Atmospheric neutrinos are produced by cosmic ray interactions in the atmosphere. The zenith-angle and energy dependence of the muon- and electron- neutrino events are observed in atmospheric neutrino experiments. Through these studies neutrino oscillations were discovered. In this article, studies of atmospheric neutrinos in the Kamiokande and Super-Kamiokande experiments are described.

]]>In this paper, we apply Osgood's criterion from the theory of ordinary differential equations to detect finite-time singularities in a spatially flat FLRW universe in the context of a perfect fluid, a perfect fluid with bulk viscosity, and a Chaplygin and anti-Chaplygin gas. In particular, we applied Osgood's criterion to demonstrate singularity behaviour for Type 0/big crunch singularities as well as Type II/sudden singularities. We show that in each case the choice of initial conditions is important as a certain number of initial conditions leads to finite-time, Type 0 singularities, while other precise choices of initial conditions which depend on the cosmological matter parameters and the cosmological constant can avoid such a finite-time singularity. Osgood's criterion provides a powerful and yet simple way of deducing the existence of these singularities, and also interestingly enough, provides clues of how to eliminate singularities from certain cosmological models.

Determining whether a cosmological solution to Einstein's field equations has any finite-time singularities is of considerable importance. Scenarios such as a “big crunch" and other ultimate fate of the universe possibilities are examples of such finite-time singularities. In this paper, a criterion is established that allows one to easily detect such finite-time singularities.

Starting from a field theory action that describes a Dirac fermion, we propose and analyze a model based on a low-relativistic Pauli equation coupled to a torsion-like term to study Spin Hall Effect (SHE). We point out a very particular connection between the modified Pauli equation and the (SHE), where what we refer to torsion as field playing an important role in the spin-orbit (SO) coupling process. In this scenario, we present a proposal of a spin-type current, considering the tiny contributions of torsion in connection with intrinsic anisotropy of the crystal electric field.

The main purpose of this work is to propose a possible connection between the non-relativistic regime of Dirac’s equation, investigated in a scenario with anisotropy, and the physics of the Spin Hall Effect.

To fully resolve Theseus' paradox in the framework of the Elementary Process Theory, this paper corrects its formalism and axiomatization.

]]>Nanosheets of bismuth telluride (Bi_{2}Te_{3}), a topological insulator material that exhibits broadband saturable absorption due to its non-trivial Dirac-cone like energy structure, are utilized to generate short pulses from Tm:ZBLAN waveguide lasers. By depositing multiple layers of a carefully prepared Bi_{2}Te_{3} solution onto a glass substrate, the modulation depth and the saturation intensity of the fabricated devices can be controlled and optimized. This approach enables the realization of saturable absorbers that feature a modulation depth of 13% and a saturation intensity of 997 kW/cm^{2}. For the first time to our knowledge, Q-switched mode-locked operation of a linearly polarized mid-IR ZBLAN waveguide chip laser was realized in an extended cavity configuration using the topological insulator Bi_{2}Te_{3}. The maximum average output power of the laser is 16.3 mW and the Q-switched and mode-locked repetition rates are 44 kHz and 436 MHz, respectively.

Novel saturable absobers for ultrashort-pulsed operation of mid-infrared waveguide chip lasers have been fabricated by depositing multiple layers of nanosheets of topological insulator material onto a glass substrate. By optimizing the number of spin coating layers, highly stable Q-switched mode-locked pulse trains have been generated in an external-cavity Thulium-ZBLAN depressed-cladding waveguide laser. These results pave the way for the realization of fully integrated ultrashort-pulsed chip lasers in the mid-infrared.

A model of nonlinear electrodynamics with the Lagrangian density is proposed. The scale invariance and the dual invariance of electromagnetic fields are broken in the model. In the limit one comes to Maxwell's electrodynamics and the scale and dual invariances are recovered. We investigate the effect of coupling electromagnetic fields with the gravitational field. The asymptotic black hole solution is found which is similar to the Reissner-Nordström solution. We obtain corrections to Coulomb's law and to the Reissner-Nordström solution in the model proposed. The existence of the regular asymptotic at was demonstrated. The mass of the black hole is calculated possessing the electromagnetic origin. It was shown that there are not superluminal fluctuations and principles of causality and unitarity take place.

A new model of nonlinear arcsin-electrodynamics with a dimensional parameter β is proposed. It was shown that in the model under consideration the scale invariance and dual invariance of electromagnetic fields are broken. In the limit when β approaches to zero arcsin-electrodynamics is converted into Maxwell's electrodynamics and the scale and dual invariances are recovered. The effect of coupling electromagnetic fields with the gravitational field is investigated. The asymptotic black hole solution is found which is similar to the Reissner-Nordström solution. Corrections to Coulomb's law and to the Reissner-Nordström solution are obtained. The existence of the regular asymptotic at big distances from the black hole was demonstrated. The mass of the black hole is calculated and is treated as the electromagnetic energy. It was shown that there are not superluminal fluctuations and principles of causality and unitarity take place.

Strength of light-matter interactions and radiative dynamics of emitters could be controlled with structuring of electromagnetic environment. While local and cross density of electromagnetic states are routinely used for predicting total and radiative decay rates in the weak coupling regime, resonant nanostructures offer going beyond this description, giving rise to new phenomena. Correlated time-evolution of a strongly coupled emitter-nanoresonator system and nonradiative channels are shown here to predefine the radiative decay dynamics and lead to substantial shortening in characteristic emission times. Quantum formalism, based on stochastic Hamiltonian treatment of radiative and nonradiative processes, was generalized for describing light-matter interactions in vicinity of open nano-resonators. The developed theory was subsequently applied to spontaneous emission dynamics of emitters, situated next to metal surfaces, supporting stopped light resonant conditions. Over four orders of magnitude lifetime shortening was predicted to be detectable in the far-field. The interplay between strong and weak coupling regimes, enabled by resonant nanostructures, could serve as a platform for ultrafast opto-electronic components, fluorescent labels and others.

Strength of light-matter interactions and radiative dynamics of emitters could be controlled with structuring of electromagnetic environment. While local and cross density of electromagnetic states are routinely used for predicting total and radiative decay rates in the weak coupling regime, resonant nanostructures offer going beyond this description, giving rise to new phenomena. The manuscript presents quantum formalism for describing light-matter interactions in vicinity of open nano-resonators.

In this study, we utilized picosecond pulses from an Nd:YAG laser to investigate the nonlinear optical characteristics of monolayer MoSe_{2}. Two-step growth involving the selenization of pulsed-laser-deposited MoO_{3} film was employed to yield the MoSe_{2} monolayer on a SiO_{2}/Si substrate. Raman scattering, photoluminescence (PL) spectroscopy, and atomic force microscopy verified the high optical quality of the monolayer. The second-order susceptibility χ^{(2)} was calculated to be ∼50 pm V^{−1} at the second harmonic wavelength ∼810 nm, which is near the optical gap of the monolayer. Interestingly, our wavelength-dependent second harmonic scan can identify the bound excitonic states including negatively charged excitons much more efficiently, compared with the PL method at room temperature. Additionally, the MoSe_{2} monolayer exhibits a strong laser-induced damage threshold ∼16 GW cm^{−2} under picosecond-pulse excitation_{.} Our findings suggest that monolayer MoSe_{2} can be considered as a promising candidate for high-power, thin-film-based nonlinear optical devices and applications.

A high second-order susceptibility and strong laser-induced damage threshold of two-step grown MoSe_{2} monolayer is reported. This superior nonlinear optical behavior is attributed to its electronic band structure modification when transition metal dichalcogenides (MX_{2}) are thinned down to the monolayer regime. Futher, wavelength-dependent second harmonic scan can identify the bound excitonic states much more efficiently, compared with the PL method at room temperature.

The feedback control scheme for a Bose-Einstein condensate (BEC) in a double-well trapping potential located in one arm of Mach-Zehnder interferometer (MZI) is investigated. The off-resonant light beam performs the phase probing in one of the wells, thus creating information about the number of atoms in this well. The parameters of the trapping potential are controlled via a feedback loop based on the measured output of the MZI. The problem is analyzed in the framework of master equations for hybrid quantum-classical systems. Significant modifications of the stationary distribution of atoms over the wells are predicted. These distributions can effectively be controlled by the tunable phase shift in the other arm of the MZI.

The feedback control scheme for a Bose-Einstein condensate in a double-well trapping potential located in one arm of Mach-Zehnder interferometer (MZI) is investigated. The parameters of the trapping potential are controlled via a feedback loop based on the measured output of the MZI. Significant modifications of the stationary populations of the wells are predicted. These distributions can effectively be controlled by the tunable phase shift in the other arm of the MZI.

We investigate the formation of Cooper pairs, bound dimers and the dimer-dimer elastic scattering of ultracold dipolar Fermi molecules confined in a 2D optical lattice bilayer configuration. While the energy and their associated bound states are determined in a variational way, the correlated two-molecule pair is addressed as in the original Cooper formulation. We demonstrate that the 2D lattice confinement favors the formation of zero center mass momentum bound states. Regarding the Cooper pairs binding energy, this depends on the molecule populations in each layer. Maximum binding energies occur for non-zero (zero) pair momentum when the Fermi system is polarized (unpolarized). We find an analytic expression for the dimer-dimer effective interaction in the deep BEC regime. The present analysis represents a route for addressing the BCS-BEC crossover in dipolar Fermi gases confined in 2D optical lattices within the current experimental panorama.

Formation of Cooper pairs, bound dimeric states and dimer-dimer scattering is investigated in Fermi molecules confined in a 2D optical lattice bilayer array. While Cooper pairs are addressed as in the original formulation, bound dimeric states are investigated in a variational way. An analytic expression for the dimer-dimer effective interaction in the deep BEC regime is found. This study is a route to address the entire BCS-BEC crossover both theoretically and experimentally.

We analyze exciting recent measurements [Phys. Rev. Lett. **114** (2015) 037202] of the magnetization, differential susceptibility and specific heat on one dimensional Heisenberg antiferromagnet Cu(C_{4}H_{4}N_{2})(NO_{3})_{2} (CuPzN) subjected to strong magnetic fields. Using the mapping between magnons (bosons) in CuPzN and fermions, we demonstrate that magnetic field tunes the insulator towards quantum critical point related to so-called fermion condensation quantum phase transition (FCQPT) at which the resulting fermion effective mass diverges kinematically. We show that the FCQPT concept permits to reveal the scaling behavior of thermodynamic characteristics, describe the experimental results quantitatively, and derive for the first time the (temperature—magnetic field) phase diagram, that contains Landau-Fermi-liquid, crossover and non-Fermi liquid parts, thus resembling that of heavy-fermion compounds.

The experimentally observed thermodynamic properties of the 1D Heisenberg antiferromagnet Cu(C_{4}H_{4}N_{2})(NO_{3})_{2} (CuPzN) are so unusual that nobody expects that it might belong to the class of heavy-fermion compounds. Here we propose the theory which permits to reveal the latter resemblance and construct for the first time the phase diagram, that contains Fermi liquid, crossover and non Fermi liquid parts.

The Sudbury Neutrino Observatory (SNO) experiment was constructed by an international scientific collaboration primarily to provide a clear determination of whether solar neutrinos change their flavor in transit from the core of the sun to the earth. The detector used 1000 tonnes of heavy water (>99.92% D2O) in an ultra-clean location 2 km underground in INCO's Creighton mine near Sudbury, Canada to observe two separate reactions of neutrinos on deuterium. The first reaction was sensitive only to electron flavor neutrinos and the second reaction was equally sensitive to all neutrino flavors. The measurements by SNO showed clearly that the hypothesis of no neutrino flavor change was ruled out by more than 5.3 standard deviations. The observation of flavor change for neutrinos implies that they have a non-zero mass. The measured total flux of active neutrinos from ^{8}B decay in the sun was found to be in excellent agreement with the predictions of solar model calculations. This paper describes the history and scientific measurements of the SNO experiment.

The intrinsic lattice thermal conductivity of MoS_{2} is an important aspect in the design of MoS_{2}-based nanoelectronic devices. We investigate the lattice dynamics properties of MoS_{2} by first-principle calculations. The intrinsic thermal conductivity of single-layer MoS_{2} is calculated using the Boltzmann transport equation for phonons. The obtained thermal conductivity agrees well with the measurements. The contributions of acoustic and optical phonons to the lattice thermal conductivity are evaluated. The size dependence of thermal conductivity is investigated as well.

The present work calculates the thermal conductivity of isotopically pure and naturally occurring single-layer (SL) MoS_{2} using the Boltzmann transport equation for phonons. The lattice dynamics properties, the scattering mechanism and the size dependence of thermal conductivity in SL MoS_{2} are investigated in details.

We theoretically present the nonlinear selective reflection spectroscopy of V-type atomic system at gas-solid interface in a pump-probe scheme. The saturation and coherence effects are distinguished by solving Liouville equation in the absence and presence of reduced density matrix element between the two excited levels. When the coherence effect exists, two peaks appear in reflection spectroscopy with asymmetry lineshape. We investigate the dependence of reflection spectroscopy on pump field intensity, frequency detuning and coherent decay rate induced by collision between atoms. The lineshape can be explained based on reflection spectroscopy contributed from atoms with negative (before collision) and positive (after collision) velocities, single-photon and two-photon processes. This study is helpful for investigating quantum coherence and dynamic processes of atoms at gas-solid interface.

The nonlinear selective reflection (SR) spectroscopy of V-type atomic system at gas-solid interface is analyzed. Saturation and coherence effects are distinguished in the absence and presence of density matrix element between two excited levels. SR spectroscopy as a function of pump field intensity, frequency detuning and coherent decay rate induced by collision between atoms are investigated. The lineshape is explained based on SR spectroscopy contributed from atoms with negative and positive velocities, single-phonon and two-photon processes.

In fiber lasers, the study of the cubic-quintic complex Ginzburg-Landau equations (CGLE) has attracted much attention. In this paper, four families (kink solitons, gray solitons, Y-type solitons and combined solitons) of exact soliton solutions for the variable-coefficient cubic-quintic CGLE are obtained via the modified Hirota method. Appropriate parameters are chosen to investigate the properties of solitons. The influences of nonlinearity and spectral filtering effect are discussed in these obtained exact soliton solutions, respectively. Methods to amplify the amplitude and compress the width of solitons are put forward. Numerical simulation with split-step Fourier method and fourth-order Runge-Kutta algorithm are carried out to validate some of the analytic results. Transformation from the variable-coefficient cubic-quintic CGLE to the constant coefficients one is proposed. The results obtained may have certain applications in soliton control in fiber lasers, and may have guiding value in experiments in the future.

Four families (kink solitons, gray solitons, Y-type solitons and combined solitons) of exact soliton solutions for the variable-coefficient cubic-quintic complex Ginzburg-Landau equation are obtained. The influences of nonlinearity and spectral filtering effect are discussed. Methods to amplify the amplitude and compress the width of solitons are put forward, with numerical simulations being added.

Numerical studies of the coupled Einstein-Klein-Gordon system have recently revealed that confined scalar fields generically collapse to form caged black holes. In the light of this finding, we analytically study the characteristic resonance spectra of the confined scalar fields in rotating linear dilaton black hole geometry. Confining mirrors (cage) are assumed to be placed in the near-horizon region of a caged rotating linear dilaton black hole ( is the radius of the cage and *r*_{2} represents the event horizon). The radial part of the Klein-Gordon equation is written as a Schrödinger-like wave equation, which reduces to a Bessel differential equation around the event horizon. Using analytical tools and proper boundary conditions, we obtain the boxed-quasinormal mode frequencies of the caged rotating linear dilaton black hole. Finally, we employ Maggiore's method, which evaluates the transition frequency in the adiabatic invariant quantity from the highly damped quasinormal modes, in order to investigate the entropy/area spectra of the rotating linear dilaton black hole.

Bekenstein conjectured that black hole area should be quantized in discrete levels (equidistant) to account for the quantized entropy. We show that when a non-asymptotically flat rotating black hole filled with dilaton and axion fields (strong candidates for the dark matter/energy) is caged by a scalar cloud, the resulting boxed quasinormal modes support Bekenstein's conjecture when Maggiore's adiabatic invariant formula is employed.

Hydrogen-rich compounds are extensively explored as candidates for high-temperature superconductors. Recently, hydrogen sulfide (H_{3}S), compressed in a diamond anvil cell, was experimentally found to show the loss of resistance at the critical temperature of 203 K. It opens the door to achieving room-temperature superconductivity in these types of materials.

This paper investigates the thermodynamic properties of superconducting hydrogen and deuterium sulfide at 150 GPa. In particular, the energy gap, specific heat jump, thermodynamic critical field and London penetration depth were calculated within the framework of the Eliashberg formalism. Then, these parameters were used to estimate the dimensionless ratios which exceed the predictions of the Bardeen-Cooper-Schrieffer theory. These discrepancies arise from the existence of the strong-coupling and retardation effects in the systems investigated. The Eliashberg theory goes beyond the BCS theory to include these effects.

The results presented in this paper are expected to stimulate experimental and theoretical exploration and discovery of new superconducting hydrogen-containing materials like H_{3}S.

Surface phonon cavities that are homogenous in both mechanical and dielectric properties are reported. The cavities are formed by the placement of a defect of a single domain within periodic domain inversion of single crystal piezoelectric lithium niobate that exhibits a surface phononic bandgap through the phonon-polariton coupling. Surface cavity resonances are observed within the bandgap, which correspond to entrapment of the phonon-polariton within the defect. In addition to demonstrating that the observed resonances are non-radiative and decoupled from bulk radiation, which is critical for high Q cavities, the possibility to tune the surface cavity resonance spectra simply by varying the defect width is also shown. Such an ability to excite a surface cavity resonance that is non-radiative with simultaneous localization of the electric field, together with the advantage of a cavity that is physically formed from a completely monolithic and uniform material, offers unique opportunities.

After an initial burst of excitement about its extraordinary implications for our concept of space and time, the theory of general relativity underwent a thirty-year period of stagnation, during which only a few specialists worked on it, achieving little progress. In the aftermath of World War II, however, general relativity gradually re-entered the mainstream of physics, attracting an increasing number of practitioners and becoming the basis for the current standard theory of gravitation and cosmology-a process Clifford Will baptized the Renaissance of General Relativity. The recent detection of gravitational radiation by the LIGO experiment can be seen as one of the most outstanding achievements in this long-lasting historical process. In the paper, we present a new multifaceted historical perspective on the causes and characteristics of the Renaissance of General Relativity, focusing in particular on the case of gravitational radiation in order to illustrate this complex and far-reaching process.

]]>Toroidal multipoles have recently been explored in various scientific communities, ranging from atomic and molecular physics, electrodynamics, and solid-state physics to biology. Here we experimentally and numerically demonstrate a three-dimensionsal toroidal metamaterial where two different toroidal dipoles along orthogonal directions have been observed. The chosen toroidal metamaterial also simultaneously supports Fano resonance and the classical analog of electromagnetically induced transparency (EIT) phenomena in the transmission spectra that originate from the electric–toroidal dipole and electric–magnetic dipole destructive interference. The intriguing properties of the toroidal resonances may open up avenues for applications in toroidal moments generator, sensing and slow-light devices.

This work experimentally and numerically presents a 3D toroidal metamaterial with two different toroidal dipoles along orthogonal directions. The chosen toroidal metamaterial also supports Fano resonance and electromagnetically induced transparency (EIT) phenomena in the transmission spectra. The radiation pattern of the multipole and the destructive interference between electric–toroidal and electric–magnetic dipoles are clearly illustrated to understand the unerlying physics of the proposed 3D toroidal metamaterial.

Hydrogen-rich compounds are extensively explored as candidates for a high-temperature superconductors. Currently, the measured critical temperature of 203 K in hydrogen sulfide (H_{3}S) is among the highest over all-known superconductors. In present paper, using the strong-coupling Eliashberg theory of superconductivity, we compared in detail the thermodynamic properties of two samples containing different hydrogen isotopes H_{3}S and D_{3}S at 150 GPa. Our research indicates that it is possible to reproduce the measured values of critical temperature 203 K and 147 K for H_{3}S and D_{3}S by using a Coulomb pseudopotential of 0.123 and 0.131, respectively. However, we also discuss a scenario in which the isotope effect is independent of pressure and the Coulomb pseudopotential for D_{3}S is smaller than for H_{3}S. For both scenarios, the energy gap, specific heat, thermodynamic critical field and related dimensionless ratios are calculated and compared with other conventional superconductors. We shown that the existence of the strong-coupling and retardation effects in the systems analysed result in significant differences between values obtained within the framework of the Eliashberg formalism and the prediction of the Bardeen-Cooper-Schrieffer theory.

Hydrogen-rich compounds are extensively explored as candidates for a high-temperature superconductors. Currently, the measured critical temperature of 203 K on H_{3}S is among the highest over all-known superconductors. This paper investigates the thermodynamic properties of superconducting hydrogen and deuterium sulfide at high pressure. In particular, the energy gap, the specific heat, the thermodynamic critical field and the London penetration depth were calculated within the framework of the Eliashberg formalism.

Surface phonon cavities that are homogenous in both mechanical and dielectric properties are reported. The cavities are formed by the placement of a defect of a single domain within periodic domain inversion of single crystal piezoelectric lithium niobate that exhibits surface phononic bandgap through the phonon-polariton coupling. Surface cavity resonances are observed within the bandgap, which manifest in entrapment of phonon-polariton within the defect. In addition to demonstrating that the observed resonances are non-radiative and decoupled to bulk radiation, which is critical for high Q cavities, it is also shown the possibility to tune the surface cavity resonance spectra simply by varying the defect width. Such an ability to excite surface cavity resonance that is non-radiative with simultaneous localization of the electric field together with the advantage of a cavity that is physically formed from a completely monolithic and uniform material offers unique opportunities for widespread applications for example in actuation, detection, and phonon lasing that can be fully integrated with other physical systems such as quantum acoustics, photonics, and microfluidics.

Surface phonon cavity formed by the placement of a defect of a single domain within periodic domain inversion of single crystal piezoelectric lithium niobate exhibiting surface phononic bandgap through the phonon-polariton coupling is introduced. It is shown that the proposed cavity can exhibit entrapment of phonon-polariton, indicated by the simultaneous localization of both surface phonons and the electric within the defect.

Motivated by recent experiments on long-lived magnetoplasmons in the presence of a perpendicular magnetic field, we investigate the dynamical dielectric response function of graphene in contact with a substrate using the random phase approximation. We add a periodically modulated magnetic field within the graphene plane and address both the inter and intra Landau band magnetoplasmons. Verification of the predicted magnetic modulation effects is possible by experiments analogous to those for the zero gap limit.

We study the inter and intra Landau band magnetoplasmons in gapped graphene by the self-consistent field approach. Long-lived magnetoplasmons are important in graphene due to the large carrier mobility and doping tunability. They facilitate a strong spatial confinement of light and the terahertz frequencies are promising for next generation optoelectronics.

The statistical mechanics of arbitrary holonomic scleronomous systems subjected to arbitrary external forces is described by specializing the Lagrange and Hamilton equations of motion to those of the Brownian motion on a manifold. In this context, the Klein-Kramers and Smoluchowski equations are derived in covariant form, and it is demonstrated that these equations have equilibrium solutions corresponding to the Gibbs distribution, in agreement with standard thermodynamics. At last, the Langevin dynamics corresponding to the Smoluchowski limit is found to exactly correspond to the Brownian motion on a smooth manifold. These results find significant applications in the study of several statistical properties of constrained molecular assemblies (e.g. polymers) of interest in chemistry, physics and biology.

The formulation of statistical mechanics based on Langevin and Fokker-Planck equations is typically developed for particles without mechanical constraints. However, the constrained statistical mechanics is indispensable to study several statistical properties of molecular assemblies of interest in chemistry, physics and biology. Therefore, in this paper we analyze arbitrary holonomic scleronomous systems by specializing the Lagrange and Hamilton equations of motion to those of the Langevin dynamics on a smooth manifold.

A single cavity photon mode is expected to modify the Coulomb interaction of an electron system in the cavity. Here we investigate this phenomena in a parallel double quantum dot system. We explore properties of the closed system and the system after it has been opened up for electron transport. We show how results for both cases support the idea that the effective electron-electron interaction becomes more repulsive in the presence of a cavity photon field. This can be understood in terms of the cavity photons dressing the polarization terms in the effective mutual electron interaction leading to nontrivial delocalization or polarization of the charge in the double parallel dot potential. In addition, we find that the effective repulsion of the electrons can be reduced by quadrupolar collective oscillations excited by an external classical dipole electric field.

The authors explore theoretically through several measurable quantities how cavity photons can enhance or reduce the interaction of two electrons in parallel quantum dots embedded in a photon cavity.

Based on Eddington affine variational principle on a locally product manifold, we derive the separate Einstein space described by its Ricci tensor. The derived field equations split into two field equations of motion that describe two maximally symmetric spaces with two cosmological constants. We argue that the invariance of the *bi-field equations* under *projections* on the separate spaces, may render *one* of the cosmological constants to zero. We also formulate the model in the presence of a scalar field. The resulted separate Einstein-Eddington spaces maybe considered as two states that describe the universe before and after inflation. A possibly interesting affine action for a general perfect fluid is also proposed. It turns out that the condition which leads to zero cosmological constant in the vacuum case, eliminates here the effects of the gravitational mass density of the perfect fluid, and the dynamic of the universe in its final state is governed by only the inertial mass density of the fluid.

In affine variational principle, a symmetric linear connection is taken as a fundamental field. The metric tensor is generated dynamically, and it appears as a canonically conjugate to the connection. From this picture, Einstein's gravity with a cosmological constant can be obtained by a covariant Legendre transformation of the affine Lagrangian. In the present paper, this formalism is applied to product spaces and the cosmological constant problem.

Localization of a particle in the wells of an asymmetric double-well (DW) potential is investigated here. Information entropy-based uncertainty measures, such as Shannon entropy, Fisher information, Onicescu energy, etc., and phase-space area, are utilized to explain the contrasting effect of localization-delocalization and role of asymmetric term in such two-well potentials. In asymmetric situation, two wells behaves like two different potentials. A general rule has been proposed for arrangement of quasi-degenerate pairs, in terms of asymmetry parameter. Further, it enables to describe the distribution of particle in either of the deeper or shallow wells in various energy states. One finds that, all states eventually get localized to the deeper well, provided the asymmetry parameter attains certain threshold value. This generalization produces symmetric DW as a natural consequence of asymmetric DW. Eigenfunctions, eigenvalues are obtained by means of a simple, accurate variation-induced exact diagonalization method. In brief, information measures and phase-space analysis can provide valuable insight toward the understanding of such potentials.

Some simple rules are proposed to analyze the distribution of a particle in an asymmetric double-well potential given by,

Straightforward modification of the same rule also enables one to explain energy distribution. Energy states become quasi-degenerate *only* at certain characteristic values of the asymmetry parameter, γ. For remaining values of γ, the two wells may be treated as two different single-well potentials.

Information-based uncertainty relations like Fisher information, Shannon entropy, Onicescu energy and Onicescu Shannon entropy, as well as conventional uncertainty product and phase space calculations also consolidate the above-mentioned rules in such potentials.