N-dimensional Schrödinger equation at finite temperature using the Nikiforov–Uvarov method

The N-radial Schrödinger equation is analytically solved. The Cornell potential is extended to finite temperature. The energy eigenvalues and the wave functions are calculated in the N-dimensional form using the Nikiforov–Uvarov (NV) method. At zero temperature, the energy eigenvalues and the wave functions are obtained in good agreement with other works. The present results are applied on the charmonium and bottomonium masses at finite temperature. The effect of dimensionality number is investigated on the quarkonium masses. A comparison is discussed with other works, which use the QCD sum rules and lattice QCD. The present approach successfully generalizes the energy eigenvalues and corresponding wave functions at finite temperature in the N-dimensional representation. In addition, the present approach can successfully be applied to the quarkonium systems at finite temperature.     

Solitary wave solution of the generalized Hirota–Satsuma coupled KdV system

In this research, we find the exact traveling wave solutions involving parameters of the generalized Hirota–Satsuma couple KdV system according to the modified simple equation method with the aid of Maple 16. When these parameters are taken special values, the solitary wave solutions are derived from the exact traveling wave solutions. It is shown that the modified simple equation method provides an effective and a more powerful mathematical tool for solving nonlinear evolution equations in mathematical physics. Comparison between our results and the well-known results will be presented.     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

The fourth power mean of the general 2-dimensional Kloostermann sums mod <Emphasis Type="Italic">p</Emphasis>

The main purpose of this paper is using the analytic methods and the properties of Gauss sums to study the computational problem of one kind fourth power mean of the general 2-dimensional Kloostermann sums mod p, and give an exact computational formula for it.     

A Near-Infrared Dye That Undergoes Multiple Interconversions through Acid–Base Equilibrium and Reversible Redox Processes

A near-infrared (NIR) polymethine dye ( 1 ), consisting of a cyclohepta[1,2-b;4,3-b′]dithiophene and two phenol moieties, was synthesized. This dye exhibited pH-responsive changes in its photophysical properties due to a two-step acid–base equilibrium that produced a protonated cation ( 1H⁺ ) and an anion ( 1⁻ ). While 1H⁺ showed an intense fluorescence in the red region of the visible spectrum, 1⁻ exhibited a strong absorption in the NIR region. The tropylium ion character in 1H⁺ induces high pK_a1 and pK_a2 values for 1 . Moreover, a stable radical ( 1^. ) was prepared, which showed a NIR absorption band with a maximum at circa 1600 nm. The cyclic voltammogram of 1^. revealed a two-step reversible redox process that produced 1⁻ and the cation 1⁺ , which is different from 1H⁺ . These redox processes accompany drastic electrochromic changes in the vis–NIR region. Overall, 1 is susceptible to multiple interconversions between five forms, due to the multifaceted character of the cycloheptadithiophene skeleton.