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An accurate description of the long-range (LR) interaction is essential for understanding the collision between cold or ultracold molecules. However, to our best knowledge, there lacks a general approach to construct the intermolecular potential energy surface (IPES) between two arbitrary molecules and/or atoms in the LR region. In this work, we derived analytical expressions of the LR interaction energy, using the multipole expansion of the electrostatic interaction Hamiltonian and the non-degenerate perturbation theory. To make these formulae practical, we also derived the independent Cartesian components of the electrostatic properties, including the multipole moments and polarizabilities, of the monomer for a given symmetry using the properties of these components and the group-theoretical methods. Based on these newly derived formulae, we developed a FORTRAN program, namely ABLRI, which is capable of calculating the interaction energy between two arbitrary monomers both in their non-degenerate electronic ground states at large separations. To test the reliability of this newly developed program, we constructed IPESs for the electronic ground state of H2O–H2 and O2–H systems in the LR region. The interaction energy computed by our program agreed well with the ab initio calculation, which shows the validity of this program. [ABSTRACT FROM AUTHOR]

An approach for identifying resonances in vibrational perturbation theory calculations is introduced. This approach makes use of the corrections to the wave functions that are obtained from non-degenerate perturbation theory calculations to identify spaces of states that must be treated with degenerate perturbation theory. Pairs of states are considered to be in resonance if the magnitude of expansion coefficients in the corrections to the wave functions in the non-degenerate perturbation theory calculation is greater than a specified threshold, χmax. This approach is applied to calculations of the vibrational spectra of CH4, H2CO, HNO3, and cc-HOONO. The question of how the identified resonances depend on the value of χmax and how the choice of the resonance spaces affects the calculated vibrational spectrum is further explored for H2CO. The approach is also compared to the Martin test [J. M. L. Martin et al., J. Chem. Phys. 103, 2589–2602 (1995)] for calculations of the vibrational spectra of H2CO and cc-HOONO. [ABSTRACT FROM AUTHOR]

Basic algebraic arguments demonstrate that the probability of radical pair recombination in low field for an arbitrary pair with Hamiltonian confined to Zeeman and isotropic hyperfine interactions contains two additive contributions linear with applied field with equal, but opposite in sign, proportionality factors. Their weights are determined by the probability of having all nuclear spins along the field in the initial electron-singlet state of the pair, and in the case of equilibrium with respect to nuclear spins, the two contributions completely compensate the field dependences of each other, producing an additive term in the singlet yield with zero derivative. However, if the nuclear set is polarized, a linear skew proportional to polarization appears, introducing anisotropy in the intrinsically spherically symmetric system. The key element in this derivation is guaranteed nondegeneracy of the eigenvalues of the "penultimate" (Mmax − 1) block of the Hamiltonian for a radical with any number of distinct spin- 1 / 2 nuclei in zero field, which leads to guaranteed applicability of the first-order nondegenerate perturbation theory with nonvanishing linear in field admixture of states, persisting all the way into the final expression for singlet yield for a pair with an arbitrary isotropic hyperfine structure. We argue that this behavior of the field dependence of recombination yield is representative of a radical pair of an arbitrarily complex hyperfine structure; this may be a possible mechanism for anisotropic response ("chemical compass") for an isotropic radical pair based system with isotropic-only internal interactions and anisotropy introduced via the initial state of nuclei, while such an approach may complement the usually needed numerical simulations. [ABSTRACT FROM AUTHOR]

We consider the canonical problem of magnetic field decay in an electrically conducting fluid ball. The problem is closely allied to the problem of the decay modes of a rigid ball, and the spatial form of the eigenmodes survives largely intact. The decaying but oscillatory behaviour of the new fluid eigenmodes first discovered by Schmitt a decade ago (and named quasi-free-decay (QFD) modes) is deduced by application of perturbation methods to the case of rapid rotation and a static applied background magnetic field that is uniform and axial. Some, but not all, of the rigid-case poloidal eigenmodes share decay rates with other toroidal modes, necessitating the use of both degenerate and non-degenerate perturbation theory within this paper. The perturbation theory is developed in terms of the Elsasser number Λ (measuring the competition between Coriolis and Lorentz forces), and the analytic results are in striking accord with numerical calculations even when Λ is of O(1). We find linear scaling of the QFD eigenfrequency with Λ and small changes in the decay rate that scale with Λ2. Although the modes are overdamped (quality factor Q<1), they are not strongly overdamped when the applied field is strong Λ∼1. [ABSTRACT FROM AUTHOR]

We solve the coupled recurrence relations for eigenenergies and -vectors in nondegenerate Rayleigh–Schrödinger perturbation theory under the constraint that the approximate eigenvector is normalised to 1 in every order. The series can be expressed in terms of diagrams that were first introduced by Bloch (Nucl Phys 6:329, 1958) for the degenerate, unnormalised case. Normalisation increases the number of terms and introduces a nontrivial dependence on the diagrams' topology to the coefficients. [ABSTRACT FROM AUTHOR]

In this study, a detailed investigation was done on the effects of changing the position of a donor impurity on the electronic and optical properties of an electron confined within a fixed size wurtzite (WZ) ZnS/ZnO/ZnS multishell quantum dot. We carried out the calculation using the framework of the effective mass approximation (EMA), relying entirely upon the non-degenerate quantum perturbation theory, using the unperturbed impurity-less eigenstates as expansion states. We computed the binding energy for the ground state and several other excited states as well as the transition dipole moment (TDM) along the z-direction for multiple electron state transitions. Finally, we concluded by computing the absorption coefficient for multiple transitions using perturbed electron states energy and transition dipole moments. The results show that the position of the donor impurity does have an impact on the electronic and optical properties of the confined electron. [ABSTRACT FROM AUTHOR]

We utilize pump/probe spectroscopy to determine the interband and intraband relaxation dynamics in InSb based quantum wells. Using non-degenerate pump/probe techniques, we observed several time scales for relaxation. One time scale τ3 ranging from 2 ps to 5 ps is due to the intraband relaxation dynamics. Here, both the emission of LO phonons (within the Γ valley) and carrier scattering between the X, L, and Γ valleys contribute to the relaxation. An observed longer relaxation time, τ2≈20 ps, is attributed to electron-hole recombination across the gap (the interband relaxation time). Finally, using a mid-infrared (MIR) degenerate pump/probe scheme, we observed a very fast relaxation time of ~1 ps, which is due to the saturation of the band-to-band absorption. Our results are important for developing concepts for InSb devices operating in the THz or MIR optical ranges with the endless need for faster response. [ABSTRACT FROM AUTHOR]

In this work, we present a rigorous mathematical scheme for the derivation of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by the potential Vper(x) = λxα, where α is a positive integer, using the non-degenerate time-independent perturbation theory. To do so, we derive a generalized formula for the integral I = ∫ − ∞ + ∞ x α ⁡ exp (− x 2 ) H n (x) H m (x) d x , where Hn(x) denotes the Hermite polynomial of degree n, using the generating function of orthogonal polynomials. Finally, the analytical results with α = 3 and α = 4 are discussed in detail and compared with the numerical calculations obtained by the Lagrange-mesh method. [ABSTRACT FROM AUTHOR]

Hydrogen iodide, a Hund's case (c) molecule, serves as a benchmark compound for studying rich molecular state interactions between Rydberg and valence states as well as between Rydberg states at high energies (72 300-74 600 cm-1) by mass resolved resonance enhanced multiphoton ionization (REMPI). Perturbations in the spectra appearing as deformations in line-positions, line-intensities, and linewidths are found to be either due to near-degenerate or non-degenerate interactions, both homogeneous and heterogeneous in nature. Perturbation analyses allow indirect observation as well as characterization of "hidden states" to some extent. Furthermore, new observable spectral features are assigned and characterized. [ABSTRACT FROM AUTHOR]

In this paper, we investigate helix and slant helices using non-degenerate curves in term of Sabban frame in de Sitter 3-space or Anti de Sitter 3-space M³(δ0). Furthermore, in M³(δ0) 3-space the necessary and sufficient conditions for the nondegenerate curves to be slant helix are given. [ABSTRACT FROM AUTHOR]

The second-order perturbative Stark effect on the ground state of hydrogen is a typical example presented in many standard texts on quantum mechanics. Some texts miss the fact that the scattering states are significant contributors to the perturbative energy correction. The inclusion of scattering states has wider applicability than to just the Stark effect. An explicit calculation involving a finite-square well with a perturbation is used to illustrate the importance of including scattering states into the calculation. The second-order correction to the ground-state energy is obtained in three distinct ways. The first involves altering the problem by imposing additional boundary conditions at large distances to make the positive-energy spectrum discrete. The second makes use of the continuum scattering states directly. The third bypasses the use of scattering states by solving a differential equation for the first-order correction to the wave function. [ABSTRACT FROM AUTHOR]

Observations of pressure-gravity mixed modes, combined with a theoretical framework for understanding mode formation, can yield a wealth of information about deep stellar interiors. In this paper, we seek to develop a formalism for treating the effects of deeply buried core magnetic fields on mixed modes in evolved stars, where the fields are moderate, i.e. not strong enough to disrupt wave propagation, but where they may be too strong for non-degenerate first-order perturbation theory to be applied. The magnetic field is incorporated in a way that avoids having to use this. Inclusion of the Lorentz force term is shown to yield a system of differential equations that allows for the magnetically affected eigenfunctions to be computed from scratch, rather than following the approach of first-order perturbation theory. For sufficiently weak fields, coupling between different spherical harmonics can be neglected, allowing for reduction to a second-order system of ordinary differential equations akin to the usual oscillation equations that can be solved analogously. We derive expressions for (i) the mixed-mode quantization condition in the presence of a field and (ii) the frequency shift associated with the magnetic field. In addition, for modes of low degree we uncover an extra offset term in the quantization condition that is sensitive to properties of the evanescent zone. These expressions may be inverted to extract information about the stellar structure and magnetic field from observational data. [ABSTRACT FROM AUTHOR]

We investigate electric field control of spin manipulation through Berry phase in III-V semiconductor quantum dots. By utilizing degenerate and non-degenerate perturbation theories, we diagonalize the total Hamiltonian of a semiconductor quantum dot and express the solution of time dependent Schrödinger equation in terms of complete and incomplete elliptic integrals of the second kind, respectively. This allows us to investigate the interplay between the Rashba and Dresselhaus spin-orbit couplings. In particular, we provide theoretical descriptions of several novel properties focusing on spin manipulation through (a) Berry phase, (b) geometric phase and (c) spin echo phenomenon followed by a strong beating patterns during the adiabatic transport of the quantum dots. [ABSTRACT FROM AUTHOR]

Abstract In [9] Etingof, Schedler and Soloviev introduced, for each non-degenerate involutive set-theoretical solution (X , σ , τ) of the Yang–Baxter equation, the equivalence relation ∼ defined on the set X and they considered a new non-degenerate involutive induced retraction solution defined on the quotient set X ∼. It is well known that translating set-theoretical non-degenerate solutions of the Yang–Baxter equation into the universal algebra language we obtain an algebra called a birack. In the paper we introduce the generalized retraction relation ≈ on a birack, which is equal to ∼ in an involutive case. We present a complete algebraic proof that the relation ≈ is a congruence of the birack. Thus we show that the retraction of a set-theoretical non-degenerate solution is well defined not only in the involutive case but also in the case of all non-involutive solutions. [ABSTRACT FROM AUTHOR]

Abstract Through desingularization of Clifford torus, we prove the existence of a sequence of nondegenerate (in the sense of Duyckaerts–Kenig–Merle ([8])) nodal nonradial solutions to the critical Yamabe problem − Δ u = n (n − 2) 4 | u | 4 n − 2 u , u ∈ D 1 , 2 (R n). The case n = 4 is the first example in the literature of a solution with maximal rank N = 2 n + 1 + n (n − 1) 2. [ABSTRACT FROM AUTHOR]

Abstract We carry out a model study on the interaction of a V -type three-level emitter with two quantized cavity modes which are weakly driven by two classical fields. The emitter may be an atom or a molecule with nondegenerate upper levels in general. The lifetimes of the two upper levels are assumed to be much longer than the lifetime of the cavity photons. We calculate the two-time second order coherence function, namely Hanbury Brown–Twiss function g (2) (τ) where τ is the time delay between the two modes. We analyze the photon–photon correlations between the two cavity modes in terms of g (2) (τ). The variation of g (2) (τ) as a function of τ exhibits collapse and revival type oscillations as well as quantum beats for relatively short τ while in the limit τ → ∞ , g (2) (τ) ≃ 1. We further show that the two cavity field modes are entangled when g (2) (0) > 2. We develop a dressed state picture with single photon in each mode to explain the results and use the negativity of the partial transpose of the reduced field density matrix to show the entanglement between field modes. The model presented in this paper may be useful for generating entangled photon pairs, and also manipulating photon–photon correlations with a cavity QED set up. [ABSTRACT FROM AUTHOR]

Abstract We consider the Gierer–Meinhardt system with small inhibitor diffusivity and very small activator diffusivity in a bounded and smooth two-dimensional domain. For any given positive integer k we construct a spike cluster consisting of k boundary spikes which all approach the same nondegenerate local maximum point of the boundary curvature. We show that this spike cluster is linearly stable. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive and the spikes are attracted towards a nondegenerate local maximum point of the boundary curvature. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable. [ABSTRACT FROM AUTHOR]

We show how non-degenerate three wave mixing with two of the optical modes asymmetrically pumped can exhibit interesting behaviours as well as producing bright entangled states with a frequency difference of up to an octave. In the travelling wave configuration, the system shows similar behaviours to the two-level Jaynes–Cummings model, with an intense field at the frequency of the difference between the other two mediating population exchange between these. The resulting dynamics are similar to Rabi oscillations. When the system is placed inside a pumped optical cavity, the fields enter a steady state in which entanglement and EPR-steering are both present. Such a system adds to the toolbox of methods used for quantum information tasks. [ABSTRACT FROM AUTHOR]

The certificateless aggregate signature (CLAS) scheme is a very important data aggregation technique that compresses a large number of signatures from different users into a short signature. CLAS can reduce the total length of a signature and the computational overhead of signature verification and is therefore highly suitable for resource-constrained network environments. Many CLAS schemes have been proposed in recent years, but the construction of a secure and efficient CLAS scheme remains important. In 2018, Li et al. found that the CLAS scheme proposed by He et al. could not resist malicious-but-passive KGC attacks, and they presented an improved CLAS scheme. Du et al. proposed a CLAS scheme with the constant aggregate signature length and claimed that their scheme was resistant to forgery attacks. Chen et al. designed a CLAS scheme with efficient verification and proved that their CLAS scheme was secure in the random oracle model. In this paper, we demonstrate that Li et al.’s CLAS scheme, Du et al.’s CLAS scheme, and Chen et al.’s CLAS scheme are insecure against coalition attacks and present concrete examples. That is, an attacker can forge a valid aggregate signature using some illegal single signatures. To withstand suck attacks, we propose an improved CLAS scheme based on Chen et al.’s CLAS scheme. [ABSTRACT FROM AUTHOR]

This note is devoted to studying the divisibility relation unk+1|um for a least positive integer m, where {un}n≥0 is a nondegenerate Lucas sequence with characteristic polynomial x² - ax - b, for some relatively prime integers a and b. [ABSTRACT FROM AUTHOR]

We solve exactly the master equation of the degenerate and nondegenerate parametric oscillator in the presence of a squeezed reservoir. We show that the formal solutions of both systems can be obtained in terms of the density operator by applying squeeze transformations and using the superoperators formalism. [ABSTRACT FROM AUTHOR]

We present a proof of the local Hölder regularity of the horizontal derivatives of weak solutions to the p-Laplace equation in the Heisenberg group ℍ1 for p > 4. [ABSTRACT FROM AUTHOR]

The aim of the paper is to give the numerical criteria for classification of different types of hexahedral cells which can emerge in a three-dimensional structured grid generation. In general, computational grids and their cells have to be nondegenerate, however, in practice, situations arise in which degenerate grids are used and computed. In these cases, to prevent lost of accuracy, special strategies must be chosen both in grid generation and physical phenomenon solution algorithms. To determine which cells need a modification in above strategies, degenerate cells have to be detected. The criteria are suggested for hexahedral cells constructed by a trilinear mapping of the unit cube. All hexahedral cells are divided into nondegenerate and degenerate. Among nondegenerate hexahedral cells, cells exotic in shape are singled out as inadmissible. Degenerate cells are divided into pyramids, prisms and tetrahedrons—types of cells which can be admissible in grid generation and solution algorithms. Inadmissible types of degenerations are also considered. An algorithm for testing three-dimensional structured grids according to suggested criteria is described. Both results of testing and examples of different types of cells are demonstrated. In conclusion, recommendations for structured grid generation with the purpose to exclude undesirable types of cells are given. [ABSTRACT FROM AUTHOR]

The Hamiltonian and hence the equation of motion of the field operators of a non-degenerate parametric oscillator (NDPO) under the influence of classical pump are formulated. The coupled operator differential equations involving the signal and idler modes are decoupled at the expense of fourth order differential equations involving the c-numbers. Without using the rotating wave approximation, the analytical solutions of the field operators are obtained. These solutions are approximated up to the second orders in dimensionless coupling constant. We investigate the squeezing, mixed mode squeezing, amplitude-squared squeezing, and the principal squeezing of the thermal and coherent light coupled to the NDPO. By using the input composite number state, we establish that the percentage and the range (interaction time) of squeezing is considerably increased with the increase of the signal photon number. For initial composite number state, the amplitude squared squeezing for Y s quadrature is obtained at the cost of canonically conjugate Z s quadrature. The percentage of amplitude squared squeezing increases significantly with the increase of signal excitation (photon). The so-called normal squeezing and the principal squeezing are also indicated for the NDPO coupled with the initially prepared composite coherent states not in the composite number states. In spite of the shortcomings of the analytical solutions, we obtain squeezing, amplitude squared squeezing and other nonclassical effects which are unavailable under the rotating wave approximation. In order to give the feelings about the analytical results (expressions), we give some symbolic calculations relevant to the possible experimental situations. [ABSTRACT FROM AUTHOR]

Nondegenerate optical parametric oscillator (NOPO) above threshold is a simple set-up to generate entangled beams. To more accurately calculate the entanglement between the generated twin beams from the NOPO, we use the Langevin equations for the NOPO with a higher-transmissivity cavity mirror. The obtained new expressions for the amplitude-difference and phase-sum noise spectra suggest that increasing the transmissivity of the output mirror can enhance the entanglement beyond the previous expectation. [ABSTRACT FROM AUTHOR]

We present an experimental study of seeded four-wave mixing (4WM) using a diamond excitation scheme (with states from theandterms) in a thermal vapour ofatoms. We investigate the 4WM spectra under the application of a strong magnetic field (0.6 T). The Zeeman interaction is strong enough to realize the hyperfine Paschen–Back regime, which has the effect of separating the optical transitions by more than the Doppler width, thereby significantly simplifying the spectral features. We show that this facilitates a quantitative comparison, even in the regime of strong dressing, between experimental data and a simple theoretical model based only on four-level optical Bloch equations. [ABSTRACT FROM AUTHOR]

Let h,hM : Fn q → Fq be quadratic forms with h not degenerate. Fix k ∊ Fq and set Cn(k,h)Fq := {h(x1, . . . ,xn) = k} = Fnq. We compute (in many cases) the image of hMCn (k,h)Fq . This question is related to a question on the numerical range of matrices over a finite field. [ABSTRACT FROM AUTHOR]

We show that for uniform domains Ω⊆Rd+1 whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to α-dimensional Hausdorff measure unless α≤d. We employ a lemma that shows that at almost every nondegenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure. [ABSTRACT FROM AUTHOR]

In this article, we consider the spectral problem for an nth-order ordinary differential operator with degenerate boundary conditions. For even n, we construct nontrivial examples of boundary-value problems which have nonclassical asymptotics on the spectrum. [ABSTRACT FROM AUTHOR]

Let R be a commutative and unital semi-local ring in which 2 is invertible. In this note, we show that anisotropic quadratic spaces over R remain anisotropic after base change to any odd-degree finite étale extension of R. This generalization of the classical Artin-Springer theorem (concerning the situation where R is a field) was previously established in the case where all residue fields of R are infinite by I. Panin and U. Rehmann. The more general result presented here permits one to extend a fundamental isotropy criterion of I. Panin and K. Pimenov for quadratic spaces over regular semi-local domains containing a field of characteristic ≠ 2 to the case where the ring has at least one residue field which is finite. [ABSTRACT FROM AUTHOR]

The impact of spatial limitation of the nonlinear interaction volume on the spontaneous parametric down-conversion spectra in a strongly nondegenerate regime is analyzed from the point of absolute calibration of the terahertz-wave nonlinear-optical detectors. We show that the idler wave angular resolution of the calibration method can be inherently low (about 1 to 10 degrees in lithium niobate crystals), and to fill all the input modes with external idler radiation, one has to take special care on geometry of the nonlinear interaction. Angular sensitivity distribution function is constructed, and a consecutive description is provided for both the parametric down-conversion and the external idler radiation detection processes. [ABSTRACT FROM AUTHOR]

We investigate the p -adic valuation of Weil sums of the form W F , d ( a ) = ∑ x ∈ F ψ ( x d − a x ) , where F is a finite field of characteristic p , ψ is the canonical additive character of F , the exponent d is relatively prime to | F × | , and a is an element of F . Such sums often arise in arithmetical calculations and also have applications in information theory. For each F and d one would like to know V F , d , the minimum p -adic valuation of W F , d ( a ) as a runs through the elements of F . We exclude exponents d that are congruent to a power of p modulo | F × | (degenerate d ), which yield trivial Weil sums. We prove that V F , d ≤ ( 2 / 3 ) [ F : F p ] for any F and any nondegenerate d , and prove that this bound is actually reached in infinitely many fields F . We also prove some stronger bounds that apply when [ F : F p ] is a power of 2 or when d is not congruent to 1 modulo p − 1 , and show that each of these bounds is reached for infinitely many F . [ABSTRACT FROM AUTHOR]

Let d ⩾ 3 be an integer, and set r = 2 + 1 for 3 ⩽ d ⩽ 4, $$\tfrac{{17}} {{32}} \cdot 2^d + 1$$ for 5 ⩽ d ⩽ 6, r = d + d+1 for 7 ⩽ d ⩽ 8, and r = d + d+2 for d ⩾ 9, respectively. Suppose that Φ( x, y) ∈ ℤ[ x, y] (1 ⩽ i ⩽ r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ, λ,..., λ are nonzero real numbers with λ/λ irrational, and λΦ( x , y ) + λΦ( x , y ) + · · · + λΦ( x , y ) is indefinite. Then for any given real η and σ with 0 < σ < 2, it is proved that the inequality has infinitely many solutions in integers x , x ,..., x , y , y ,..., y . This result constitutes an improvement upon that of B. Q. Xue. [ABSTRACT FROM AUTHOR]

We provide a criterion in order to decide the stability of non-degenerate equilibrium states of completely integrable systems. More precisely, given a Hamilton–Poisson realization of a completely integrable system generated by a smooth n -dimensional vector field, X , and a non-degenerate regular (in the Poisson sense) equilibrium state, x ‾ e , we define a scalar quantity, I X ( x ‾ e ) , whose sign determines the stability of the equilibrium. Moreover, if I X ( x ‾ e ) > 0 , then around x ‾ e , there exist one-parameter families of periodic orbits shrinking to { x ‾ e } , whose periods approach 2 π / I X ( x ‾ e ) as the parameter goes to zero. The theoretical results are illustrated in the case of the Rikitake dynamical system. [ABSTRACT FROM AUTHOR]

This paper deals with the study of the behavior of the wave functions of a two-component Bose-Einstein condensate near the interface, in the case of strong segregation. This yields a system of two coupled ordinary differential equations for which we want to have estimates on the asymptotic behavior, as the strength of the coupling tends to infinity. As in phase separation models, the leading order profile is a hyperbolic tangent. We construct an approximate solution and use the properties of the associated linearized operator to perturb it into a genuine solution for which we have an asymptotic expansion. We prove that the constructed heteroclinic solutions are linearly nondegenerate, in the natural sense, and that there is a spectral gap, independent of the large interaction parameter, between the zero eigenvalue (due to translations) at the bottom of the spectrum and the rest of the spectrum. Moreover, we prove a uniqueness result which implies that, in fact, the constructed heteroclinic is the unique minimizer (modulo translations) of the associated energy, for which we provide an expansion. [ABSTRACT FROM AUTHOR]

The essence of the rotating wave approximation (RWA) is to eliminate the non-conserving energy terms from the interaction Hamiltonian. The cost of using RWA is heavy if the frequency of the input radiation field is low (e.g. below optical region). The well known Bloch-Siegert effect is the out come of the inclusion of the terms which are normally neglected under RWA. We investigate the fluctuations of the quantum phase of the coherent light and the thermal light coupled to a nondegenerate parametric oscillator (NDPO). The Hamiltonian and hence the equations of motion involving the signal and idler modes are framed by using the strong (classical) pump condition. These differential equations are nonlinear in nature and are found coupled to each other. Without using the RWA, we obtain the analytical solutions for the signal and idler fields. These solutions are obtained up to the second orders in dimensionless coupling constants. The analytical expressions for the quantum phase fluctuation parameters due to Carruther's and Nieto are obtained in terms of the coupling constants and the initial photon numbers of the input radiation field. Moreover, we keep ourselves confined to the Pegg-Barnett formalism for measured phase operators. With and without using the RWA, we compare the quantum phase fluctuations for coherent and thermal light coupled to the NDPO. In spite of the significant departures (quantitative), the qualitative features of the phase fluctuation parameters for the input thermal light are identical for NDPO with and without RWA. On the other hand, we report some interesting results of input coherent light coupled to the NDPO which are substantially different from their RWA counterpart. In spite of the various quantum optical phenomena in a NDPO, we claim that it is the first effort where the complete analytical approach towards the solutions and hence the quantum phase fluctuations of input radiation fields coupled to it are obtained beyond rotating wave approximation. To have the feelings of the analytical solutions, we give few numerical estimates of the quantum phase fluctuation parameters relevant to a real experimental situation. [ABSTRACT FROM AUTHOR]

The nonlinear systems obtained by discretizing degenerate parabolic equations may be hard to solve, especially with Newton's method. In this paper, we apply to the Richards equation, a strategy that consists of defining a new primary unknown for the continuous equation in order to stabilize Newton's method by parametrizing the graph linking the pressure and the saturation. The resulting form of the Richards equation is then discretized thanks to a monotone finite volume scheme. We prove the well-posedness of the numerical scheme. Then we show under appropriate nondegeneracy conditions on the parametrization that Newton's method converges locally and quadratically. Finally, we provide numerical evidences of the efficiency of our approach. [ABSTRACT FROM AUTHOR]

The large order Rayleigh–Schrödinger perturbation theory (RSPT) was applied for calculating vibrational states of linear molecules. Two molecules (CO 2 and C 2 H 2) were used as test cases with using of isomorphic Watson Hamiltonian and quartic force fields. For CO 2 the Sayvetz condition can remove all degeneracies for purely vibrational states and the non-degenerate perturbation theory can be applied. However, an existence of two degenerate modes in C 2 H 2 requires using the upgraded degenerate version of RSPT that was employed in this context for the first time. The dominating divergent behavior of such series requires the resummation technique that mimics the multivalued nature of the underlying solutions, and the applied quartic Padé–Hermite approximants (QPHA) provided full solution of the problem. Moreover, some mathematical properties of QPHA proved to be an efficient tool for studying resonance effects through the Katz theorem that controls the singular points of the eigenvalues on the complex plane. In the case of C 2 H 2 , not only all earlier observed classical resonances were confirmed and quantified, but also subtle interpolyad resonances ( K 2 / 55 , K 3 / 4555 ), proposed recently by Herman (2011) were described as well. Following the analysis, we found several novel resonances, of which we proposed one independent interpolyad resonance K 2 / 4444. The complete analysis of such critical points provided the full resonance picture of all studied molecules. [Display omitted] • Singularity of eigenvalues is obtained from a resummation of perturbation series. • Analysis of singularity provides a novel way to understand vibrational resonances. • Degenerate RSPT extends the scope of the method to all classes of semi-rigid molecules. • The resonance structures of CO 2 and C 2 H 2 are fully revealed by the applied method. [ABSTRACT FROM AUTHOR]

We prove existence, non-degeneracy, and exponential decay at infinity of a non-trivial solution to Emden's equation on an unbounded -shaped domain, subject to Dirichlet boundary conditions. Besides the direct value of this result, we also regard this solution as a building block for solutions on expanding bounded domains with corners, to be established in future work. Our proof makes heavy use of computer assistance. Starting from a numerical approximate solution, we use a fixed-point argument to prove existence of a near-by exact solution. The eigenvalue bounds established in the course of this proof also imply non-degeneracy of the solution. [ABSTRACT FROM AUTHOR]

This paper is concerned with the exponential convergence rates of solutions to non-degenerate traveling fronts in some exponentially weighted spaces and the algebraic convergence rates of solutions to non-degenerate and degenerate traveling fronts in some algebraically weighted spaces forquasi-linear relaxation systems with small viscous using weighted energy method, where the small initial perturbations of the traveling fronts are in some appropriate weighted spaces and the wave strengths are small. [ABSTRACT FROM AUTHOR]

The Fourier-Bessel (FFB) numerical solver is a useful tool for obtaining the steady states of resonator structures that conform to a cylindrical symmetry. Recently the FFB solver has been greatly simplified by reconfiguring the matrix generating expressions using Maxwell's curl expressions rather than the standard wave equations. This presentation provides a numerical framework suitable for the application on non-degenerate perturbation theory within the theoretical structure of the reconfigured FFB computation environment. It is shown that the resonator structure's perturbation contribution can be isolated as a separate matrix which dictates the shift in resonator state properties. Two distinct application examples are provide; the first has the perturbation possess the same rotational symmetry as the original structure and preserves azimuthal mode order families; the second perturbation has a symmetry different than the original structure and promotes a mixing between azimuthal mode order families. The perturbation extension promises to amplify the potential usefulness of the FFB technique when theoretically considering photonic sensors such as whispering-gallery mode, photonic crystal hole infiltration and a host of others in which the measurand undergoes small changes in its optical properties. [ABSTRACT FROM AUTHOR]

This paper deals with left non-degenerate set-theoretic solutions to the Yang–Baxter equation (= LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution there is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties. [ABSTRACT FROM AUTHOR]

We give a new geometric construction of the big projective module in the principal block of the BGG category $$\mathscr {O}$$ , or rather the corresponding $$\mathscr {D}$$ -module on the flag variety. Namely, given a one-parameter family of nondegenerate additive characters of the unipotent radical of a Borel subgroup which degenerate to the trivial character, there is a corresponding one-parameter family of Whittaker sheaves. We show that the unipotent nearby cycles functor applied to this family yields the big projective $$\mathscr {D}$$ -module. [ABSTRACT FROM AUTHOR]

We present a new analytical solution for Fokker–Planck equation of phase-mismatched frequency non-degenerate optical parametric amplifier (FNOPA). We investigate the squeezing and continuous variable (CV) entanglement characteristic of a FNOPA working at phase-mismatched condition. We numerically calculate the dependence of the CV entanglement on the relative loss parameterξand the phase-mismatched parameter. The smallercan bring a long-time entanglement. This method can also be applied to multipartite entanglement situation. [ABSTRACT FROM AUTHOR]

In this paper, we have investigated the effects of Bohm potential on the low frequency wave propagation in a dense non-degenerate dusty magnetoplasma. We have employed two potential theory to study the quantum modification of kinetic Alfvén wave on a dust acoustic velocity branch. The linear dispersion relation and growth rate of instability of dust kinetic Alfvén wave are obtained by incorporating Bohm quantum potential in the linearized Vlasov model. It is found that unlike in classical plasma, the presence of CQ which arises through the Bohm potential, tends to enhance the dispersion of the particles, while the growth rates are suppressed. The importance and relevance to compact astrophysical objects and space dusty plasma environments is also pointed out. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of doubly-connected rotating patches for Euler equations when the classical non-degeneracy conditions are not satisfied. We prove the bifurcation of the V-states with two-fold symmetry, however for higher m -fold symmetry with m ≥ 3 the bifurcation does not occur. This answers to a problem left open in [10] . Note that, contrary to the known results for simply-connected and doubly-connected cases where the bifurcation is pitchfork, we show that the degenerate bifurcation is actually transcritical. These results are in agreement with the numerical observations recently discussed in [10] . The proofs stem from the local structure of the quadratic form associated to the reduced bifurcation equation. [ABSTRACT FROM AUTHOR]

A boundary control problem for a nonlinear steady-state heat transfer model accounting for heat radiation effects is considered. The problem consists in the minimization of a cost functional by controlling the reflection properties of the boundary. The solvability of the control problem is proven, an optimality system is derived, and the nondegeneracy of optimality conditions is established. The results of numerical simulations are presented. [ABSTRACT FROM AUTHOR]

We first revisit the question of existence and uniqueness of maximal weight solutions of the truncated trigonometric matrix moment problem (Problem (TTM)) in the nondegenerate case. The starting point of the consideration is a specialized representation of the matrix weight assigned at each point on the unit circle for the solutions of (nondegenerate) Problem (TTM). Our approach is essentially self-contained and different from the existing methods. Moreover, based on this approach we also derive some important weight features of the so-called N-extremal solutions within the whole solution set of (nondegenerate) Problem (TTM) and the discrete structure of them. In the second part, these observations inspire us to look for the corresponding weight features of the solutions of a (nondegenerate) moment problem for rational matrix-valued functions (Problem (R)), where only the theory of orthogonal matrix polynomials on the unit circle is involved. [ABSTRACT FROM AUTHOR]

We prove that if E and F are large ideals of B(G) for which the associated coaction functors are exact, then the same is true for E ∩ F. We also give an example of a coaction functor whose restriction to the maximal coactions does not come from any large ideal. [ABSTRACT FROM AUTHOR]