in a previous work with representations of the group rather than those of the semigroup. (shrink)

We use a new, distinctly “geometrical” interpretation of non-relativistic quantum mechanics (NRQM) to argue for the fundamentality of the 4D blockworld ontology. We argue for a geometrical interpretation whose fundamental ontology is one of spacetime relations as opposed to constructive entities whose time-dependent behavior is governed by dynamical laws. Our view rests on two formal results: Kaiser (1981 & 1990), Bohr & Ulfbeck (1995) and Anandan, (2003) showed independently that the Heisenberg commutation relations of NRQM follow from the relativity of (...) simultaneity (RoS) per the Poincaré Lie algebra. And, Bohr, Ulfbeck & Mottelson (2004a & 2004b) showed that the density matrix for a particular NRQM experimental outcome may be obtained from the spacetime symmetry group of the experimental configuration. This shows how the blockworld view is not only consistent with NRQM, not only an implication of our geometrical interpretation of NRQM, but it is necessary in a non-trivial way for explaining quantum interference and “non-locality” from the spacetime perspective. Together the formal results imply that contrary to accepted wisdom, NRQM, the measurement problem and so-called quantum non-locality do not provide reasons to abandon the 4D blockworld implication of RoS. But rather, the deep non-commutative structure of the quantum and the deep structure of spacetime as given by the Minkowski interpretation of special relativity (STR) are deeply unified in a 4D spacetime regime that lies between Galilean spacetime (G4) and Minkowski spacetime (M4). Taken together the aforementioned formal results allow us to model NRQM phenomena such as interference without the need for realism about 3N Hilbert space, establishing that the world is really 4D and that configuration space is nothing more than a calculational device. Our new geometrical interpretation of NRQM provides a geometric account of quantum entanglement and so-called non-locality free of conflict with STR and free of interpretative mystery. In section 2 we discuss the various tensions between STR and NRQM with respect to the dimensionality of the world. Section 3 is devoted to an explication of the Kaiser et al. results and their philosophical implications. Likewise, the Bohr et al. results and their implications are the subject of section 4. In section 5, we present our geometric interpretation of quantum entanglement and “non-locality.”. (shrink)

In presence of predator population, the prey population may significantly change their behavior. Fear for predator population enhances the survival probability of prey population, and it can greatly reduce the reproduction of prey population. In this study, we propose a predator-prey fishery model introducing the cost of fear into prey reproduction with Holling type-II functional response and prey-dependent harvesting and investigate the global dynamics of the proposed model. For the system without harvest, it is shown that the level of fear (...) may alter the stability of the positive equilibrium, and an expression of fear critical level is characterized. For the harvest system, the existence of the semitrivial order-1 periodic solution and positive order- q periodic solution is discussed by the construction of a Poincaré map on the phase set, and the threshold conditions are given, which can not only transform state-dependent harvesting into a cycle one but also provide a possibility to determine the harvest frequency. In addition, to ensure a certain robustness of the adopted harvest policy, the threshold condition for the stability of the order- q periodic solution is given. Meanwhile, to achieve a good economic profit, an optimization problem is formulated and the optimum harvest level is obtained. Mathematical findings have been validated in numerical simulation by MATLAB. Different effects of different harvest levels and different fear levels have been demonstrated by depicting figures in numerical simulation using MATLAB. (shrink)

In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings of matter, (...) gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics. (shrink)

We argue that, under certain plausible assumptions, de Sitter space settles into a quiescent vacuum in which there are no dynamical quantum fluctuations. Such fluctuations require either an evolving microstate, or time-dependent histories of out-of-equilibrium recording devices, which we argue are absent in stationary states. For a massive scalar field in a fixed de Sitter background, the cosmic no-hair theorem implies that the state of the patch approaches the vacuum, where there are no fluctuations. We argue that an analogous conclusion (...) holds whenever a patch of de Sitter is embedded in a larger theory with an infinite-dimensional Hilbert space, including semiclassical quantum gravity with false vacua or complementarity in theories with at least one Minkowski vacuum. This reasoning provides an escape from the Boltzmann brain problem in such theories. It also implies that vacuum states do not uptunnel to higher-energy vacua and that perturbations do not decohere while slow-roll inflation occurs, suggesting that eternal inflation is much less common than often supposed. On the other hand, if a de Sitter patch is a closed system with a finite-dimensional Hilbert space, there will be Poincaré recurrences and dynamical Boltzmann fluctuations into lower-entropy states. Our analysis does not alter the conventional understanding of the origin of density fluctuations from primordial inflation, since reheating naturally generates a high-entropy environment and leads to decoherence, nor does it affect the existence of non-dynamical vacuum fluctuations such as those that give rise to the Casimir effect. (shrink)

Here, a novel conservative chaotic oscillator is presented. Various dynamics of the oscillator are examined. Studying the dynamical properties of the oscillator reveals its unique behaviors. The oscillator is multistable with symmetric dynamics. Equilibrium points of the oscillator are investigated. Bifurcations, Lyapunov exponents, and the Poincare section of the oscillator’s dynamics are analyzed. Also, the oscillator is investigated from the viewpoint of initial conditions. The study results show that the oscillator is conservative and has no dissipation. It also has various (...) dynamics, such as equilibrium point and chaos. The stability analysis of equilibrium points shows there are both stable and unstable fixed points. (shrink)

In the present paper, we introduce new models of pendulum motions for two cases: the first model consists of a pendulum with mass M moving at the end of a string with a suspended point moving on an ellipse and the second one consists of a pendulum with mass M moving at the end of a spring with a suspended point on an ellipse. In both models, we use the Lagrangian functions for deriving the equations of motions. The derived equations (...) are reduced to a quasilinear system of the second order. We use a new mathematical technique named a large parameter method for solving both models’ systems. The analytical solutions are obtained in terms of the generalized coordinates. We use the numerical techniques represented by the fourth-order Runge–Kutta method to solve the autonomous system for both cases. The stabilities of the obtained solutions are studied using the phase diagram procedure. The obtained numerical solutions and analytical ones are compared to examine the accuracy of the mathematical and numerical techniques. The large parameter technique gives us the advantage to obtain the solutions at infinity in opposite with the famous Poincare’s method which was used by many outstanding scientists in the last two centuries. (shrink)

Let K be affine, that is, \\in {\mathbb {R}}^{n+m}: y_{1}=\cdots =y_{m}=0\}\). We compute the sharp constant of Hardy inequality related to the distance d for polyharmonic operator. Moreover, we show that there exists a constant \ such that for each \\), there holds $$\begin{aligned} \int _{{\mathbb {R}}^{n+m}}|\nabla ^{k} u|^{2}\mathrm{d}x\mathrm{d}y-c_{m,k}\int _{{\mathbb {R}}^{n+m}}\frac{u^{2}}{|y|^{2k}}\mathrm{d}x\mathrm{d}y\ge C\left ^{\frac{n+m-2k}{n+m}}, \end{aligned}$$where \, \}{n+m-2k}\), \p}{2}-n-m\) and \ is the sharp Hardy constant. These inequalities generalize the result of Maz’ya ) and Lu and the second author for polyharmonic operators). (...) In order to prove the main result, we establish some Poincaré–Sobolev inequalities on hyperbolic space which is of independent interest. (shrink)

The most promising accounts of ontic probability include the Spielraum conception of probabilities, which can be traced back to J. von Kries and H. Poincaré, and the best system account by D. Lewis. This paper aims at comparing both accounts and at combining them to obtain the best of both worlds. The extensions of both Spielraum and best system probabilities do not coincide because the former only apply to systems with a special dynamics. Conversely, Spielraum probabilities may not be part (...) of the best system, e.g. if a broad class of random devices is not often used. Spielraum probabilities have the potential to provide illuminating explanations of frequencies with which outcomes of trials on gambling devices arise. They ultimately fail to account for such frequencies though because they are compatible with frequencies that grossly differ from the values of the respective probabilities. I thus follow recent proposals by M. Strevens and M. Abrams and strengthen the definition of Spielraum probabilities by adding a further condition that restricts some actual frequencies. The resulting account remains limited in scope, but stands objections raised against it in the recent literature: it is neither circular nor based upon arbitrary choices of a measure or of physical variables. Nor does it lack any explanatory power. (shrink)

A is for Alice and astronomers arguing about acceleration -- B is for Bernard's body-exchange machine -- C is for the Catholic cannibal -- D is for Maxwell's demon -- E is for evolution (and an embarrassing problem with it) -- F is for the forms lost forever to the prisoners of the cave -- G is for Galileo's gravitational balls -- H is for Hume's shades -- I is for the identity of indiscernibles -- J is for Henri Poincaré (...) and alternative geometries -- K is for the Kritik and Kant's kind of thought experiments -- L is for Lucretius' spear -- M is for Mach's motionless chain -- N is for Newton's bucket -- O is for Olbers' paradox -- P is for Parfit's person -- Q is for the questions raised by thought experiments quotidiennes -- R is for the rule-ruled room -- S is for Salvatius' ship, sailing along its own space-time line -- T is for the time-travelling twins -- U is for the universe, and Einstein's attempts to understand it -- V is for the vexed case of the violinist -- W is for Wittgenstein's beetle -- X is for xenophanes and thinking by examples -- Y is for counterfactuals and a backwards approach to history -- Z is for Zeno and the mysteries of infinity. (shrink)

Excerpt from La Pensee Contemporaine: Les Grands Problemes En effet, M. Henri Poincare demontre que la science vit non seulement d'hypotheses, autre ment dit de suppositions plus ou moins verifiables sur les rapports phenomenaux qu'elle vise exclu sivement, les lois scientifiques n'etant que leur affirmation confirmee, mais encore de con ventions ou, plus precisement, d'artifices, de propositions admises comme vraies, bien qu'on les sache ne point meriter ce titre, uniquement pour les commodites de l'etude, voire de la pratique, et le (...) profit materiel que nous en pou vons tirer. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Nous reproduisons ici, avec l’aimable autorisation de la Hofbibliothek Aschaffenburg, la courte correspondance inédite entre Dingler et Henri Poincaré. Cette correspondance, conservée aux Archives Dingler dans la Hofbibliothek Aschaffenburg, consiste en trois documents : un brouillon en allemand de la main de Dingler, sa traduction française par une autre plume, mais avec des corrections ef...

Nous reproduisons ici, avec l’aimable autorisation de la Hofbibliothek Aschaffenburg, la courte correspondance inédite entre Dingler et Henri Poincaré. Cette correspondance, conservée aux Archives Dingler dans la Hofbibliothek Aschaffenburg, consiste en trois documents : un brouillon en allemand de la main de Dingler, sa traduction française par une autre plume, mais avec des corrections ef...