Search Results (31)

Phase space methods, particularly Wigner functions, provide intuitive tools for representing and analyzing quantum states. We focus on systems with SU(2) dynamical symmetry, which naturally describes spin and a wide range of two-mode quantum models. We present a unified phase space framework tailored to these systems, highlighting its broad applicability in quantum optics, metrology, and information. After reviewing the core SU(2) phase-space formalism, we apply it to states designed for optimal quantum sensing, where their nonclassical features are clearly revealed in the Wigner representation. We then extend the approach to systems with an indefinite number of excitations, introducing a generalized framework that captures correlations across multiple SU(2)-invariant subspaces. These results offer practical tools for understanding both theoretical and experimental developments in quantum science. Full article

The gypsum crystals (CaSO₄·2H₂O) crystallizes in a low symmetry system (monoclinic) and shows a marked layered structure along with a perfect cleavage parallel to the {010} faces. Owing to its widespread occurrence, as a single or twinned crystal, here the gypsum equilibrium (E.S.) and growth shapes (G.S.) have been re-visited. In making the distinction among E.S. and G.S., in the present work, the basic difference between epitaxy and homo-taxy is clearly evidenced. Gypsum has also been a fruitful occasion to recollect the general rules concerning either contact or penetration twins, for free growing and for twinned crystals nucleating onto pre-existing substrates. Both geometric and crystal growth aspects have been considered as well, by unifying theory and experiments of crystallography and crystal growth through the intervention of ${\mathsf{\beta }}_{\mathrm{adh}}$, the physical quantity representing the specific adhesion energy between gypsum and other phases. Hence, the adhesion energy allowed us to systematically use the Dupré’s formula. In the final part of the paper, peculiar attention has been paid to sediments (or solution growth) where the crystal size is very small, in order to offer a new simple way to afford classical (CNT) and non-classical nucleation (NCNT) theories, both ruling two quantities commonly used in the industrial crystallization: the total induction times (${\mathrm{t}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$) and crystal size distribution (CSD). Full article

The basic idea of the ‘partially nonclassical method’, developed in the present paper, is to apply the invariance requirement of the Lie group method using not all differential consequences of the invariant surface condition but only part of them. It differs from the ‘classical’ method, in which the invariant surface condition is not used, and from the ‘nonclassical’ method, in which all the differential consequences are used. It provides additional possibilities for the symmetry analysis of partial differential equations (PDEs), as compared with the ‘classical’ and ‘nonclassical’ methods, in the so-named no-go case when the group generator, associated with one of the independent variables, is identically zero. The method is applied to the flat steady-state boundary layer problem, reduced to an equation for the stream function, and it is found that applying the partially nonclassical method in the no-go case yields new symmetry reductions and new exact solutions of the boundary layer equations. A computationally convenient unified framework for the classical, nonclassical and partially nonclassical methods ($\lambda$-formulation) is developed. The issue of conformal invariance in the context of the Lie group method is considered, stemming from the observation that the classical Lie method procedure yields transformations not leaving the differential polynomial of the PDE invariant but modifying it by a conformal factor. The physical contexts, in which that observation could be important, are discussed using the derivation of the Lorentz transformations of special relativity as an example. Full article

Non-Lie reduction operators, also known as nonclassical symmetries, are derived for special systems that appear in Plasma Physics. These operators are used to construct similarity mappings, which reduce the systems under study into systems of ordinary differential equations. Furthermore, potential equivalence transformations are presented. Based on these results, a number of exact solutions are constructed. Full article

The postulate of universal Weyl conformal symmetry for all elementary physical fields introduces nonclassical gravitational effects in both conformal gravitation (CG) and the conformal Higgs model (CHM). The resulting theory is found to explain major observed phenomena, including excessive galactic rotation velocities and accelerating Hubble expansion, without invoking dark matter (DM). The recent history of this development is surveyed here. The argument is confined to implications of classical field theory, which include galactic baryonic Tully–Fisher relationships and dark galactic haloes of a definite large radius. Cosmological CHM parameters exclude a massive Higgs boson but are consistent with a novel alternative particle of the observed mass. Full article

In this paper, a non-classical symmetry method for obtaining the symmetries of differential–difference equations is proposed. The non-classical symmetry method introduces an additional constraint known as the invariant surface condition, which is applied after the infinitesimal transformation. By solving the governing equations that satisfy this condition, we can obtain the corresponding reduced equation. This allows us to determine the non-classical symmetry of the differential–difference equation. This method avoids the complicated calculation involved in extending the infinitesimal generator and allows for a wider range of symmetry forms. As a result, it enables the derivation of a greater number of differential–difference equations. In this paper, two kinds of (2+1)-dimensional Toda-like lattice equations are taken as examples, and their corresponding symmetric and reduced equations are obtained using the non-classical symmetry method. Full article

In this article, the generalized Jacobi elliptic function expansion method with four new Jacobi elliptic functions was used to the generalized fractional (3 + 1)-dimensional Kadomtsev–Petviashvili (GFKP) equation with the Atangana-Baleanu-Riemann fractional derivative, and abundant new types of analytical solutions to the GFKP were obtained. It is well known that there is a tight connection between symmetry and travelling wave solutions. Most of the existing techniques to handle the PDEs for finding the exact solitary wave solutions are, in essence, a case of symmetry reduction, including nonclassical symmetry and Lie symmetries etc. Some 3D plots, 2D plots, and contour plots of these solutions were simulated to reveal the inner structure of the equation, which showed that the efficient method is sufficient to seek exact solutions of the nonlinear partial differential models arising in mathematical physics. Full article

The purpose of this article is to achieve new soliton solutions of the Gilson–Pickering equation (GPE) with the assistance of Sardar’s subequation method (SSM) and Jacobi elliptic function method (JEFM). The applications of the GPE is wider because we study some valuable and vital equations such as Fornberg–Whitham equation (FWE), Rosenau–Hyman equation (RHE) and Fuchssteiner–Fokas–Camassa–Holm equation (FFCHE) obtained by particular choices of parameters involved in the GPE. Many techniques are available to convert PDEs into ODEs for extracting wave solutions. Most of these techniques are a case of symmetry reduction, known as nonclassical symmetry. In our work, this approach is used to convert a PDE to an ODE and obtain the exact solutions of the NLPDE. The solutions obtained are unique, remarkable, and significant for readers. Mathematica 11 software is used to derive the solutions of the presented model. Moreover, the diagrams of the acquired solutions for distinct values of parameters were demonstrated in two and three dimensions along with contour plots. Full article

Reductions make it possible to reduce the solution of a PDE to solving an ODE. The best known are the traveling wave, self-similar and symmetry reductions. Classical and non-classical symmetries are also used to construct reductions, as is the Clarkson–Kruskal direct method. Recently, authors have proposed a method for constructing reductions of PDEs with two independent variables based on the idea of invariance. The proposed method in this work is a modification of the Clarkson–Kruskal direct method and expands the possibilities for its application. The main result of this article consists of a method for constructing reductions that generalizes the previously proposed approach to the case of three independent variables. The proposed method is used to construct reductions of the unsteady axisymmetric boundary layer equation to ODEs and simpler PDEs. All reductions of this equation were obtained. Full article

The postulate of universal conformal (local Weyl scaling) symmetry modifies both general relativity and the Higgs scalar field model. The conformal Higgs model (CHM) generates an effective cosmological constant that fits the observed accelerating Hubble expansion for redshifts $z\le 1$ (7.33 Gyr) accurately with only one free parameter. Growth of a galaxy is modeled by the central accumulation of matter from an enclosing empty spherical halo whose radius expands with depletion. Details of this process account for the nonclassical, radial centripetal acceleration observed at excessive orbital velocities in galactic haloes. There is no need for dark matter. Full article

We present an experimental study on the optical quality of InAs/InP quantum dots (QDs). Investigated structures have application relevance due to emission in the 3rd telecommunication window. The nanostructures are grown by ripening-assisted molecular beam epitaxy. This leads to their unique properties, i.e., low spatial density and in-plane shape symmetry. These are advantageous for non-classical light generation for quantum technologies applications. As a measure of the internal quantum efficiency, the discrepancy between calculated and experimentally determined photon extraction efficiency is used. The investigated nanostructures exhibit close to ideal emission efficiency proving their high structural quality. The thermal stability of emission is investigated by means of microphotoluminescence. This allows to determine the maximal operation temperature of the device and reveal the main emission quenching channels. Emission quenching is predominantly caused by the transition of holes and electrons to higher QD’s levels. Additionally, these carriers could further leave the confinement potential via the dense ladder of QD states. Single QD emission is observed up to temperatures of about 100 K, comparable to the best results obtained for epitaxial QDs in this spectral range. The fundamental limit for the emission rate is the excitation radiative lifetime, which spreads from below 0.5 to almost 1.9 ns (GHz operation) without any clear spectral dispersion. Furthermore, carrier dynamics is also determined using time-correlated single-photon counting. Full article

The diffusive Lotka–Volterra system arising in an enormous number of mathematical models in biology, physics, ecology, chemistry and society is under study. New Q-conditional (nonclassical) symmetries are derived and applied to search for exact solutions in an explicit form. A family of exact solutions is examined in detail in order to provide an application for describing the competition of two species in population dynamics. The results obtained are compared with those published earlier as well. Full article

The dual XH (OH and CH) hydrogen-bond-donating property of 1,1,1,3,3,3-hexafluoroisopropanol (HFIP) and the strong dual XH–π interaction with arenes were firstly disclosed by theoretical studies. Here, the high accuracy post-Hartree–Fock methods, CCSD(T)/CBS, reveal the interaction energy of HFIP/benzene complex (−7.22 kcal/mol) and the contribution of the electronic correlation energy in the total interaction energy. Strong orbital interaction between HFIP and benzene was found by using the DFT method in this work to disclose the dual XH–π intermolecular orbital interaction of HFIP with benzene-forming bonding and antibonding orbitals resulting from the orbital symmetry of HFIP. The density of states and charge decomposition analyses were used to investigate the orbital interactions. Isopropanol (IP), an analogue of HFIP, and chloroform (CHCl₃) were studied to compare them with the classical OH–π, and non-classical CH–π interactions. In addition, the influence of the aggregating effect of HFIP, and the numbers of substituted methyl groups in benzene rings were also studied. The interaction energies of HFIP with the selected 24 common organic compounds were calculated to understand the role of HFIP as solvent or additive in organic transformation in a more detailed manner. A single-crystal X-ray diffraction study of hexafluoroisopropyl benzoate further disclosed and confirmed that the CH of HFIP shows the non-classical hydrogen-bond-donating behavior. Full article

In this paper, we consider a member of an integrable family of generalized Camassa–Holm (GCH) equations. We make an analysis of the point Lie symmetries of these equations by using the Lie method of infinitesimals. We derive nonclassical symmetries and we find new symmetries via the nonclassical method, which cannot be obtained by Lie symmetry method. We employ the multiplier method to construct conservation laws for this family of GCH equations. Using the conservation laws of the underlying equation, double reduction is also constructed. Finally, we investigate traveling waves of the GCH equations. We derive convergent series solutions both for the homoclinic and heteroclinic orbits of the traveling-wave equations, which correspond to pulse and front solutions of the original GCH equations, respectively. Full article

In order to be able to use methods of universal algebra for investigating posets, we assigned to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset, a certain algebra (based on a commutative directoid or on a $\lambda$-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and $\lambda$-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics. Full article