Quantum Harmonic Oscillators with Nonlinear Effective Masses in the weak density approximation

We study the eigen-energy and eigen-function of a quantum particle acquiring the probability density-dependent effective mass (DDEM) in harmonic oscillators. Instead of discrete eigen-energies, continuous energy spectra are revealed due to the introduction of a nonlinear effective mass. Analytically, we map this problem into an infinite discrete dynamical system and obtain the stationary solutions in the weak density approximation, along with the proof on the monotonicity in the perturbed eigen-energies. Numerical results not only give agreement to the asymptotic solutions stemmed from the expansion of Hermite-Gaussian functions, but also unveil a family of peakon-like solutions without linear counterparts. As nonlinear Schr ¨odinger wave equation has served as an important model equation in various sub-fields in physics, our proposed generalized quantum harmonic oscillator opens an unexplored area for quantum particles with nonlinear effective masses.

Symmetries, conservation laws and dust acoustic solitons of two-temperature ion in inhomogeneous plasma

In this paper, we investigate dust acoustic solitary waves in dusty inhomogeneous plasma with a weak density gradient. For this, Lie analysis is utilized and solitary waves are obtained with the aid of the newly extended direct algebraic method. The solutions so obtained carry a variety of some new families including dark-bright, dark, dark-singular, and singular solutions of types [Formula: see text] and [Formula: see text]. The sufficient conditions for the existence of these structures are given. Meantime, the effect of parameters like dust charge, cold ion density, and number density on the soliton structures are investigated in detail, which shows that dust charge affects the amplitude but density does not put any impact on soliton’s profile. A graphical approach is practiced to discuss the physical impact of the problem. Finally, by using a multiplier approach, conserved quantities are reported.

ON LOCAL WEAK τ-DENSITY OF TOPOLOGICAL SPACES

A topological space X is locally weakly separable [3] at a point x∈X if x has a weakly separable neighbourhood. A topological space X is locally weakly separable if X is locally weakly separable at every point x∈X. The notion of local weak separability can be generalized for any cardinal τ ≥ℵ0 . A topological space X is locally weakly τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a weak τ-dense neighborhood in X [4]. The local weak density at a point x is denoted as lwd(x). The local weak density of a topological space X is defined in following way: lwd ( X ) = sup{ lwd ( x) : x∈ X } . A topological space X is locally τ-dense at a point x∈X if τ is the smallest cardinal number such that x has a τ-dense neighborhood in X [4]. The local density at a point x is denoted as ld(x). The local density of a topological space X is defined in following way: ld ( X ) = sup{ ld ( x) : x∈ X } . It is known that for any topological space we have ld(X ) ≤ d(X ) . In this paper, we study questions of the local weak τ-density of topological spaces and establish sufficient conditions for the preservation of the property of a local weak τ-density of subsets of topological spaces. It is proved that a subset of a locally τ-dense space is also locally weakly τ-dense if it satisfies at least one of the following conditions: (a) the subset is open in the space; (b) the subset is everywhere dense in space; (c) the subset is canonically closed in space. A proof is given that the sum, intersection, and product of locally weakly τ-dense spaces are also locally weakly τ-dense spaces. And also questions of local τ-density and local weak τ-density are considered in locally compact spaces. It is proved that these two concepts coincide in locally compact spaces.

THE LOCAL DENSITY AND THE LOCAL WEAK DENSITY IN THE SPACE OF PERMUTATION DEGREE AND IN HATTORI SPACE

In this paper, the local density \((l d)\) and the local weak density \((l w d)\) in the space of permutation degree as well as the cardinal and topological properties of Hattori spaces are studied. In other words, we study the properties of the functor of permutation degree \(S P^{n}\) and the subfunctor of permutation degree \(S P_{G}^{n}\), \(P\) is the cardinal number of topological spaces. Let \(X\) be an infinite \(T_{1}\)-space. We prove that the following propositions hold.(1) Let \(Y^{n} \subset X^{n}\); (A) if \(d\, \left(Y^{n} \right)=d\, \left(X^{n} \right)\), then \(d\, \left(S P^{n} Y\right)=d\, \left(SP^{n} X\right)\); (B) if \(l w d\, \left(Y^{n} \right)=l w d\, \left(X^{n} \right)\), then \(l w d\, \left(S P^{n} Y\right)=l w d\, \left(S P^{n} X\right)\). (2) Let \(Y\subset X\); (A) if \(l d \,(Y)=l d \,(X)\), then \(l d\, \left(S P^{n} Y\right)=l d\, \left(S P^{n} X\right)\); (B) if \(w d \,(Y)=w d \,(X)\), then \(w d\, \left(S P^{n} Y\right)=w d\, \left(S P^{n} X\right)\).(3) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is a locally compact \(T_{1}\)-space, then \(S P^{n} X, \, S P_{G}^{n} X\), and \(\exp _{n} X\) are \(k\)-spaces.(4) Let \(n\) be a positive integer, and let \(G\) be a subgroup of the permutation group \(S_{n}\). If \(X\) is an infinite \(T_{1}\)-space, then \(n \,\pi \,w \left(X\right)=n \, \pi \,w \left(S P^{n} X \right)=n \,\pi \,w \left(S P_{G}^{n} X \right)=n \,\pi \,w \left(\exp _{n} X \right)\).We also have studied that the functors \(SP^{n},\) \(SP_{G}^{n} ,\) and \(\exp _{n} \) preserve any \(k\)-space. The functors \(SP^{2}\) and \(SP_{G}^{3}\) do not preserve Hattori spaces on the real line. Besides, it is proved that the density of an infinite \(T_{1}\)-space \(X\) coincides with the densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\). It is also shown that the weak density of an infinite \(T_{1}\)-space \(X\) coincides with the weak densities of the spaces \(X^{n}\), \(\,S P^{n} X\), and \(\exp _{n} X\).

Synthesis of N-(Anthracen-9-ylmethyl)-N-methyl-2-(phenylsulfonyl)ethanamine via Microwave Green Synthesis Method: X-ray Characterization, DFT and Hirshfeld Analysis

N-(Anthracen-9-ylmethyl)-N-methyl-2-(phenylsulfonyl)ethanamine 3 has been synthesized via the aza-Michael addition approach by reaction of the corresponding amine with the vinyl sulfone derivative under microwave conditions. The structure of the aza-Michael product 3 is elucidated by X-ray crystallography. The study of molecular packing by employing the Hirshfeld analysis indicates that the percentages of O…H, C…H and H…H contacts are 16.8%, 34.1% and 48.6%, respectively, where the O...H hydrogen bonds have the characteristics of short and strong contacts while the C...H contacts are considered weak. Density functional theory (DFT) investigations show that the aza-Michael product 3 is polar with a net dipole moment of 5.2315 debye.