Words that almost commute

The Hamming distance $\mathrm{ham}(u,v)$ between two equal-length words u, v is the number of positions where u and v differ. The words u and v are said to be conjugates if there exist non-empty words $x,y$ such that $u=xy$ and $v=yx$. The smallest value $\mathrm{ham}(xy,yx)$ can take on is 0, when x and y commute. But, interestingly, the next smallest value $\mathrm{ham}(xy,yx)$ can take on is 2 and not 1. In this paper, we consider conjugates $u=xy$ and $v=yx$ where $\mathrm{ham}(xy,yx)=2$. More specifically, we provide an efficient formula to count the number $h(n)$ of length-n words $u=xy$ over a k-letter alphabet that have a conjugate $v=yx$ such that $\mathrm{ham}(xy,yx)=2$. We also provide efficient formulae for other quantities closely related to $h(n)$. Finally, we show that $h(n)$ grows erratically: cubically for n prime, but exponentially for n even.     

A bound on the Wasserstein-2 distance between linear combinations of independent random variables

We provide a bound on a distance between finitely supported elements and general elements of the unit sphere of ${?}^2\left({\mathbb{N}}^1\right)$. We use this bound to estimate the Wasserstein-2 distance between random variables represented by linear combinations of independent random variables. Our results are expressed in terms of a discrepancy measure related to Nourdin–Peccati’s Malliavin–Stein method. The main application is towards the computation of quantitative rates of convergence to elements of the second Wiener chaos. In particular, we explicit these rates for non-central asymptotic of sequences of quadratic forms and the behavior of the generalized Rosenblatt process at extreme critical exponent.     

Ground state solutions for Klein–Gordon–Maxwell system with steep potential well

In this paper, we study the following Klein–Gordon–Maxwell system $\left\{\begin{array}{c}-\Delta u+(\lambda a\left(x\right)+1)u-(2\omega +\varphi )\varphi u=f(x,u),\mathrm{in}{\mathbb{R}}^3,\hfill \\ -\Delta \varphi =-(\omega +\varphi )u^2,\mathrm{in}{\mathbb{R}}^3.\hfill \end{array}\right.$Using variational methods, we obtain the existence of ground state solutions under some appropriate assumptions on $a$ and $f$.     

Anisotropy Effects and Observational Data on the Constraints of Evolution Dark Energy Models

We investigate cosmological dark energy models where the accelerated expansion of the universe is driven by a field with an anisotropic universe. The constraints on the parameters are obtained by maximum likelihood analysis using observational of 194 Type Ia supernovae(SNIa) and the most recent joint light-curve analysis(JLA) sample. In particular we reconstruct the dark energy equation of state parameter w(z) and the deceleration parameter q(z). We find that the best fit dynamical w(z) obtained from the 194 SNIa dataset does not cross the phantom divide line w(z) =-1 and remains above and close to w(z)≈-0.92 line for the whole redshift range 0 ≤ z ≤ 1.75 showing no evidence for phantom behavior. By applying the anisotropy effect on the ΛCDM model, the joint analysis indicates that ?_(σ0)= 0.0163 ± 0.03,with 194 SNIa, ?_(σ0)=-0.0032 ± 0.032 with 238 the SiFTO sample of JLA and ?_(σ0)= 0.011 ± 0.0117 with 1048 the SALT2 sample of Pantheon at 1σ′confidence interval. The analysis shows that by considering the anisotropy, it leads to more best fit parameters in all models with JLA SNe datasets. Furthermore, we use two statistical tests such as the usual χ_(min)~2/dof and p-test to compare two dark energy models with ΛCDM model. Finally we show that the presence of anisotropy is confirmed in mentioned models via SNIa dataset.     

Ab initio structure and vibration-rotation dynamics of germylene,GeH2

Accurate structure and potential energy surface of germylene, GeH₂, in its ground electronic state ¹A₁ were determined from ab initio calculations using the coupled-cluster approach in conjunction with the correlation-consistent basis sets up to sextuple-zeta quality. The Born-Oppenheimer equilibrium structural parameters for the ¹A₁ state are estimated to be r_e(GeH) = 1.5793 Å and ∠_e(HGeH) = 91.19^∘. The term value T_e for the lowest excited electronic state ã³B₁ of GeH₂ is predicted to be 9140 cm^–1. The vibration-rotation energy levels for the ¹A₁ state of the ⁷⁴GeH₂, ⁷⁴GeD₂, ⁷²GeH₂, and ⁷⁰GeH₂ isotopologues were determined using a variational approach and compared with the experimental data. The role of the core-electron correlation, higher-order valence-electron correlation, scalar relativistic, spin-orbit, and adiabatic effects for prediction of the structure and vibration-rotation dynamics of the GeH₂ molecule is discussed. © 2019 Wiley Periodicals, Inc.     

Distribution dependent stochastic differential equations

Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs) have been intensively investigated. In this paper, we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.     

Wave packet dynamics of nonlinear Gazeau–Klauder coherent states of a position-dependent mass system in a Coulomb-like potential

AD = 1 position-dependent mass approach to constructing nonlinear quantum states for a modified Coulomb potential is used to generate Gazeau–Klauder coherent states. It appears that their energy eigenvalues are scaled down by the quantum number and the nonlinearity coefficient. We study the basic properties of these states, which are found to be undefined on the whole complex plane, and some details of their revival structure are discussed.