p(x) and Tc(x) characteristics in the Y 1−b(Ca)bBa2Cu3O6+x cuprate family

A model has been proposed to calculate the $p\left(x\right)$ and $T_c\left(x\right)$ dependences in the Y _1?b(Ca)_bBa₂Cu₃O_6+x high-$T_c$ cuprate family and applied to $b=0$, $b=0.1$, and $b=0.2$ cases, for which experimental data exist in the literature. The results obtained imply that the Ca efficiency to provide holes is independent of the basal plane oxygen concentration, which is consistent with a view that electrons from CuO₂ layers would go primarily to Ca since it is twice closer than oxygen (in addition, the chain oxygen is screened by a layer made up of Ba and O(4) ions). It is shown that, in fully oxygenized compounds ($x=1$) the average efficiency, $\chi$, of a chain oxygen to attract an electron from the two nearby layers is reduced by the Ca insertion, though not because the charge transfer mechanism is in itself weakened by Ca, but because a part of electrons that are otherwise available in CuO${}_2$ layers has already been removed by the substitution of Y ³⁺ with Ca²⁺. It has been found that the $b$-dependence of the average oxygen doping efficiency can be fairly accurately described by the following relation: $\chi \left(b\right)=0.39\times (1?0.78b)$. The calculated $p\left(x\right)$ and $T_c\left(x\right)$ dependences are in very good agreement with the available experimental data.     

Klein–Gordon equation for a charged particle in space-varying electromagnetic fields–A systematic study via the Laplace transform

Exact solutions of the Klein–Gordon equation for a charged particle in the presence of three spatially varying electromagnetic fields, namely, (i) $\overrightarrow{E}=\alpha {\beta }_0{e}^{?\alpha x_2}{\widehat{x}}_2,$$\overrightarrow{B}=\alpha {\beta }_1{e}^{?\alpha x_2}{\widehat{x}}_3$ (ii) $\overrightarrow{E}=\frac{{\beta }_0^{{}^{\prime }}}{x_2^2}{\widehat{x}}_2,$$\overrightarrow{B}=\frac{{\beta }_1^{{}^{\prime }}}{x_2^2}{\widehat{x}}_3,$ and (iii) $\overrightarrow{E}=\frac{2{\beta }_0^{{}^{\prime }}}{x_2^3}{\widehat{x}}_2,$$\overrightarrow{B}=\frac{2{\beta }_1^{{}^{\prime }}}{x_2^3}{\widehat{x}}_3,$ are studied. All these fields are generated from a systematic study of a particular type of differential equation whose coefficients are linear in the independent variable. The Laplace transform approach is used to find the solutions, and the corresponding eigenfunctions are expressed in terms of the hypergeometric functions ?₁F₁(a′, b′; x) for the first two cases of the above configurations, while the same are expressed in terms of the Bessel functions of first kind, J_n(x), for the last case.     

A fresh look into m¯c,b(m¯c,b) and precise fD(s),B(s) from heavy–light QCD spectral sum rules

Using recent values of the QCD (non-)perturbative parameters given in Table 1 and an estimate of the N3LO QCD perturbative contributions based on the geometric growth of the PT series, we re-use QCD spectral sum rules (QSSR) known to N2LO PT series and including all dimension-six NP condensate contributions in the full QCD theory, for improving the existing estimates of ${\overline{m}}_{c,b}$ and $f_{D_{(s)},B_{(s)}}$ from the open charm and beauty systems. We especially study the effects of the subtraction point on “different QSSR data” and use (for the first time) the Renormalization Group Invariant (RGI) scale-independent quark masses in the analysis. The estimates [rigourous model-independent upper bounds within the SVZ framework] reported in Table 8: $f_D/f_{\pi }=1.56(5)$ $[?1.68(1)]$, $f_B/f_{\pi }=1.58(5)$ $[?1.80(3)]$ and $f_{D_s}/f_K=1.58(4)$ $[?1.63(1)]$, $f_{B_s}/f_K=1.50(3)$ $[?1.61(3.5)]$, which improve previous QSSR estimates, are in perfect agreement (in values and precisions) with some of the experimental data on $f_{D,D_s}$ and on recent lattice simulations within dynamical quarks. These remarkable agreements confirm both the success of the QSSR semi-approximate approach based on the OPE in terms of the quark and gluon condensates and of the Minimal Duality Ansatz (MDA) for parametrizing the hadronic spectral function which we have tested from the complete data of the $J/\psi$ and ? systems. The values of the running quark masses ${\overline{m}}_c(m_c)=1286(66)\text{MeV}$ and ${\overline{m}}_b(m_b)=4236(69)\text{MeV}$ from $M_{D,B}$ are in good agreement though less accurate than the ones from recent $J/\psi$ and ? sum rules.     

Electromagnetic radiation in a semi-compact space

In this note, we investigate the electromagnetic radiation emitted from a revolving point charge in a compact space. If the point charge is circulating with an angular frequency ${\omega }_0$ on the $(x,y)$-plane at $z=0$ with boundary conditions, $x～x+2\pi R$ and $y～y+2\pi R$, it emits radiation into the z-direction of $z\in [?\infty ,+\infty ]$. We find that the radiation shows discontinuities as a function of ${\omega }_{0}R$ at which a new propagating mode with a different Fourier component appears. For a small radius limit ${\omega }_{0}R?1$, all the Fourier modes except the zero mode on $(x,y)$-plane are killed, but an effect of squeezing the electric field totally enhances the radiation. In the large volume limit ${\omega }_{0}R\to \infty$, the energy flux of the radiation reduces to the expected Larmor formula.     

An efficient series approximation for the Lévy α-stable symmetric distribution

A relevant problem in the statistical and mathematical physics literature is to derive numerically accurate expressions to calculate Lévy α-stable distributions $P_{\alpha }(x;\beta )$. On the formal side, important exact results rely on special functions, such as Meijer-G, Fox-H and finite sums of hypergeometric functions, with only a few exceptional cases expressed in terms of elementary functions. Hence, from a more practical point of view, methods such as series expansions are in order, e.g., to allow for the estimation of the Lévy distribution with high numerical precision, even though most of the existing approaches are restricted to a subset of the distribution parameters and/or usually demand relatively time-consuming sophisticated algorithms. Here we present a rather simple truncated expansion for the case of symmetric Lévy distributions $P_{\alpha }(x)$ ($\beta =0$). This is achieved by dividing the full range of integration into windows, performing proper series expansion inside each, and then calculating the integrals term by term. The obtained representation is convergent for any $0<\alpha \le 2$. Moreover, its accuracy is directly controlled by the number of terms in the truncated expression, being straightforward to implement numerically. As we show with different examples, for almost all allowable α's the calculations lead to $P_{\alpha }(x)$ with reasonable low absolute error for computationally inexpensive simulations.     

The pentagram integrals for Poncelet families

The pentagram map is now known to be a discrete integrable system. We show that the integrals for the pentagram map are constant along Poncelet families. That is, if $P_1$ and $P_2$ are two polygons in the same Poncelet family, and $f$ is a monodromy invariant for the pentagram map, then $f\left(P_1\right)=f\left(P_2\right)$. Our proof combines complex analysis with an analysis of the geometry of a degenerating sequence of Poncelet polygons.     

Numerically anticanonical divisors on Kato surfaces

The compact curves of an intermediate Kato surface $S$ form a basis of $H^2(S,\mathbb{Q})$. We present a way to compute the associated rational coefficients of the first Chern class $c_1\left(S\right)$. We get in particular a simple geometric obstruction for $c_1\left(S\right)$ to be an integral class, or equivalently index $\left(S\right)=1$. In the final part we discuss relations with some recent work of Dloussky (2011) and Oeljeklaus and Toma (2009).     

The first principle study of the electronic structure of SixGe(1−x) alloy films

First principles calculation based on density functional theory (DFT) with the generalized gradient approximation (GGA) are carried out to investigate the electronic band structures of ${\text{Si}}_x{\text{Ge}}_{(1?x)}$ alloys nanofilms. The calculation results show that the band gaps of (100), (110) and (111) surfaces ${\text{Si}}_x{\text{Ge}}_{(1?x)}$ alloy films with different thickness first increase with the increase of Si content, then flatten out, and finally decrease. At the same time, the transformation of direct band gap and indirect band gap occurs when the thickness of films and Si content of the three surface ${\text{Si}}_x{\text{Ge}}_{(1?x)}$ alloy films changes to a certain critical condition. It will be a good way to obtain direct-gap band emission in ${\text{Si}}_x{\text{Ge}}_{(1?x)}$ alloys materials.