Adele Naddeo

INFN - Istituto Nazionale di Fisica Nucleare | INFN · Naples

The onset and the development of the concept of exchange force in quantum physics are historically reconstructed, starting from Heisenberg’s seminal contributions in 1926 and going through the great developments in nuclear physics, which allowed the emergence of the idea of force mediating virtual quanta. Although most of such work was performed in...

A teaching-learning module, aimed at introducing basic concepts of general relativity at the high school level, is proposed. Emphasis is on conceptual rather than technical aspects, and only familiarity with simple calculus is required on the mathematical side. The starting point is a critical overview of the principles of Newtonian mechanics, in p...

Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck's version of the correspondence principle, classical mechanics...

We demonstrate the construction and utilization of an affordable apparatus using readily available materials to accurately measure in a quantitative manner the wavelengths reflected by a compact disc (CD) under skimming light rays. In fact, only a limited number of wavelengths can be revealed when light rays from a white lamp are directed at a CD (...

Although the common lore unofficially (and erroneously) credits Feynman with the famous quote “shut up and calculate”, he did not avoid foundational questions in quantum mechanics. The first records of his considerations on such issues are in the report of the 1957 Chapel Hill conference on “The Role of Gravitation in Physics”, where foundational q...

Some of Stanley Deser's multifaceted contributions to quantum gravity are here analysed, starting from his contribution to the famous 1957 Chapel Hill conference on “The Role of Gravitation in Physics”. The main idea of his talk, as also summarized in Peter G. Bergmann's introduction, was the possibility offered by the development of quantum gravit...

During typical general relativity courses, the so-called frame dragging effect is explained by emphasizing the presence of a gravitational Coriolis-like force term. The key difference is that, unlike the usual Coriolis force, this is not a fictitious force but a real gravitational force caused by the rotating body. In general, textbooks do not disc...

It is shown that the non-unitary Newtonian gravity (NNG) model admits a simple interpretation in terms of the Feynman path integral, in which the sum over all possible histories is replaced by a summation over pairs of paths. Correlations between different paths are allowed by a fundamental decoherence mechanism of gravitational origin and can be i...

It is shown that Non-unitary Newtonian Gravity (NNG) model admits a simple interpretation in terms of Feynman path integral, in which the sum over all possible histories is replaced by a summation over pairs of paths. Correlations between different paths are allowed by a fundamental decoherence mechanism of gravitational origin and can be interpret...

Many topics in modern physics are currently included in the curricula of the last year of high school in many countries, as for example in Italy. A consistent part of the curriculum should be devoted to the special theory of relativity. A particularly interesting phenomenon in this framework is the so-called Terrell–Penrose effect (TPE), which the...

We discuss part of Richard Feynman’s work in classical and quantum gravity, first presented at the Chapel Hill Conference in 1957. Being concerned with the relation of gravitation with the rest of physics, Feynman embraced a field-theoretical and non-geometric approach to general relativity where, after the recognition that the gravitational intera...

An interesting issue in history and philosophy of science concerns possible relations between scientific theories, in particular reductions of theories. According to Thomas Nickles, a possible notion of reduction is the physicist’s one, according to which a theory is reduced to another theory by a mathematical limiting process on a parameter. When...

The response of a detector carried out by an observer in a circular motion at a constant angular velocity to an incident plane wave is considered here. It is shown that, despite a formal analogy with Unruh effect, a power spectrum is obtained, which is very different from the thermal one. We find a discrete spectrum depending on the frequency of th...

A teaching-learning sequence designed to introduce some fundamental concepts of quantum physics to high school teachers is proposed. Some parts of the proposal can be adapted to be taught to advanced high school students themselves. The inspiration came from the recognition of the fact that the roots of many pivotal concepts of quantum physics, nam...

L’originale e poliedrico approccio di Richard P. Feynman all’elettromagnetismo classico viene qui presentato nella sua generalità, ripercorrendo l’intero percorso storico, dalla iniziale “teoria dell’assorbitore” all’approccio originale e poco noto delineato in appunti manoscritti del 1963 e sviluppato poi nelle Hughes Lectures del 1967-68. Alcune...

A detailed and updated account is given of De Filippo’s non-unitary fourth-derivative gravity and its Newtonian limit, by pointing out the crucial role of non-unitarity in addressing transition to classicality and specifically localization of macroscopic bodies, microscopic foundation of the second law of thermodynamics, and measurement problem; fu...

The broad debate on foundational issues in quantum mechanics, which took place at the famous 1957 Chapel Hill conference on \textit{The Role of Gravitation in Physics}, is here critically analyzed with an emphasis on Richard Feynman's contributions. One of the most debated questions at Chapel Hill was whether the gravitational field had to be quant...

An account of Richard Feynman's work on gravitational waves is given. Feynman's involvement with this subject can be traced backto 1957, when he attended the famous Chapel Hill conference on the Role of Gravitation in Physics. At that conference, he presented in particular the celebrated sticky bead argument, which was devised to intuitively argue...

In this contribution we report about Feynman's approach to gravitation, starting from the records of his interventions at the Chapel Hill Conference of 1957. As well known, Feynman was concerned about the relation of gravitation with the rest of physics. Probably for this reason, he promoted an unusual, field theoretical approach to general relativ...

In this contribution we present an unpublished formulation of electromagnetism, motivated by Feynman’s involvement in undergraduate teaching, as sketched by him in handwritten notes, and subsequently completed in unpublished lectures given at the Hughes Aircraft Company in the late sixties. Starting from the relativity principle, Lorentz invariance...

In the past century research programs for unifying science (Hilbert, neo-positivists, physicalists) grew up. Many scholars pursue the goal of a Theory of Everything. They have to prove the reduction of theories insisting on the same field of phenomena. On the other hand, historians and philosophers of science introduced the notion of a mutual incom...

Richard P. Feynman’s work on gravitation, as can be inferred from several published and unpublished sources, is reviewed. Feynman was involved with this subject at least from late 1954 to the late 1960s, giving several pivotal contributions to it. Even though he published only three papers, much more material is available, beginning with the record...

Dopo aver passato in rassegna il lavoro svolto da Feynman nell’ambito del progetto Manhattan, viene studiata in dettaglio una parte di tale lavoro, descritta in un rapporto tecnico scritto da Feynman nel 1946. Tale attività riguardò lo studio teorico di amplificatori lineari, con lo scopo di confrontare tali modelli teorici con i circuiti reali uti...

Introducing some fundamental concepts of quantum physics to high school students, and to their teachers, is a timely challenge. In this paper we describe ongoing research, in which a teaching–learning sequence for teaching quantum physics, whose inspiration comes from some of the fundamental papers about the quantum theory of radiation by Albert Ei...

The coefficient of restitution of the bounces of a marble on a table is here studied and measured in terms of the total bouncing time. Interesting links arise with mathematics (geometric series) as well as philosophy (Zeno’s paradox), which make our proposal appealing for an interdisciplinary teaching-learning sequence. The relevance of this work f...

Feynman's multifaceted contributions to gravitation, as can be inferred from several published and unpublished sources, are here reviewed. Feynman thought about this subject at least from late 1954 to the late '60s, giving several contributions to it. Even if he only published three papers on gravity, much more material is available, especially the...

At high school as well as undergraduate level, Quantum Mechanics (QM) is usually introduced through an overview of the main experiments and theoretical attempts which took place at the beginning of 20-th century. Even if retracing the historical path which led to the introduction of the new conceptual and mathematical framework has undoubted advant...

A timely challenge in current physics education is to develop educational tracks aimed at introducing advanced high school students to the main concepts of quantum theory. While standard tracks are historical in nature, going from Planck’s hypothesis to the Schrödinger equation, several points of this history tend to be left out. However, the richn...

In Cosmology and in Fundamental Physics there is a crucial question like: where the elusive substance that we call Dark Matter is hidden in the Universe and what is it made of? that, even after 40 years from the Vera Rubin seminal discovery [ 1 ] does not have a proper answer. Actually, the more we have investigated, the more this issue has become...

We retrace an ab initio relativistic derivation of the inhomogeneous Maxwell’s equations that was developed by Feynman in unpublished notes, clarifying the analogies and the differences with analogous treatments present in the literature. Unlike the latter, Feynman’s approach stands out because it considers electromagnetic potentials as primary, re...

In Cosmology and in Fundamental Physics there is a crucial question like: where the elusive substance that we call Dark Matter is hidden in the Universe and what is it made of?, that, even after 40 years from the Vera Rubin seminal discovery does not have a proper answer. Actually, the more we have investigated, the more this issue has become stron...

In this contribution we describe an original formulation of electromag-netism which was sketched by Feynman in a set of unpublished notes, and alluded to by him in the preface of his Caltech lectures and in a famous interview to C. Weiner. Feynman further refined his view when lecturing on electrodynamics at the Hughes Aircraft Company some years l...

A detailed and updated review is given of De Filippo's Non-unitary $4$-th Derivative Gravity and its Newtonian limit, by pointing out the crucial role of non-unitarity in addressing transition to classicality and specifically localization of macroscopic bodies, microscopic foundation of the second law of thermodynamics, measurement problem; further...

Observing the bouncing of a marble on a table is a rather common experience. The tic-tac sound of the rigid ball, nevertheless, carries quite a pleasant surprise. In fact, by measuring the total time of bouncing Δt, the coefficient of restitution can be estimated. As is well known, in an inelastic collision the kinetic energy is not conserved, and...

Richard Feynman’s involvement in the Manhattan Project during the World War II is well known. He studied instruments and experimental devices, being directly involved, for instance, in the study of the “water boiler”, a small nuclear reactor designed for experimenting on fundamental properties of the chain reaction. In most of such experiments, the...

Viene descritta una formulazione alternativa, mai pubblicata, dell’elettromagnetismo classico dovuta a Richard P. Feynman. Tale approccio è delineato e iniziato in alcuni appunti manoscritti, recentemente rinvenuti da M. A. Gottlieb negli archivi del Caltech, e successivamente completata da Feynman in altri contributi inediti. Partendo dal solo pri...

We retrace an ab initio relativistic derivation of Maxwell's equations that was developed by Feynman in unpublished notes, clarifying the analogies and the differences with analogous treatments present in the literature. Unlike the latter, Feynman's approach stands out because it considers electromagnetic potentials as primary, reflecting his ideas...

The quest for a microscopic foundation of thermodynamics is addressed within the Nonunitary Newtonian Gravity (NNG) model through the study of a specific closed system, namely, a three-dimensional harmonic nanocrystal. A numerical calculation of the nanocrystal von Neumann entropy as a function of time is performed, showing a sharp monotonic increa...

We study the analog of Snell's law for particles moving across the interface of two regions with two different potential energies, from two different points of view. First, a simple demonstration involving marbles and a potential step is shown. Then, from a theoretical point of view, this law describing mechanical refraction is derived from the Mau...

We analyze Feynman's work on the response of an amplifier performed at Los Alamos and described in a technical report of 1946, as well as lectured on at the Cornell University in 1946-47 during his course on Mathematical Methods. The motivation for such a work was Feynman's involvement in the Manhattan Project, for which the necessity emerged of fe...

The quest for a microscopic foundation of Thermodynamics is addressed within the Nonunitary Newtonian Gravity model through the study of a specific closed system, namely a three-dimensional harmonic nanocrystal. A numerical calculation of the nanocrystal von Neumann entropy as a function of time is performed, showing a sharp monotonic increase, fol...

We analyze Feynman’s work on the response of an amplifier performed at Los Alamos and described in a technical report of 1946, as well as lectured on at the Cornell University in 1946–47 during his course on Mathematical Methods. The motivation for such a work was Feynman’s involvement in the Manhattan Project, for which the necessity emerged of fe...

It is explicitly shown that a one-family parameter model reproducing the nonlinear Newton–Schrödinger equation as the parameter goes to infinity is free from any causality violation problem for any finite value of it. This circumstance arises from the intrinsic mechanism of spontaneous state reduction of the model, absent in the Newton–Schrödinger...

We discuss a previously unpublished description of electromagnetism developed by Richard P. Feynman in the 1960s. Though similar to the existing approaches deriving electromagnetism from special relativity, the present one extends a long way towards the derivation of Maxwell's equations with minimal physical assumptions (in particular, without post...

It is explicitly shown that a one-family parameter model reproducing the nonlinear Newton-Schr\"odinger equation as the parameter goes to infinity is free from any causality violation problem for any finite value of it. This circumstance arises from the intrinsic mechanism of spontaneous state reduction of the model, absent in the Newton-Schr\"odin...

In this paper we study the small oscillations of a charged pendulum in the electric field generated by a couple of point charges. The complexity of the system makes the use of the Lagrangian formalism very convenient. The effect of an oscillating pivot in the spirit of Kapitza is discussed by introducing a rapidly oscillating forcing term and compu...

We discuss a less known aspect of Feynman’s multifaceted scientific work, centered about his interest in molecular biology, which came out around 1959 and lasted for several years. After a quick historical reconstruction about the birth of molecular biology, we focus on Feynman’s work on genetics with Robert S. Edgar in the laboratory of Max Delbru...

We discuss a less known aspect of Feynman's multifaceted scientific work, centered about his interest in molecular biology, which came out around 1959 and lasted for several years. After a quick historical reconstruction about the birth of molecular biology, we focus on Feynman's work on genetics with Robert S. Edgar in the laboratory of Max Delbru...

A simple version due to Feynman of Fermats principle is analyzed. It deals with the path a lifeguard on a beach must follow to reach a drowning swimmer. The solution for the exact point, P(×, 0), at the beach-sea boundary, corresponding to the fastest path to the swimmer, is worked out in detail and the analogy with light traveling at the air-water...

The optical properties of a compact disc (CD) under “skimming” light rays have been analyzed. We have noticed that a clear green line can be detected when the disc is irradiated with light rays coming from a lamp in such a way that only those skimming the CD, held horizontally, are selected. We provide a physical interpretation of this phenomenon o...

The result of a numerical simulation of two interacting particles in the framework of Nonunitary Newtonian Gravity is presented here. Particles are held close together by a 3-d harmonic trap and interact with each other via an 'electrical' delta-like potential and via the ordinary Newtonian term, together with a fluctuational nonunitary counterpart...

We study the unconventional transmission properties of a wave packet through a PT-symmetric potential region as describing the actual electromagnetic wave propagation along a waveguide filled with gain and loss media. The nontrivial behavior of the transmission probability manifests in the giant amplification of the incident electromagnetic signal...

The equation of the motion of a point charge q repelled by a fixed point-like charge Q is derived and studied. In solving this problem useful concepts in classical and relativistic kinematics, in Newtonian mechanics and in non-linear ordinary differential equations are revised. The validity of the approximations is discussed from the physical point...

The goal of international trade theories is to explain the exchange of goods and services between different countries, aiming to benefit from it. Albeit the idea is very simple and known since ancient history, smart policy and business strategies need to be implemented by each subject, resulting in a complex as well as not obvious interplay. In ord...

An important issue within the present economic crisis is understanding the dynamics of the public debt of a given country, and how the behavior of average consumers and tax payers in that country affects it. Starting from a model of the average consumer behavior introduced earlier by the authors, we propose a simple model to quantitatively address...

In the present paper we investigate the paramagnetic susceptibility and the short-range order correlation functions of a d-dimensional classical isotropic ferromagnetic Heisenberg model with short-range exchange interactions by employing the two-time Green function method in classical statistical mechanics. Here we use Tyablikov–Callen-like decoupl...

Understanding consumption dynamics and its impact on the whole economy and welfare within the present economic crisis is not an easy task. Indeed the level of consumer demand for different goods varies with the prices, consumer incomes and demographic factors. Furthermore crisis may trigger different behaviors which result in distortions and amplif...

We study multi-soliton solutions of the four-dimensional SU(N) Skyrme model by combining the hedgehog ansatz for SU(N) based on the harmonic maps of $S^{2}$ into $CP^{N-1}$ and a geometrical trick which allows to analyze explicitly finite-volume effects without breaking the relevant symmetries of the ansatz. The geometric setup allows to introduce...

We discuss in detail very little known results obtained by Majorana as early as 1931, regarding the quantum theory of the chemical bond in homopolar molecules, based on the key concept of exchange interaction. After a brief historical overview of the quantum homopolar valence theory, we address the intriguing issues of the formation of the helium m...

Relevant contributions by Majorana regarding Compton scattering off free or bound electrons are considered in detail, where a (full quantum) generalization of the Kramers-Heisenberg dispersion formula is derived. The role of intermediate electronic states is appropriately pointed out in recovering the standard Klein-Nishina formula (for free electr...

An historical overview is given of the relevant steps that allowed the genesis of the quantum theory of the chemical bond, starting from the appearance of the new quantum mechanics and following later developments till approximately 1931. General ideas and some important details are discussed concerning molecular spectroscopy, as well as quantum co...

We study the quantum dynamics of a spin-orbit (SO) coupled Bose-Einstein condensate (BEC) in a double-well potential inspired by the experimental protocol recently developed by NIST group. We focus on the regime where the number of atoms is very large and perform a two-mode approximation. An analytical solution of the two-site Bose-Hubbard-like Ham...

The temperature dependence of magnetization in ferromagnetic nanostructures (e.g., nanoparticles or nanoclusters) is usually analyzed by means of an empirical extension of the Bloch law sufficiently flexible for a good fitting to the observed data and indicates a strong softening of magnetic coupling compared to the bulk material. We analytically d...

We study nonequilibrium properties of an atomic quantum dot (AQD) coupled to a Bose-Einstein condensate (BEC) within Keldysh-Green's function formalism when the AQD level is varied harmonically in time. Nonequilibrium features in the AQD energy absorption spectrum are the side peaks that develop as an effect of photon absorption and emission. We sh...

We review our recent results on the physics of quantum Hall fluids at Jain and non conventional fillings within a general field theoretic framework. We focus on a peculiar conformal field theory (CFT), the one obtained by means of the m-reduction technique, and stress its power in describing strongly correlated low dimensional condensed matter syst...

The two-electron atom is the simplest nontrivial quantum system not amenable to exact solutions. Today, its relevance in the development of quantum mechanics and its pedagogical value within the realm of atomic physics are widely recognized. In this work, an historical review of the known different methods and results devised to study such a proble...

a b s t r a c t Complex phenomena occurring in natural sciences are usually characterized by a non triv-ial interplay between microscopic and macroscopic dynamics, which can be successfully captured by the cellular automata (CA) computational paradigm [1]. In this paper we show that some approximation of the classical CA paradigm is needed in order...

We explore the low-temperature thermodynamic properties and crossovers of a d-dimensional classical planar Heisenberg ferromagnet in a longitudinal magnetic field close to its field-induced zero-temperature critical point by employing the two-time Green's function formalism in classical statistical mechanics. By means of a classical Callen-like met...

a b s t r a c t A study of the d-dimensional classical Heisenberg ferromagnetic model in the presence of a magnetic field is performed within the two-time Green function's framework in classical statistical physics. We extend the well known quantum Callen method to derive analytically a new formula for magnetization. Although this formula is valid...

In a recent paper, by exploiting the notion of Morita equivalence for field theories on noncommutative tori and choosing rational values of the noncommutativity parameter θ (in appropriate units), a general one-to-one correspondence between the m-reduced conformal field theory (CFT) describing a quantum Hall fluid (QHF) at paired states fillings1,2...

The quantum tricriticality of d-dimensional transverse Ising-like systems is studied by means of a perturbative renormalization group approach focusing on static susceptibility. This allows us to obtain the phase diagram for 3 ≤ d < 4, with a clear location of the critical lines ending in the conventional quantum critical points and in the quantum...

A new analogy between superfluid systems and cosmology is here presented, which relies strongly on the following ingredient: the back-reaction of the vacuum to the quanta of sound waves. We show how the presence of thermal phonons, the excitations above the quantum vacuum for T > 0, enable us to deduce an hydrodynamical equation formally similar to...

The new formal analogy between superfluid systems and cosmology, which emerges by taking into account the back-reaction of the vacuum to the quanta of sound waves,1 enables us to put forward some common features between these two different areas of physics. We find the condition that allows us to justify a General Relativity (GR) derivation of the...

We study the quantum dynamics of a binary mixture of Bose–Einstein condensates (BECs) in a double-well potential starting from a two-mode Bose–Hubbard Hamiltonian. Focussing on the regime where the number of atoms is very large, a mapping onto an SU (2) spin problem together with a Holstein–Primakoff transformation is performed. The quantum evoluti...

The aim of this work is to calculate explicitly the result of the experiment of superposition of a mirror in the Michelson photon cavities interferometric device proposed by Marshall, Simon, Penrose, and Bouwmeester, as expected within a recently proposed model of nonunitary self-gravity-inducing localization. As for other proposals of modification...

Cellular Automata (CA), one of the most challenging computational paradigms in microscopic and macroscopic complex systems simulation, can be successfully addressed also by using a modified CA classical approach. In this contribution we discuss related aspects in applying the CANv2 approach in examples of micro and macro dynamics such as: supercond...

By exploiting the notion of Morita equivalence for field theories on noncommutative tori and choosing rational values of the noncommutativity parameter θ (in appropriate units), a one-to-one correspondence between an Abelian noncommutative field theory (NCFT) and a non-Abelian theory of twisted fields on ordinary space can be established. Starting...

We study the quantum behaviour of a binary mixture of Bose–Einstein condensates in a double-well potential starting from a two-mode Bose–Hubbard Hamiltonian. We focus on the small tunnelling amplitude regime and apply perturbation theory up to second order. Analytical expressions for the energy eigenvalues and eigenstates are obtained. Then, the qu...

We consider the extended Hubbard model in the atomic limit on a Bethe lattice with coordination number z. By using the equations of motion formalism, the model is exactly solved for both attractive and repulsive intersite potential V. By focusing on the case of negative V, i.e., attractive intersite interaction, we study the phase diagram at finite...

The Morita equivalence for field theories on noncommutative two-tori is analysed in detail for rational values of the noncommutativity parameter θ (in appropriate units): an isomorphism is established between an Abelian noncommutative field theory (NCFT) and a non-Abelian theory of twisted fields on ordinary space. We focus on a particular conforma...

We show how the low energy properties of the two-leg XXZ spin-1/2 ladders with general anisotropy parameter Δ on closed geometries can be accounted for in the framework of the m-reduction procedure developed previously (Cristofano et al 2000 Mod. Phys. Lett. A 15 547; Cristofano et al 2000 Mod. Phys. Lett. A 15 1679; Cristofano et al 2002 Nucl. Phy...

shown to develop the phenomenon of flux fractionalization [G. Cristofano, V. Marotta, A. Naddeo, G. Niccoli, Eur. Phys. J. B 49 (2006) 83]. That led us to predict the emergence of a topological order in such a system [G. Cristofano, V. Marotta, A. Naddeo, J. Stat. Mech.: Theory Exp. (2005) P03006]. In this Letter we analyze the ground states and th...

We study the Morita equivalence for field theories on noncommutative two-tori. For rational values of the noncommutativity parameter θ (in appropriate units) we show the equivalence between an abelian noncommutative field theory and a nonabelian theory of twisted fields on ordinary space. We concentrate on a particular conformal field theory (CFT),...

We show that a system of spinless Fermi particles, localized on the sites of the Bethe lattice with coordination number z and interacting through a repulsive nearest-neighbor interaction, exhibits a phase transition to a charge-ordered state. The phase diagram in the n-T plane is derived. Relevant thermodynamic quantities, such as the free energy,...

We study the phase diagram at finite temperature of a system of Fermi particles on the sites of the Bethe lattice with coordination number z and interacting through onsite U and nearest-neighbor V interactions. This is a physical realization of the extended Hubbard model in the atomic limit. By using the equations of motion method, we exactly solve...

We show how to realize a "protected" qubit by using a fully frustrated Josephson junction ladder (JJL) with Mobius boundary conditions. Such a system has been recently studied within a twisted conformal field theory (CFT) approach [G. shown to develop the phenomenon of flux fractionalization [G. Cristofano, V. Marotta, A. Naddeo, G. Niccoli, Eur. P...

Reversibility is a concept widely studied in several research fields. Sev-eral contributions have been provided in physics and computer science. Reversible computation is characterized by means of invertible proper-ties [1]. Reversibility is studied in computer science also in order to design less dissipative computers. The evolution of quantum sys...

The issue of the number, nature and sequence of phase transitions in the fully frustrated XY (FFXY) model is a highly non trivial one due to the complex interplay between its continuous and discrete degrees of freedom. In this contribution we attack such a problem by means of a twisted conformal field theory (CFT) approach and show how it gives ris...

A fermionic model, built up of q species of localized Fermi particles, interacting by charge correlations, is isomorphic to a spin-q 2 Ising model. However, the equivalence is only formal and the two systems exhibit a different physical behavior. By considering a Bethe lattice with q = 1, we have exactly solved the models. There exists a critical t...