Fast proton conduction path in {\text{Ni}} - {\text{BaCe}}_{{0.8}} {\text{Y}}_{{0.2}} {\text{O}}_{{2.9 - \delta }} membrane

The effect of metal-to-oxide grain boundary layer in $ {\text{Ni}} - {\text{BaCe}}_{{0.8}} {\text{Y}}_{{0.2}} {\text{O}}_{{3 - \delta }} $ (BCY) cermet membrane on hydrogen permeation was studied by applying the different size of oxide grain on Ni-BCY membranes. Two types of cermet membranes having different grain size of oxide were prepared by using different starting particle size of oxide powder. The hydrogen flux of coarse-oxide-grain membrane showed higher flux than that of small-oxide-grain membrane. It was understood that the negative potential at metal-to-oxide grain boundary, reference to the bulk oxide ( $ \phi _{0} < \phi _{\infty } = 0 $ ), was developed, and the accumulation of the effectively positively charged protons may occur at the grain boundary layer (space charge layer), which may result in providing highly conductive proton path by shifting the charge neutrality condition from $ {\left[ {OH^{ \bullet }_{O} } \right]} = {\left[ {Y^{/}_{{Ce}} } \right]} $ to $ {\left[ {OH^{ \bullet }_{O} } \right]} = n $ .     

Gleichzeitige Messung von Hyperfeinstruktur,Stark-Effekt und Zeeman-Effekt des39K19F mit einer Molekülstrahlresonanzapparatur

An electric Molecular-Beam-Resonance-Spectrometer has been used to measure simultanously the Zeeman- and Stark-effect splitting of the hyperfine structure of³⁹K¹⁹ F. Electric four pole lenses served as focusing and refocusing fields of the spectrometer. A homogenous magnetic field (Zeeman field) was superimposed to the electric field (Stark field) in the transition region of the apparatus. The observed (Δm _J =±1)-transitions were induced electrically. Completely resolved spectra of KF in theJ=1 rotational state have been measured. The obtained quantities are: The electric dipolmomentμ _e l of the molecul forv=0,1 and 2; the rotational magnetic dipolmomentμ _J forv=0,1; the difference of the magnetic shielding (σ_⊥ ? σ_∥) by the electrons of both nuclei as well as the difference of the molecular susceptibility (ξ_⊥ ? ξ_∥). The numerical values are

$$\begin{array}{*{20}c} {\mu _{e1} = 8,585(4)deb,} \\ {\frac{{(\mu _{e1} )_{\upsilon = 1} }}{{(\mu _{e1} )_{\upsilon = 0} }} = 1,0080,} \\ {{{\mu _J } \mathord{\left/ {\vphantom {{\mu _J } J}} \right. \kern-\nulldelimiterspace} J} = ( - )2352(10) \cdot 10^{ - 6} \mu _B ,} \\ {(\sigma _ \bot - \sigma _\parallel )F = ( - )2,19(9) \cdot 10^{ - 4} ,} \\ {(\sigma _ \bot - \sigma _\parallel )K = ( - )12(9) \cdot 10^{ - 4} ,} \\ {(\xi _ \bot - \xi _\parallel ) = 3 (1) \cdot 10^{ - 30} {{erg} \mathord{\left/ {\vphantom {{erg} {Gau\beta ^2 }}} \right. \kern-\nulldelimiterspace} {Gau\beta ^2 }}} \\ \end{array} $$     

A New Set of Limiting Gibbs Measures for the Ising Model on a Cayley Tree

For the Ising model (with interaction constant J>0) on the Cayley tree of order k≥2 it is known that for the temperature T≥T _c,k =J/arctan?(1/k) the limiting Gibbs measure is unique, and for T<T _c,k there are uncountably many extreme Gibbs measures. In the Letter we show that if \(T\in(T_{c,\sqrt{k}}, T_{c,k_{0}})\), with \(\sqrt{k} then there is a new uncountable set \({\mathcal{G}}_{k,k_{0}}\) of Gibbs measures. Moreover \({\mathcal{G}}_{k,k_{0}}\ne {\mathcal{G}}_{k,k'_{0}}\), for k ₀≠k′₀. Therefore if \(T\in (T_{c,\sqrt{k}}, T_{c,\sqrt{k}+1})\), \(T_{c,\sqrt{k}+1} then the set of limiting Gibbs measures of the Ising model contains the set {known Gibbs measures}\(\cup(\bigcup_{k_{0}:\sqrt{k}.     

Suche nach Strahlungsprozessen zweiter Ordnung beim Zerfall von Ag109m

The decay of an excited state by the emission of twoγ-quanta (γ γ-transitions) or two conversion electrons (e e-transitions) or oneγ-quantum and one conversion electron (γ e-transitions) is expected as a second order radiation process. The decay of Ag^109m was examined for such events using a special arrangement of two NaJ-scintillation counters in coincidence. The energies of coincident quanta were displayed on the two axes of an “X-Y”-Oscilloscope respectively. For the ratio ofγ γ-transitions to one-quantum transitions an upper limit of\(\frac{{W_{\gamma \gamma } }}{{W_\gamma }} \leqq 1,9 \cdot 10^{ - 5} \) was obtained. Furthermore theγ-spectrum in coincidence withK X-rays was studied. From these measurementse e- andγ e-transition rates can be calculated for the case ofK shell conversion. The results obtained are:

$$\frac{{W_{^e K^e K} }}{{W_\gamma }} = \left( {8,1_{ - 1,7}^{ + 0,6} } \right) \cdot 10^{ - 3} and\frac{{W_{\gamma ^e K} }}{{W_\gamma }}< 1,5 \cdot 10^{ - 3} .$$     

Isomeric states in134Ba

Several new levels including two isomeric states have been established in¹³⁴Ba. Spin and parity assignments of 10⁺ and 5^? are proposed for the isomers. The former may have a \(\left( {vh_{1 1/2} } \right)_{10^ + } \) configuration while the latter may be either \((vs_{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2}} vh_{{{11} \mathord{\left/ {\vphantom {{11} 2}} \right. \kern-0em} 2}} )_{5 - } \) or \(\left( {vd_{3/2} vh_{1 1/2} } \right)_{5^ - } \) .     

Heuristic Relative Entropy Principles with Complex Measures: Large-Degree Asymptotics of a Family of Multi-variate Normal Random Polynomials

Let \(z\in \mathbb {C}\), let \(\sigma ^2>0\) be a variance, and for \(N\in \mathbb {N}\) define the integrals

$$\begin{aligned} E_N^{}(z;\sigma ) := \left\{ \begin{array}{ll} {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}}\! (x^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x^2}}{\sqrt{2\pi }}dx&{}\quad \text{ if }\, N=1,\\ {\frac{1}{\sigma }} \!\!\!\displaystyle \int _{\mathbb {R}^N}\! \prod \prod \limits _{1\le k<l\le N}\!\! e^{-\frac{1}{2N}(1-\sigma ^{-2}) (x_k-x_l)^2} \prod _{1\le n\le N}\!\!\!\!(x_n^2+z^2) \frac{e^{-\frac{1}{2\sigma ^2} x_n^2}}{\sqrt{2\pi }}dx_n &{}\quad \text{ if }\, N>1. \end{array}\right. \!\!\! \end{aligned}$$

These are expected values of the polynomials \(P_N^{}(z)=\prod _{1\le n\le N}(X_n^2+z^2)\) whose 2N zeros \(\{\pm i X_k\}^{}_{k=1,\ldots ,N}\) are generated by N identically distributed multi-variate mean-zero normal random variables \(\{X_k\}^{N}_{k=1}\) with co-variance \(\mathrm{{Cov}}_N^{}(X_k,X_l)=(1+\frac{\sigma ^2-1}{N})\delta _{k,l}+\frac{\sigma ^2-1}{N}(1-\delta _{k,l})\). The \(E_N^{}(z;\sigma )\) are polynomials in \(z^2\), explicitly computable for arbitrary N, yet a list of the first three \(E_N^{}(z;\sigma )\) shows that the expressions become unwieldy already for moderate N—unless \(\sigma = 1\), in which case \(E_N^{}(z;1) = (1+z^2)^N\) for all \(z\in \mathbb {C}\) and \(N\in \mathbb {N}\). (Incidentally, commonly available computer algebra evaluates the integrals \(E_N^{}(z;\sigma )\) only for N up to a dozen, due to memory constraints). Asymptotic evaluations are needed for the large-N regime. For general complex z these have traditionally been limited to analytic expansion techniques; several rigorous results are proved for complex z near 0. Yet if \(z\in \mathbb {R}\) one can also compute this “infinite-degree” limit with the help of the familiar relative entropy principle for probability measures; a rigorous proof of this fact is supplied. Computer algebra-generated evidence is presented in support of a conjecture that a generalization of the relative entropy principle to signed or complex measures governs the \(N\rightarrow \infty \) asymptotics of the regime \(iz\in \mathbb {R}\). Potential generalizations, in particular to point vortex ensembles and the prescribed Gauss curvature problem, and to random matrix ensembles, are emphasized.

     

Polarization-optical and X-ray diffraction investigations on the symmetry of distorted phases of ammonium cryolite (NH4)3 ScF6

Single-crystal plates of different sections of the (NH₄)₃ScF₆ crystal have been investigated by polarization-optical microscopy and X-ray diffraction over a wide temperature range, including the temperatures of two known phase transitions and the third transition found recently. It is established that the symmetry of 5 phases changes in the following sequence: $\begin{gathered} O_h^5 - Fm3m(Z = 4) \leftrightarrow C_{2h}^5 - {{P12_1 } \mathord{\left/ {\vphantom {{P12_1 } {n1}}} \right. \kern-0em} {n1}}(Z = 2) \leftrightarrow C_{2h}^3 - {{I12} \mathord{\left/ {\vphantom {{I12} {m1}}} \right. \kern-0em} {m1}} \\ (Z = 16) \leftrightarrow C_i^1 - I\bar 1(Z = 16) \\ \end{gathered} $ .     

Rules of correspondence in atomic physics

The Smirnov method of analytic continuation (B.M. Smirnov, Sov. Phys. JETP 20, 345 (1964)) has been justified and developed for atomic physics. It has been shown that the polarizability of alkali atoms α, their van der Waals interaction constant C ₆, and the oscillator strength of the transition to the first P state f ₀₁ are related to the parameter 〈r ²〉 and gap in the spectrum \(\frac{3}{2}\frac{f}{\Delta } \approx \frac{3}{2}\alpha \Delta \approx {\left( {3{C_6}\Delta } \right)^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}} \approx \left\langle {{r^2}} \right\rangle \). The average square of the coordinate of the valence electron 〈r ²〉 in the first approximation has a hydrogen dependence \({J_1} = \frac{1}{{2{v^2}}}.\) on the filling factor ν, which is defined in terms of the first ionization potential: xxxxxxxxx     

Krein formula with compensated singularities for the ND-mapping and the generalized Kirchhoff condition at the Neumann Schrödinger junction

The Neumann Schrödinger operator \(\mathcal{L}\) is considered on a thin 2D star-shaped junction, composed of a vertex domain Ω_int and a few semi-infinite straight leads ω _m , m = 1, 2, ..., M, of width δ, δ ? diam Ω_int, attached to Ω_int at Γ ? ?Ω_int. The potential of the Schrödinger operator l ^ω on the leads vanishes, hence there are only a finite number of eigenvalues of the Neumann Schrödinger operator L _int on Ω_int embedded into the open spectral branches of l ^ω with oscillating solutions χ _±(x, p) = \(e^{ \pm iK_ + x} e_m \) of l ^ω χ _± = p ² χ _±. The exponent of the open channels in the wires is

$K_ + (\lambda ) = p\sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = \sqrt \lambda P_ + $

, with constant e _m , on a relatively small essential spectral interval Δ ? [0, π ² δ ^?2). The scattering matrix of the junction is represented on Δ in terms of the ND mapping

$\mathcal{N} = \frac{{\partial P_ + \Psi }}{{\partial x}}(0,\lambda )\left| {_\Gamma \to P_ + \Psi _ + (0,\lambda )} \right|_\Gamma $

as

$S(\lambda ) = (ip\mathcal{N} + I_ + )^{ - 1} (ip\mathcal{N} - I_ + ), I_ + = \sum\limits_{m = 1}^M {e^m } \rangle \langle e^m = P_ + $

. We derive an approximate formula for \(\mathcal{N}\) in terms of the Neumann-to-Dirichlet mapping \(\mathcal{N}_{\operatorname{int} } \) of L _int and the exponent K _? of the closed channels of l ^ω . If there is only one simple eigenvalue λ ₀ ∈ Δ, L _intφ0 = λ _0φ0 then, for a thin junction, \(\mathcal{N} \approx |\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} \) with

$\vec \phi _0 = P_ + \phi _0 = (\delta ^{ - 1} \int_{\Gamma _1 } {\phi _0 (\gamma )} d\gamma ,\delta ^{ - 1} \int_{\Gamma _2 } {\phi _0 (\gamma )} d\gamma , \ldots \delta ^{ - 1} \int_{\Gamma _M } {\phi _0 (\gamma )} d\gamma )$

and \(P_0 = \vec \phi _0 \rangle |\vec \phi _0 |^{ - 2} \langle \vec \phi _0 \),

$S(\lambda ) \approx \frac{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} - I_ + }}{{ip|\vec \phi _0 |^2 P_0 (\lambda _0 - \lambda )^{ - 1} + I_ + }} = :S_{appr} (\lambda )$

. The related boundary condition for the components P ₊Ψ(0) and P ₊Ψ′(0) of the scattering Ansatz in the open channel \(P_ + \Psi (0) = (\bar \Psi _1 ,\bar \Psi _2 , \ldots ,\bar \Psi _M ), P_ + \Psi '(0) = (\bar \Psi '_1 , \bar \Psi '_2 , \ldots , \bar \Psi '_M )\) includes the weighted continuity (1) of the scattering Ansatz Ψ at the vertex and the weighted balance of the currents (2), where

$\frac{{\bar \Psi _m }}{{\bar \phi _0^m }} = \frac{{\delta \sum\nolimits_{t = 1}^M { \bar \Psi _t \bar \phi _0^t } }}{{|\vec \phi _0 |^2 }} = \frac{{\bar \Psi _r }}{{\bar \phi _0^r }} = :\bar \Psi (0)/\bar \phi (0), 1 \leqslant m,r \leqslant M$

(1)

,

$\sum\limits_{m = 1}^M {\bar \Psi '_m } \bar \phi _0^m + \delta ^{ - 1} (\lambda - \lambda _0 )\bar \Psi /\bar \phi (0) = 0$

(1)

. Conditions (1) and (2) constitute the generalized Kirchhoff boundary condition at the vertex for the Schrödinger operator on a thin junction and remain valid for the corresponding 1D model. We compare this with the previous result by Kuchment and Zeng obtained by the variational technique for the Neumann Laplacian on a shrinking quantum network.

     

A Heisenberg Double Addition to the Logarithmic Kazhdan–Lusztig Duality

For a Hopf algebra B, we endow the Heisenberg double \({\mathcal{H}(B^*)}\) with the structure of a module algebra over the Drinfeld double \({\mathcal{D}(B)}\). Based on this property, we propose that \({\mathcal{H}(B^*)}\) is to be the counterpart of the algebra of fields on the quantum-group side of the Kazhdan–Lusztig duality between logarithmic conformal field theories and quantum groups. As an example, we work out the case where B is the Taft Hopf algebra related to the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) quantum group that is Kazhdan–Lusztig-dual to (p,1) logarithmic conformal models. The corresponding pair \({(\mathcal{D}(B),\mathcal{H}(B^*))}\) is “truncated” to \({(\overline{\mathcal{U}}_{\mathfrak{q}} s\ell2,\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2))}\), where \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)}\) is a \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\) module algebra that turns out to have the form \({\overline{\mathcal{H}}_{\mathfrak{q}} s\ell(2)=\mathbb{C}_{\mathfrak{q}}[z,\partial]\otimes\mathbb{C}[\lambda]/(\lambda^{2p}-1)}\), where \({\mathbb{C}_{\mathfrak{q}}[z,\partial]}\) is the \({\overline{\mathcal{U}}_{\mathfrak{q}} s\ell(2)}\)-module algebra with the relations z ^p  = 0, ? ^p  = 0, and \({\partial z = \mathfrak{q}-\mathfrak{q}^{-1} + \mathfrak{q}^{-2} z\partial}\).     

Stability of {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} and {\text{SrZr}}_{{0.2}} {\text{Ce}}_{{0.7}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} under H2 atmospheres

Previous H₂ permeation tests showed a degradation of H₂ permeation flux with time. To understand the cause of degradation and develop a solution, the stability of $ {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ and $ {\text{SrZr}}_{{0.2}} {\text{Ce}}_{{0.7}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ samples were studied under dry and wet H₂ atmospheres. Total conductivity of $ {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ increased with time in dry H₂. The X-ray diffraction pattern of $ {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ after dry hydrogen atmosphere heat treatments show CeO₂ peaks indicating that $ {\text{SrCe}}_{{0.9}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ decomposes under dry H₂ atmospheres; scanning electron microscopy and energy dispersive X-ray spectroscopy analyses prove that decomposition proceeded along the grain boundaries. $ {\text{SrZr}}_{{0.2}} {\text{Ce}}_{{0.7}} {\text{Eu}}_{{0.1}} {\text{O}}_{{3 - \delta }} $ was investigated and demonstrated greater stability under dry hydrogen atmospheres. However, Zr substitution results in a tradeoff with electrical properties.     

Selectivity of anion exchange membrane modified with polyethyleneimine

Previous works have been made on the improvement of selectivity of ion exchange membranes using adsorption of polyelectrolyte on the surface of the materials. The modification of the surface material in the case of an anion exchange membrane concerns the hydrophilic/hydrophobic balance properties and its relationship with the hydration state. Starting from this goal, the AMX membrane has been modified, in this work, by adsorption of polyethyleneimine on its surface. Many conditions of modification of the AMX membrane surface were studied. A factorial experimental design was used for determining the influent parameters on the AMX membrane modification. The results obtained have shown that the initial concentration of polyethyleneimine and the pH of solution were the main influent parameters on the adsorption of polyethyleneimine on the membrane surface. Competitive ion exchange reactions were studied for the modified and the unmodified membrane involving $ {\text{C}}{{\text{l}}^{ - }} $ , $ {\text{NO}}_3^{ - } $ and $ {\text{SO}}_4^{{2 - }} $ ions. All experiments were carried out at constant concentration of 0.3?mol?L^?1 and at 25?°C. Ion exchange isotherms for the binary systems $ \left( {{\text{C}}{{\text{l}}^{ - }}/{\text{NO}}_3^{ - }} \right) $ , $ \left( {{\text{C}}{{\text{l}}^{ - }}/{\text{SO}}_4^{{2 - }}} \right) $ and $ \left( {{\text{NO}}_3^{ - }/{\text{SO}}_4^{{2 - }}} \right) $ were studied. The obtained results show that chloride was the most sorbed and the selectivity order both for the modified membrane and the unmodified one is: $ {\text{Cl}} > {\text{NO}}_3^{ - } > {\text{SO}}_4^{{2 - }} $ , under the experimental conditions. Selectivity coefficients $ {\text{K}}_{{{\text{C}}{{\text{l}}^{ - }}}}^{{{\text{NO}}_3^{ - }}} $ , $ {\text{K}}_{{2{\text{C}}{{\text{l}}^{ - }}}}^{{{\text{SO}}_4^{{2 - }}}} $ and $ {\text{K}}_{{2{\text{NO}}_3^{ - }}}^{{{\text{SO}}_4^{{2 - }}}} $ for the three binary systems and for the two membranes were determined. It was also observed that for the modified membrane the selectivity towards sulfate ion decrease and the modified membrane became more selective towards monovalent anions.     

Asymptotic Behavior of the Selberg Zeta Functions for Degenerating Families of Hyperbolic Manifolds

Let {M _k } be a degenerating sequence of finite volume, hyperbolic manifolds of dimension d, with d = 2 or d = 3, with finite volume limit M _∞. Let \({Z_{M_{k}} (s)}\) be the associated sequence of Selberg zeta functions, and let \({{\mathcal{Z}}_{k} (s)}\) be the product of local factors in the Euler product expansion of \({Z_{M_{k}} (s)}\) corresponding to the pinching geodesics on M _k . The main result in this article is to prove that \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) converges to \({Z_{M_{\infty}} (s)}\) for all \({s \in \mathbf{C}}\)with Re(s) > (d ? 1)/2. The significant feature of our analysis is that the convergence of \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) to \({Z_{M_{\infty}} (s)}\) is obtained up to the critical line, including the right half of the critical strip, a region where the Euler product definition of the Selberg zeta function does not converge. In the case d = 2, our result reproves by different means the main theorem in Schulze (J Funct Anal 236:120–160, 2006).     

Higher recurrences for Apostol-Bernoulli-Euler numbers

We present explicit formulas for sums of products of Apostol-Bernoulli and Apostol-Euler numbers of the form

$\sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)B_{m_1 } (q) \cdots B_{m_N } (q),} \sum\limits_{_{m_1 , \cdots ,m_N \geqslant n}^{m_1 + \cdots + m_N = n} } {\left( {_{m_1 , \cdots m_N }^n } \right)E_{m_1 } (q) \cdots E_{m_N } (q),}$

where N and n are positive integers, B _m (q) _n stand for the Apostol-Bernoulli numbers, E _m (q) for the Apostol-Euler numbers, and \(\left( {\begin{array}{*{20}c} n \\ {m_1 , \cdots ,m_N } \\ \end{array} } \right) = \frac{{n!}}{{m_1 ! \cdots m_N !}}.\) Our formulas involve Stirling numbers of the first kind. We also derive results for Apostol-Bernoulli and Apostol-Euler polynomials. As an application, for q = 1 we recover results of Dilcher, and our paper can be regarded as a q-extension of that of Dilcher.

     

Method of uniform effective field in structure-dynamic approach of nanoionics

In the structure-dynamic approach of nanoionics, the method of a uniform effective field \( {F}_{\mathrm{eff}}^{j,k} \) of a crystallographic planeX ^j has been substantiated for solid electrolyte nanostructures. The \( {F}_{\mathrm{eff}}^{j,k} \)is defined as an approximation of a non-uniform field \( {F}_{\mathrm{dis}}^j \)of X ^j with a discrete- random distribution of excess point charges. The parameters of \( {F}_{\mathrm{eff}}^{j,k} \)are calculated by correction of the uniform Gauss field \( {F}_{\mathrm{G}}^j \) of X ^j . The change in an average frequency of ionic jumps X ^k ?→?X ^k?+?1 between adjacent planes of nanostructure is determined by the sum of field additives to the barrier heights η _k , _k?+?1, and for \( {F}_{\mathrm{G}}^j \) and \( {F}_{\mathrm{dis}}^j \), these sums are the same decimal order of magnitude. For nanostructures with length ~4 nm, the application of \( {F}_{\mathrm{G}}^j \) (as \( {F}_{\mathrm{eff}}^{j,k} \)) gives the accuracy ~20 % in calculations of ion transport characteristics. The computer explorations of the “universal” dynamic response (Reσ ^??∝?ω ⁿ ) show an approximately the same power n < ≈1 for\( {F}_{\mathrm{G}}^j \) and \( {F}_{\mathrm{eff}}^{j,k} \).     

Critical behavior of La0.67−y (Sr, Ba, Ca)0.33+y Mn1−x Sn x O3 (x = 0.01, 0.02, y = 0, 0.07) perovskites

The magnetic critical behavior of the manganese perovskite series $ {\text{La}}_{{0.67 - y}} {\left( {{\text{Sr,}}\,\,{\text{Ba,}}\,\,{\text{Ca}}} \right)}_{{0.33 + y}} {\text{Mn}}_{{1 - x}} {\text{Sn}}_{x} {\text{O}}_{3} The magnetic critical behavior of the manganese perovskite series (x = 0.01, 0.02, y = 0, 0.07) is studied by means of dc magnetic measurements and ¹¹⁹Sn M?ssbauer spectroscopy. The structure can be described by a rhombohedral unit cell (space group R–3C) for the samples where the A-site is occupied by La and Sr or La and Ba ions and orthorhombic unit cell (space group Pnma) for the samples where the A-site is occupied by La and Ca ions. Arrott and scaling plots show that the samples, where the A-site is occupied by La and Sr or La and Ba ions, follow the behavior of a conventional second-order ferromagnetic transition. In contrast, the samples that contain La and Ca ions in the A-site show anomalous behavior around Curie point. M?ssbauer measurements show two magnetic phases below T _c. One of them exhibits stronger exchange interactions with more rapid electron transfer between Mn³⁺/Mn⁴⁺, compared to the other.     

Theoretical study of collision and collision-free radiative lifetimes of {{\text{S}}_2}\left( {{{\text{B}}^3}\Sigma_{\text{u}}^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right)

We calculate multireference configuration-interaction wavefunctions and the potential-energy curves for the $ {B^3}\Sigma_u^{-} $ and $ {X^3}\Sigma_g^{-} $ states of the collision-free S₂ molecule and the T-shape collision complex S₂?CHe using cc-pVQZ basis sets. We obtain the transition dipole moments of the $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \to {X^3}\Sigma_g^{-} } \right) $ and the Franck?CCondon factors between the vibrational levels of this two states. We evaluate the radiative lifetimes of $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right) $ levels of the collision complex and the collision-free molecule and compare them with the experiments. The collision provides little change in the radiative lifetimes of $ {{\text{S}}_2}\left( {{B^3}\Sigma_u^{-} \left( {{\upsilon^{\prime}} = 0 - 9} \right)} \right) $ according to the previous calculations. We obtain excellent agreement between the theoretical results and the experiments. The data calculated are very useful in the study of the microwave-driven high-pressure sulfur lamp and an S₂ laser pumped by a transverse fast discharge.     

Parametrization of supersingular perturbations in the method of rigged Hilbert spaces

A classification of bounded below supersingular perturbations à of a self-adjoint operator A ? 1 is suggested. In the A-scale of Hilbert spaces \(\mathcal{H}_{ - k} \sqsupset \mathcal{H} \sqsupset \mathcal{H}_k \) = Dom A ^k/2, k > 0, a parametrization of operators à in terms of bounded mappings S: \(\mathcal{H}_k \to \mathcal{H}_{ - k} \) such that ker S is dense in \(\mathcal{H}_{k/2} \) is obtained.     

Exact solutions of the D-dimensional Schrödinger equation for a ring-shaped pseudoharmonic potential

A new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form $ V(r,\theta ) = \tfrac{1} {8}\kappa r_e^2 \left( {\tfrac{r} {{r_e }} - \tfrac{{r_e }} {r}} \right)^2 + \tfrac{{\beta cos^2 \theta }} {{r^2 sin^2 \theta }} A new non-central potential, consisting of a pseudoharmonic potential plus another recently proposed ring-shaped potential, is solved. It has the form . The energy eigenvalues and eigenfunctions of the bound-states for the Schr?dinger equation in D-dimensions for this potential are obtained analytically by using the Nikiforov-Uvarov method. The radial and angular parts of the wave functions are obtained in terms of orthogonal Laguerre and Jacobi polynomials. We also find that the energy of the particle and the wave functions reduce to the energy and the wave functions of the bound-states in three dimensions.      

On Blow-up Profile of Ground States of Boson Stars with External Potential

We study minimizers of the pseudo-relativistic Hartree functional \({\mathcal {E}}_{a}(u):=\Vert (-\varDelta +m^{2})^{1/4}u\Vert _{L^{2}}^{2}+\int _{{\mathbb {R}}^{3}}V(x)|u(x)|^{2}\mathrm{d}x-\frac{a}{2}\int _{{\mathbb {R}}^{3}}(\left| \cdot \right| ^{-1}\star |u|^{2})(x)|u(x)|^{2}\mathrm{d}x\) under the mass constraint \(\int _{{\mathbb {R}}^3}|u(x)|^2\mathrm{d}x=1\). Here \(m>0\) is the mass of particles and \(V\ge 0\) is an external potential. We prove that minimizers exist if and only if a satisfies \(0\le a<a^{*}\), and there is no minimizer if \(a\ge a^*\), where \(a^*\) is called the Chandrasekhar limit. When a approaches \(a^*\) from below, the blow-up behavior of minimizers is derived under some general external potentials V. Here we consider three cases of V: trapping potential, i.e. \(V\in L_{\mathrm{loc}}^{\infty }({\mathbb {R}}^3)\) satisfies \(\lim _{|x|\rightarrow \infty }V(x)=\infty \); periodic potential, i.e. \(V\in C({\mathbb {R}}^3)\) satisfies \(V(x+z)=V(x)\) for all \(z\in \mathbb {Z}^3\); and ring-shaped potential, e.g. \( V(x)=||x|-1|^p\) for some \(p>0\).