Mixed lump-stripe,bright rogue wave-stripe,dark rogue wave-stripe and dark rogue wave solutions of a generalized Kadomtsev–Petviashvili equation in fluid mechanics

Under investigation in this paper is a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation in fluid mechanics. We derive the mixed lump-stripe waves, bright mixed rogue wave-stripe, dark mixed rogue wave-stripe and dark rogue waves solutions by virtue of the symbolic computation. We observe the fission and fusion phenomena between the lump and one-stripe wave through the mixed-stripe wave solutions. Then, we observe that the influence of l₁, l₂, l₃, l₄, l₅, l₆, l₇ and l₈ on the mixed lump-stripe waves, where l₁ and l₂ represent the dispersion and nonlinear effects, l₃, l₆, l₇ and l₈ are the perturbed effects, while l₄ and l₅ stand for the disturbed wave velocities along the transverse spatial coordinates y and z, respectively. We graphically present the interaction between a rogue wave and a pair of stripe waves through the mixed rogue wave-stripe solutions. We derive a dark mixed rogue wave-stripe when l₁ < 0. We study the influence of l₁, l₂, l₃, l₄, l₅, l₆, l₇ and l₈ on the rogue wave and a pair of stripe waves. We present the dark rogue wave with certain parameters and observe that two stripe waves merge into one stripe wave.     

Symmetric and traceless tensors on Minkowski space

The symmetric traceless projection of a tensor of rank 2l on Minkowski space is determined. These tensors form an invariant subspace under transformations by the 2l-fold product of an element of the Lorentz group SO₀(1, 3). As is well known, this representation is irreducible and equivalent to the representation (l₁,l₂) of the group $\text{SL(}\text{2,C}\text{)}$ for l₁ = l₂ = l. We derive the explicit form of the equivalence operator and its main properties. With these investigations we extend results obtained by S. Weinberg.     

Relativistic shift of the linear Zeeman effect

The efficiency with which a magnetic field acts on sublevels of the first and second doublets of a hydrogen-like atom is shown to vary because of relativistic effects by a factor of 1 ? 2γ(l + 1)/(2l + 3) and 1 ? 2γl/(2l ? 1), respectively, where γ is the ratio of the electron binding energy to the rest energy and l is the quantum number of the orbital angular momentum.     

The perturbative approach to the critical behaviour of two-matrix models in the limit N → ∞

We construct representations of the Heisenberg algebra by pushing the perturbation expansion to high orders. If the multiplication operators B_1,2 tend to differential operators of order l_2,1, respectively, the singularity is characterized by (l₁, l₂). Let l₁ ⩾ l₂. Then the two cases, (A) “l₂ does not divide l₁” and (B) “l₂ divides l₁”, need a different treatment. The universality classes are labelled [p, q] where [p, q] = [l₁,l₂] in case (A) and [p, q]=[l₁ + 1,l₂] in case (B).     

Investigation of nucleon electromagnetic form factors in the unphysical region by means of the ̄NN → πl + l − reactions

A theoretical investigation of ?NN → πl + l ? processes is carried out. First, the general structure of the differential annihilation rate of very slow antinucleons on nucleons at rest into pion and lepton pairs is derived, then the structure of the electromagnetic current of the ?NN → πγ* transition in the case of S-state annihilation is restored and the general properties of the corresponding form factors are demonstrated. Next, by using the treediagram approximation of the amplitude, those form factors are calculated explicitly and for the special process (?pp) → π⁰γ* → π⁰ l ⁺ l ^? they are shown to be completely described by the magnetic form factor of the proton in the unphysical region. Finally, the effective mass spectra of lepton pairs and the integral coefficients of internal conversion for the ?pp → π⁰ l + l ^? and ?pn → π^? l ⁺ l ^? processes are predicted.     

Localization and transmission coefficient for two coupled metal chains with disorder

The equation for distribution of probabilities for the transmission coefficient T has been obtained for an electron passing through the finite section of the length L, consisting of two coupled disordered chains. The behavior of the mean 〈lnT〉 = - L/l_loc suggests localization of the electron on the length l, which depends on the coupling energy t between the chains. The ratio l_loc(t =)/l_loc(t?v_F/l) is found to be equal to 1 ? 1/π.     

Application of optimization methods to vibration control of stretched strings

In this paper, non-linear programming techniques are applied to the problem of controlling the vibration pattern of a stretched string. First, the problem of finding the magnitudes of two control forces applied at two points l₁ and l₂ on the string to reduce the energy of vibration over the interval (l₁, l₂) relative to the energy outside the interval (l₁, l₂) is considered. For this problem the relative merits of various methods of non-linear programming are compared. The more complicated problem of finding the positions and magnitudes of two control forces to obtain the desired energy pattern is then solved by using the slack unconstrained minimization technique with the Fletcher-Powell search. In the discussion of the results it is shown that the position of the control force is very important in controlling the energy pattern of the string.     

Structural features of a Lennard-Jones system at melting and crystallization

The structural features of the crystallization and melting of a system of particles whose pair interaction is described by the Lennard-Jones potential have been considered. The bond order parameter method is used to quantitatively describe the orientational short-range order. The rotational invariants of the second (q _l ) and third (w _l ) orders are calculated for each particle of the system. These calculations require only information on the snapshot of atom positions, which is quite easily obtained in experiments, and provide the distribution functions of particles in q _l and w _l (where l is the rank of an invariant; the results for l = 4, 6 are presented), which are important characteristics of the phase state of the system. It has been shown that the cumulant of the distribution of particles in w ₆ is very sensitive to the destruction/formation of the short-range orientational order in the Lennard-Jones system and, correspondingly, can be used as a criterion of the melting and crystallization of this system.     

Magnetization process in cluster spin glass model in the low temperature region

An ensemble of magnetic clusters is approximated by an ensemble of two-singlet systems: at each lattice site l there are two levels, ?_l,1 and ?_l,2. ?_l,1 is associated with spin zero and ?_l,2 with spin 1 (or vice versa). Δ = ?_l,2??_l,1 originates from some intra-cluster interaction and some effective magnetic field; for the both some distribution is assumed. An exchange interaction between different clusters is also taken into account. The excitation spectrum of this ensemble as function of H_ext is calculated and applied to discuss the response of the system to some external magnetic field H_ext in the low-temperature region. Typical instabilities of magnetization as function of H_ext are obtained. Finally, possible applications of our results to spin-glass systems are discussed.     

Breaking of the SO(n) symmetry with vector and adjoint higgses

The spontaneous symmetry breaking of SO(n) is investigated by studying the most general quartic SO(n)-invariant Higgs potential with two multiplets of scalars belonging to a vector and to an adjoint representation. In the most general cases largest residual symmetry is found to be SU(l), SU(l-1), SO(l), SO(l-1) or SO(n-2) wherel is the rank of SO(n). In particular, the breaking of SO(n) into SU(n ₁)×SU(n ₂) is found to occur only in special cases.     

Anisotropic longitudinal electron diffusion coefficient in wurtzite gallium nitride

The longitudinal electron diffusion coefficient (D _l ) of wurtzite (WZ) gallium nitride (GaN) is calculated by an ensemble Monte Carlo (EMC) method. By using the power spectral density associated with velocity fluctuation, the relationship between D _l and electric field strength, frequency, doping concentration and temperature is presented. The anisotropic D _l of GaN impacted by anisotropy of the electronic dispersion is also investigated. It has been found that the D _l in Γ–A direction (c-direction) is larger than that in Γ–M direction (basal plane) in most cases. For lower electric field, the D _l keeps constant at first, then decreases with increasing frequency. However, for higher electric field, the D _l firstly approaches a peak value, then decreases with increasing frequency. When the frequency is zero, the D _l decreases with the increasing electric field, and then increases until a peak value. Finally, it decreases with increasing electric field again. When the temperature increases, the D _l decreases in both directions for increasing scattering rate. A comparison between our calculated diffusion coefficient and the mobility under low electric field by Einstein equation is presented.     

2-mm resonance oscillator operating on the second spatial harmonic of a two-row periodic orotron structure

Orotron generation in the resonance BWO mode on the second spatial harmonic of a two-row periodic structure with the parameters l= 0.6 mm and d/l ～ 0.8 (l is the period and d is the groove width) has been observed for the first time. With a change in the voltage from 4.7 to 11 kV and the corresponding adjustment of the open resonator, frequency tuning in the range 94–140 GHz was obtained.     

Parabolic drift towards homogeneity in large-scale structures of galaxies

To describe the progressive transition in large-scale structures of galaxies from a seemingly fractal behavior at small scales to a homogeneous distribution at large scales, we use a new geometrical framework called entropic-skins geometry which is based on a diffusion equation of scale entropy through scale space. In the case of an equipartition of scale entropy losses in scale space, it is shown that fractal dimension (varying from 0 to 3) depends linearly on the logarithm of scale from the average size l_c of galaxies until a characteristic length scale l₀ beyond which distribution becomes homogeneous. A simple parabolic expression for correlation function can be derived: ln(1+ξ_i)=(β/2)ln²(l_o/l_i) with β=3/ln(l₀/l_c)≈0.32 and . This law has been verified using correlation functions measured on several redshift surveys.     

Role of the substrate thickness for the structural properties of organic-organic heterostructures

F₁₆CuPc deposited on pentacene is characterized by the coexistence of two different configurations: F₁₆CuPc is found in the standing up phase (“s-configuration”) on top of pentacene terraces and in a lying down phase (“l-configuration”) at pentacene step edges. By combining AFM and grazing incidence X-ray diffraction we show that the ratio between F₁₆CuPc in l- and s-configurations increases with thickness of the pentacene substrate film, demonstrating the role of the pentacene steps as nucleation centers for the F₁₆CuPc l-configuration. Experiments performed with ultra-thin pentacene thicknesses disclose that the F₁₆CuPc l-configuration does not grow on top of the first and second pentacene layers, pointing to the action of long-range interactions with the substrate.     

The nonparaxial propagation of the TEM0l doughnut beam with the orbital angular momentum in the far field

The analytical expression for nonparaxial electric field amplitude of the TEM_0l^? doughnut beam with the orbital angular momentum quantum number l is derived in the far field by means of the angular spectrum representation and the stationary phase method. It is shown that the divergent angle of the far field of TEM_0l^? doughnut beam will be smaller with the decreasing of the orbital angular momentum quantum number l or the increasing of the beam waist width w of the initial beam. And the maximal radial intensity of the beam is decreased with its propagation at different rates for different l and w.     

Quantitative investigation of critical slowing down in all-optical bistability

We have observed optical bistability in rubidium vapor enclosed in a Fabry-Perot cavity for λ ～ 7779 Å (two-photon transition) and λ ～ 7800 Å (one-photon transition). We have measured the switching time τ_S as a function of (l?l_c) where l_c is the last value of the length of the cavity for which switching occurs. In all the experimental situations, we have found a similar law τ_S ～ (l?l_c)^-α with α ～ 0.5. This result is in agreement with theoretical predictions.     

Improved dynamic MRI reconstruction by exploiting sparsity and rank-deficiency

In this paper we address the problem of dynamic MRI reconstruction from partially sampled K-space data. Our work is motivated by previous studies in this area that proposed exploiting the spatiotemporal correlation of the dynamic MRI sequence by posing the reconstruction problem as a least squares minimization regularized by sparsity and low-rank penalties. Ideally the sparsity and low-rank penalties should be represented by the l₀-norm and the rank of a matrix; however both are NP hard penalties. The previous studies used the convex l₁-norm as a surrogate for the l₀-norm and the non-convex Schatten-q norm (0 < q ≤ 1) as a surrogate for the rank of matrix. Following past research in sparse recovery, we know that non-convex l_p-norm (0 < p ≤ 1) is a better substitute for the NP hard l₀-norm than the convex l₁-norm. Motivated by these studies, we propose improvements over the previous studies by replacing the l₁-norm sparsity penalty by the l_p-norm. Thus, we reconstruct the dynamic MRI sequence by solving a least squares minimization problem regularized by l_p-norm as the sparsity penalty and Schatten-q norm as the low-rank penalty. There are no efficient algorithms to solve the said problems. In this paper, we derive efficient algorithms to solve them. The experiments have been carried out on Dynamic Contrast Enhanced (DCE) MRI datasets. Both quantitative and qualitative analysis indicates the superiority of our proposed improvement over the existing methods.     

Analytical structure and properties of Coulomb wave functions for real and complex energies

The radical Coulomb wave functions are analysed in their dependence on the energy E considered as a complex parameter. Repulsive and attractive fields are both considered. First turning to the function Φ_l ∝ r^?l?1F_l introduced by Briet, slightly modifying its definition, and assuming that the angular momentum is also a complex parameter, for which the notation L is used, it is proved that Φ_L is an entire function of both E and L. From an expansion of the regular Whittaker function given by Buchholz, the Taylor expansion of Φ_L in powers of E and a simple recurrence relation for its coefficients are easily obtained. The expansion of the regular function F_l is readily obtained from that of Φ_L for L = l, but the irregular function G_l contains Φ_l and $\text{?Φ}{}_{\mathrm{L}}\text{?L}$ for L = l and ?l?1. Having proved that the expansion obtained for Φ_L in powers of E can also be regarded as a uniformly convergent series of entire functions of L, the derivative $\text{?Φ}{}_{\mathrm{L}}\text{?L}$ can be obtained by term-by-term derivation. This method for obtaining the expansion of G_l is straightforward and leads to a final result involving essentially: (i) the conventional function $\text{h(η) =}\text{1}\text{2}\text{ψ(1 + iη) +}\text{1}\text{2}\text{ψ(1 ? iη) ?}\text{ln}\text{η}$ which is singular at η = ∞, i.e., at k = 0; (ii) two entire functions of E, namely Φ_l and Ψ_l; the terms of the expansion of the latter in powers of E contain only Bessel functions multiplied by Bernoulli numbers and coefficients easily obtained from a simple recurrence relation. As an application of the above results, the last sections contain: (i) an alternate from of G_l expansion useful in numerical computations; (ii) the definition and expansion of two linearly independent solutions of the Coulomb equation which are entire in E; (iii) the expansion and threshold properties of the outgoing and incoming solutions, O_l and I_l, corresponding to those we have obtained for F_l and G_l.     

Anisotropy in the length change of gallium single crystals at their superconducting transition

The magnetostriction of gallium single crystals has been measured for the three principal axes between 0.3K and 1.1 K. It is found that the length change at the transition from the superconducting to the normal state is given approximately by (Δl/l)_a = − 3.10⁻⁹, (Δl/l)_b = + 4.4 · 10⁻⁹ and (Δl/l)_c = + 1.8 · 10⁻⁹, for the a, b and c axes at T = 0 K. A short discussion of the results with respect to microscopic theory is presented.     

Spacetime symplectic extension

It is conjectured that in the origin of spacetime there lies a symplectic rather than metric structure. The complex symplectic symmetry Sp(2l, C), l≥1 instead of the pseudoorthogonal one SO(1, d?1), d≥4 is proposed as the spacetime local structure group. A discrete sequence of the metric spacetimes of the fixed dimensionalities d=(2l)² and signatures, with l(2l?1) timelike and l(2l+1) spacelike directions, defined over the set of Hermitian second-rank spin tensors, is considered as an alternative to the pseudo-Euclidean extra dimensional spacetimes. The basic concepts of the symplectic framework are developed in general, and the ordinary and next-to-ordinary spacetime cases with l=1, 2, respectively, are elaborated in more detail. In particular, the scheme provides the rationale for the four-dimensionality and 1+3 signature of the ordinary spacetime.