Erdős–Gallai stability theorem for linear forests

The Erd?s–Gallai Theorem states that every graph of average degree more than $l?2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erd?s–Gallai Theorem in terms of minimum degree. Let $G$ be a connected graph of order $n$ and $F=\left({?}_{i=1}^k{P}_{2a_i}\right)?\left({?}_{i=1}^l{P}_{2b_i+1}\right)$ be $k+l$ disjoint paths of order $2a_1,\dots ,2a_k,2b_1+1,\dots ,2b_l+1,$ respectively, where $k\ge 0$, $0\le l\le 2$, and $k+l\ge 2$. If the minimum degree $\delta \left(G\right)\ge {\sum }_{i=1}^k{a}_i+{\sum }_{i=1}^l{b}_i?1$, then $F?G$ except several classes of graphs for sufficiently large $n$, which extends and strengths the results of Ali and Staton for an even path and Yuan and Nikiforov for an odd path.     

Kersten-Krasil’shchik耦合KdV-mKdV方程的达布-贝克隆变换

从矩阵形式的线性系统出发, 构造了Kersten-Krasil’shchik耦合KdV-mKdV方程的达布变换和相应的贝克隆变换, 并由一对达布-贝克隆变换的相容性得到了非线性叠加公式.     

Approximate analytical solutions and mean energies of stationary Schrödinger equation for general molecular potential

The Schrödinger equation is solved with general molecular potential via the improved quantization rule. Expression for bound state energy eigenvalues, radial eigenfunctions, mean kinetic energy, and potential energy are obtained in compact form. In modeling the centrifugal term of the effective potential, a Pekeris-like approximation scheme is applied. Also, we use the Hellmann-Feynman theorem to derive the relation for expectation values. Bound state energy eigenvalues, wave functions and meanenergies of Woods-Saxon potential, Morse potential, Möbius squared and Tietz-Hua oscillators are deduced from the general molecular potential. In addition, we use our equations to compute the bound state energy eigenvalues and expectation values for four diatomic molecules viz. H₂, CO, HF, and O₂. Results obtained are in perfect agreement with the data available from the literature for the potentials and molecules. Studies also show that as the vibrational quantum number increases, the mean kinetic energy for the system in a Tietz-Hua potential increases slowly to a threshold value and then decreases. But in a Morse potential, the mean kinetic energy increases linearly with vibrational quantum number increasing.     

Linearity studies of HD-810 dosimeters by light ion beams

Radiochromic films (RCF), also called GafChromic? films, represent a performant material for accurate quantitative radiation dosimetry. Their compositions allow high dose sensitivity and fewer environmental dependence, giving a good feedback to the absorbed dose value and to the active media absorption, turning color upon being irradiated. The RCF take into account their reduced response near the Bragg peak due to a high linear energy transfer (LET). HD-810 GafChromic? films are tissue-equivalent, have easy optical readings and can be employed for ion dosimetry in radio diagnostic and therapy and for industrial applications. Such dosimeters were employed at Tandetron-Nuclear Physics Institute (?e?, Czech Republic) to study the responses of helium, proton and carbon beams, commonly employed in radiotherapy and microelectronics. The sensitivity of the detector is low enough to measure multiple-beam exposures. The induced effects by the ions in the energy range of 600?keV to 2.0?MeV were investigated in terms of optical absorbance, measured in the irradiated active region of the polymer. The employed ion dose range was between 40?Gy and 2.5?kGy. The experimental results show that the absorbance increases with the irradiation time (i.e. with the absorbed dose). The absorbance induced in the radio chromic film was measured at 673?nm, at which is observed the highest sensitivity of the films. Such data, together with the dose linearity and the dependence on the ion stopping power will be presented and discussed.