Sequential Product and Jordan Product of Quantum Effects

The quantum effects for a physical system can be described by the set E(H)\mathcal{E(H)} of positive operators on a complex Hilbert space H\mathcal{H} that are bounded above by the identity operator I. We denote the set of sharp effects by P(H){\mathcal{P(H) }}. For A,B ? E(H)A,B\in\mathcal{E(H)}, the operation of sequential product A°B=A^\frac12BA^\frac12A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}} was proposed as a model for sequential quantum measurements. Denote by A*B=\fracAB+BA2A\ast B=\frac{AB+BA}{2} the Jordan product of A,B ? E(H)A,B\in\mathcal{E(H)}. The main purpose of this note is to study some of the algebraic properties of the Jordan product of effects. Many of our results show that algebraic conditions on A∗B imply that A and B commute for the usual operator product. And there are many common properties between Jordan product and sequential product of effects. For example, if A∗B satisfies certain associative laws, then AB=BA. Moreover, A*B ? P(H)A\ast B\in{\mathcal{P(H) }} if and only if A°B ? P(H)A\circ B\in{\mathcal{P(H)}}.     

Global Solutions to the 3-D Incompressible Anisotropic Navier-Stokes System in the Critical Spaces

In this paper, we consider the global wellposedness of the 3-D incompressible anisotropic Navier-Stokes equations with initial data in the critical Besov-Sobolev type spaces B{\mathcal{B}} and B^{-\frac12,\frac12}₄{\mathcal{B}^{-\frac12,\frac12}_4} (see Definitions 1.1 and 1.2 below). In particular, we proved that there exists a positive constant C such that (ANS _ν ) has a unique global solution with initial data u₀ = (u₀^h, u₀³){u_0 = (u_0^h, u_0^3)} which satisfies ||u₀^h||_B exp(\fracCn⁴ ||u₀³||_B⁴) ￡ c₀n{\|u_0^h\|_{\mathcal{B}} \exp\bigl(\frac{C}{\nu^4} \|u_0^3\|_{\mathcal{B}}^4\bigr) \leq c_0\nu} or ||u₀^h||_{B^{-\frac12,\frac12}4} exp(\fracCn⁴ ||u₀³||_{B^{-\frac12,\frac12}4}⁴) ￡ c₀n{\|u_0^h\|_{\mathcal{B}^{-\frac12,\frac12}_{4}} \exp \bigl(\frac{C}{\nu^4} \|u_0^3\|_{\mathcal{B}^{-\frac12,\frac12}_{4}}^4\bigr)\leq c_0\nu} for some c ₀ sufficiently small. To overcome the difficulty that Gronwall’s inequality can not be applied in the framework of Chemin-Lerner type spaces, [(L^p_t)\tilde](B){\widetilde{L^p_t}(\mathcal{B})}, we introduced here sort of weighted Chemin-Lerner type spaces, [(L²_{t, f})\tilde](B){\widetilde{L^2_{t, f}}(\mathcal{B})} for some apropriate L ¹ function f(t).     

Dielectric relaxation and ionic conductivity studies of Ag<Subscript>2</Subscript>ZnP<Subscript>2</Subscript>O<Subscript>7</Subscript>

The complex impedance of the Ag₂ZnP₂O₇ compound has been investigated in the temperature range 419–557 K and in the frequency range 200 Hz–5 MHz. The Z′ and Z′ versus frequency plots are well fitted to an equivalent circuit model. Dielectric data were analyzed using complex electrical modulus M* for the sample at various temperatures. The modulus plot can be characterized by full width at half-height or in terms of a non-exponential decay function f( \textt ) = exp( - \textt/t )^b \phi \left( {\text{t}} \right) = \exp {\left( { - {\text{t}}/\tau } \right)^\beta } . The frequency dependence of the conductivity is interpreted in terms of Jonscher’s law: s( w) = s_\textdc + \textAwⁿ \sigma \left( \omega \right) = {\sigma_{\text{dc}}} + {\text{A}}{\omega^n} . The conductivity σ _dc follows the Arrhenius relation. The near value of activation energies obtained from the analysis of M″, conductivity data, and equivalent circuit confirms that the transport is through ion hopping mechanism dominated by the motion of the Ag⁺ ions in the structure of the investigated material.     

Analysis of the <Emphasis Type="Italic">Y</Emphasis>(4140) and related molecular states with QCD sum rules

In this article, we assume that there exist scalar D^*[`(D)]<sup>*</sup>{D}^{\ast}{\bar {D}}^{\ast}, D<sub>s</sub><sup>*</sup>[`(D)]_s^*{D}_{s}^{\ast}{\bar{D}}_{s}^{\ast}, B^*[`(B)]<sup>*</sup>{B}^{\ast}{\bar {B}}^{\ast} and B<sub>s</sub><sup>*</sup>[`(B)]_s^*{B}_{s}^{\ast}{\bar{B}}_{s}^{\ast} molecular states, and study their masses using the QCD sum rules. The numerical results indicate that the masses are about (250–500) MeV above the corresponding D ^*–[`(D)]<sup>*</sup>{\bar{D}}^{\ast}, D <sub>s</sub> <sup>*</sup>–[`(D)]_s^*{\bar {D}}_{s}^{\ast}, B ^*–[`(B)]<sup>*</sup>{\bar{B}}^{\ast} and B <sub>s</sub> <sup>*</sup>–[`(B)]_s^*{\bar {B}}_{s}^{\ast} thresholds, the Y(4140) is unlikely a scalar D_s^*[`(D)]<sub>s</sub><sup>*</sup>{D}_{s}^{\ast}{\bar{D}}_{s}^{\ast} molecular state. The scalar D<sup>*</sup>[`(D)]^*D^{\ast}{\bar{D}}^{\ast}, D_s^*[`(D)]<sub>s</sub><sup>*</sup>D_{s}^{\ast}{\bar{D}}_{s}^{\ast}, B<sup>*</sup>[`(B)]^*B^{\ast}{\bar{B}}^{\ast} and B_s^*[`(B)]<sub>s</sub><sup>*</sup>B_{s}^{\ast}{\bar{B}}_{s}^{\ast} molecular states maybe not exist, while the scalar D￠<sup>*</sup>[`(D)]^￠*{D'}^{\ast}{\bar{D}}^{\prime\ast}, D_s^￠*[`(D)]<sub>s</sub><sup>￠*</sup>{D}_{s}^{\prime\ast}{\bar{D}}_{s}^{\prime\ast}, B<sup>￠*</sup>[`(B)]^￠*{B}^{\prime\ast}{\bar{B}}^{\prime\ast} and B_s^￠*[`(B)]_s^￠*{B}_{s}^{\prime\ast}{\bar{B}}_{s}^{\prime\ast} molecular states maybe exist.     

A new approach to diffusion-limited reaction rate theory

A new approach to the diffusion-limited reaction rate theory is developed on the base of a similar approach to consideration of Brownian coagulation, recently proposed by the author. The traditional diffusion approach to calculation of the reaction rate is critically analyzed. In particular, it is shown that the traditional approach is applicable only to the special case of reactions with a large reaction radius, [`(r)]<sub>A</sub> << R<sub>AB</sub> << [`(r)]_B\bar r_A \ll R_{AB} \ll \bar r_B (where [`(r)]<sub>A</sub>\bar r_A, [`(r)]_B\bar r_B are the mean interparticle distances), and becomes inappropriate to calculation of the reaction rate in the case of a relatively small reaction radius, R_AB << [`(r)]<sub>A</sub>R_{AB} \ll \bar r_A, [`(r)]_B\bar r_B. In the latter, most general case particles collisions occurmainly in the kinetic regime (rather than in the diffusion one) characterized by a homogeneous (at random) spatial distribution of particles. Homogenization of particles distribution occurs owing to particles diffusion mixing on the length scale of the mean interparticle distance with the characteristic diffusion time being small in comparison with the characteristic reaction time. The calculated reaction rate for a small reaction radius in 3D formally (and casually) coincides with the expression derived in the traditional approach for reactions with a large reaction radius, however, notably deviates at large times from the traditional result in the plane (2D) geometry.     

Stochastic Porous Media Equations and Self-Organized Criticality: Convergence to the Critical State in all Dimensions

If X = X(t, ξ) is the solution to the stochastic porous media equation in O ì R^d, 1 ￡ d ￡ 3,{\mathcal{O}\subset \mathbf{R}^d, 1\le d\le 3,} modelling the self-organized criticality (Barbu et al. in Commun Math Phys 285:901–923, 2009) and X _c is the critical state, then it is proved that ò^￥₀m(O\O^t₀)dt < ￥,\mathbbP-a.s.{\int^{\infty}_0m(\mathcal{O}{\setminus}\mathcal{O}^t_0)dt<{\infty},\mathbb{P}\hbox{-a.s.}} and lim_t?￥ ò_O|X(t)-X_c|dx = l < ￥, \mathbbP-a.s.{\lim_{t\to{\infty}} \int_\mathcal{O}|X(t)-X_c|d\xi=\ell<{\infty},\ \mathbb{P}\hbox{-a.s.}} Here, m is the Lebesgue measure and O^t_c{\mathcal{O}^t_c} is the critical region {x ? O; X(t,x)=X_c(x)}{\{\xi\in\mathcal{O}; X(t,\xi)=X_c(\xi)\}} and X _c (ξ) ≤ X(0, ξ) a.e. x ? O{\xi\in\mathcal{O}}. If the stochastic Gaussian perturbation has only finitely many modes (but is still function-valued), lim_{t ? ￥} ò_K|X(t)-X_c|dx = 0{\lim_{t \to {\infty}} \int_K|X(t)-X_c|d\xi=0} exponentially fast for all compact K ì O{K\subset\mathcal{O}} with probability one, if the noise is sufficiently strong. We also recover that in the deterministic case ℓ = 0.     

Anomalous dipion invariant mass distribution of the ?(4S) decays into ?(1S)π+π? and ?(2S)π+π?

To solve the discrepancy between the experimental data on the partial widths and lineshapes of the dipion emission of ϒ(4S) and the theoretical predictions, we suggest that there is an additional contribution, which had not been taken into account in previous calculations. Noticing that the mass of ϒ(4S) is above the production threshold of B[`(B)]B\bar{B}, the contribution of the sequential process \varUpsilon(4S)? B[`(B)]? \varUpsilon(nS)+S?\varUpsilon(nS)+p⁺p^-\varUpsilon(4S)\to B\bar{B}\to \varUpsilon(nS)+S\to\varUpsilon(nS)+\pi^{+}\pi^{-} (n=1,2) may be sizable, and its interference with that from the direct production would be important. The goal of this work is to investigate if a sum of the two contributions with a relative phase indeed reproduces the data. Our numerical results on the partial widths and the lineshapes d\varGamma(\varUpsilon(4S)?\varUpsilon(2S,1S)p⁺p^-)/d(m_p⁺p^-)d\varGamma(\varUpsilon(4S)\to\varUpsilon(2S,1S)\pi^{+}\pi^{-})/d(m_{\pi ^{+}\pi^{-}}) are satisfactorily consistent with the measurements; thus the role of this mechanism is confirmed. Moreover, with the parameters obtained by fitting the data of the Belle and BaBar collaborations, we predict the distributions dΓ(ϒ(4S)→ϒ(2S,1S)π ⁺ π ⁻)/dcosθ, which have not been measured yet.     

Ionization of Coulomb Systems in {\mathbb{R}^3} by Time Periodic Forcings of Arbitrary Size

We analyze the long time behavior of solutions of the Schrödinger equation ${i\psi_t=(-\Delta-b/r+V(t,x))\psi}We analyze the long time behavior of solutions of the Schr?dinger equation iy_t=(-D-b/r+V(t,x))y{i\psi_t=(-\Delta-b/r+V(t,x))\psi}, x ? \mathbbR³{x\in\mathbb{R}^3}, r =  |x|, describing a Coulomb system subjected to a spatially compactly supported time periodic potential V(t, x) =  V(t +  2π/ω, x) with zero time average.     

Asymptotics for Large Time of Global Solutions to the Generalized Kadomtsev–Petviashvili Equation

We study the large time asymptotic behavior of solutions to the generalized Kadomtsev-Petviashvili (KP) equations $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP $ \left\{\alignedat2 &u_t + u_{xxx} + \sigma\partial_x^{-1}u_{yy}= - (u^{\rho})_x, &;&;\qquad (t,x,y) \in {\bold R}\times {\bold R}^2,\\ \vspace{.5\jot} &u(0,x,y) = u_0 (x,y),&;&; \qquad (x,y) \in{\bold R}^2, \endalignedat \right. \TAG KP where † = 1 or † = m 1. When „ = 2 and † = m 1, (KP) is known as the KPI equation, while „ = 2, † = + 1 corresponds to the KPII equation. The KP equation models the propagation along the x-axis of nonlinear dispersive long waves on the surface of a fluid, when the variation along the y-axis proceeds slowly [10]. The case „ = 3, † = m 1 has been found in the modeling of sound waves in antiferromagnetics [15]. We prove that if „ S 3 is an integer and the initial data are sufficiently small, then the solution u of (KP) satisfies the following estimates: ||u(t)||_￥ ￡ C (1 + |t|)^-1 (log(2+|t|))^k, ||u_x(t)||_￥ ￡ C (1 + |t|)^-1 \|u(t)\|_\infty \le C (1 + |t|)^{-1} (\log (2+|t|))^{\kappa}, \|u_x(t)\|_\infty \le C (1 + |t|)^{-1} for all t ] R, where s = 1 if „ = 3 and s = 0 if „ S 4. We also find the large time asymptotics for the solution.     

Chiral color quark symmetry and possible constraints on the <Emphasis Type="Italic">G</Emphasis>′-boson mass from tevatron and LHC data

A gauge model featuring a chiral color symmetry of quarks was considered, and possible manifestations of this symmetry in proton-antiproton and proton-proton collisions at the Tevatron and LHC energies were studied. The cross section s_{t[`(t)]</sub>\sigma _{t\bar t} for the production of t[`(t)]t\bar t quark pairs at the Tevatron and the forward-backward asymmetry A_FB<sup>p[`(p)]</sup>A_{FB}^{p\bar p} in this process were calculated and analyzed with allowance for the contributions of the G′-boson predicted by the chiral color symmetry of quarks, the G′-boson massm <sub>G′</sub> and the mixing angle θ <sub>G</sub> being treated as free parameters of the model. Limits on m <sub>G′</sub> versus θ <sub>G</sub> were studied on the basis of data from the Tevatron on s<sub>t[`(t)]}\sigma _{t\bar t} and A_FB^p[`(p)]A_{FB}^{p\bar p}, and the region compatible with these data within one standard deviation was found in the m _G′-θ _G plane. The region ofm _G′-mass values that is appropriate for observing the G′-boson at LHC is discussed.     

Benford’s law and cross-sections of A(n,$ \alpha$)B reactions

Benford’s law, also called the first-digit law, states that in lists of numbers from many quite disparate databases, the leading digit is distributed in a non-uniform but actually logarithmic way. We have investigated the first-digit distribution of experimental cross-sections of A(n,a \alpha)B reactions. In the case of below-barrier a \alpha -particle emission from compound nucleus, it is found that the (n,a \alpha) reaction cross-sections approximately follow the first-digit distribution indicated by Benford’s law. The origin of this first-digit distribution is discussed within the framework of the statistical model. In addition, Benford’s law is used to test the evaluated cross-sections of A(n,a \alpha)B reactions.     

Propagation of electromagnetic waves generated by moving sources in dispersive media

The propagation of electromagnetic waves issued by modulated moving sources of the form j( t,x ) = a( t )e ^{- iw₀ t} [(x)\dot]₀ ( t )d( x - x₀ ( t ) )j\left( {t,x} \right) = a\left( t \right)e^{ - i\omega _0 t} \dot x_0 \left( t \right)\delta \left( {x - x_0 \left( t \right)} \right) is considered, where j(t, x) stands for the current density vector, x = (x ₁, x ₂, x ₃) ∈ ℝ³ for the space variables, t ∈ ℝ for time, t → x ₀(t) ∈ ℝ³ for the vector function defining the motion of the source, ω ₀ for the eigenfrequency of the source, a(t) for a narrow-band amplitude, and δ for the standard δ function. Suppose that the media under consideration are dispersive. This means that the electric and magnetic permittivity ɛ(ω), μ(ω) depends on the frequency ω. We obtain a representation of electromagnetic fields in the form of time-frequency oscillating integrals whose phase contains a large parameter λ > 0 characterizing the slowness of the change of the amplitude a(t) and the velocity [(x)\dot]₀ ( t )\dot x_0 \left( t \right) and a large distance between positions of the source and the receiver. Applying the two-dimensional stationary phase method to the integrals, we obtain explicit formulas for the electromagnetic field and for the Doppler effects. As an application of our approach, we consider the propagation of electromagnetic waves produced by moving source in a cold nonmagnetized plasma and the Cherenkov radiation in dispersive media.     

Analysis of the X(1835) and related baryonium states with Bethe-Salpeter equation

In this article, we study the mass spectrum of the baryon-antibaryon bound states p [`(p)] \bar{{p}} , S \Sigma [`(S)] \bar{{\Sigma}} , X \Xi [`(X)] \bar{{\Xi}} , L \Lambda [`(L)] \bar{{\Lambda}} , p [`(N)] \bar{{N}}(1440) , S \Sigma [`(S)] \bar{{\Sigma}}(1660) , X \Xi [`(X)]<sup>￠</sup> \bar{{\Xi}}^{{\prime}}_{} and L \Lambda [`(L)] \bar{{\Lambda}}(1600) with the Bethe-Salpeter equation. The numerical results indicate that the p [`(p)] \bar{{p}} , S \Sigma [`(S)] \bar{{\Sigma}} , X \Xi [`(X)] \bar{{\Xi}} , p [`(N)] \bar{{N}}(1440) , S \Sigma [`(S)] \bar{{\Sigma}}(1660) , X \Xi [`(X)]^￠ \bar{{\Xi}}^{{\prime}}_{} bound states maybe exist, and the new resonances X(1835) and X(2370) can be tentatively identified as the p [`(p)] \bar{{p}} and p [`(N)] \bar{{N}}(1440) (or N(1400)[`(p)] \bar{{p}} bound states, respectively, with some gluon constituents, and the new resonance X(2120) may be a pseudoscalar glueball. On the other hand, the Regge trajectory favors identifying the X(1835) , X(2120) and X(2370) as the excited h^￠ \eta^{{\prime}}_{}(958) mesons with the radial quantum numbers n = 3 , 4 and 5, respectively.     

Production of the P-wave excited Bc-states through the Z0 boson decays

In Deng et al. (Eur. Phys. J. C 70:113, 2010), we have dealt with the production of the two color-singlet S-wave (c[`(b)])(c\bar{b})-quarkonium states B<sub>c</sub>(|(c[`(b)])₁[¹S₀]?)B_{c}(|(c\bar {b})_{\mathbf{1}}[^{1}S_{0}]\rangle) and B^*_c(|(c[`(b)])<sub>1</sub>[<sup>3</sup>S<sub>1</sub>]?)B^{*}_{c}(|(c\bar{b})_{\mathbf{1}}[^{3}S_{1}]\rangle) through the Z <sup>0</sup> boson decays. As an important sequential work, we make a further discussion on the production of the more complicated P-wave excited (c[`(b)])(c\bar{b})-quarkonium states, i.e. |(c[`(b)])<sub>1</sub>[<sup>1</sup>P<sub>1</sub>]?|(c\bar{b})_{\mathbf{1}}[^{1}P_{1}]\rangle and |(c[`(b)])₁[³P_J]?|(c\bar{b})_{\mathbf{1}}[^{3}P_{J}]\rangle (with J=(1,2,3)). More over, we also calculate the channel with the two color-octet quarkonium states |(c[`(b)])<sub>8</sub>[<sup>1</sup>S<sub>0</sub>]g?|(c\bar{b})_{\mathbf{8}}[^{1}S_{0}]g\rangle and |(c[`(b)])₈[³S₁]g?|(c\bar{b})_{\mathbf{8}}[^{3}S_{1}]g\rangle, whose contributions to the decay width maybe at the same order of magnitude as that of the color-singlet P-wave states according to the naive nonrelativistic quantum chromodynamics scaling rules. The P-wave states shall provide sizable contributions to the B _c production, whose decay width is about 20% of the total decay width \varGamma _{Z⁰? Bc}\varGamma _{Z^{0}\to B_{c}}. After summing up all the mentioned (c[`(b)])(c\bar {b})-quarkonium states’ contributions, we obtain \varGamma _{Z⁰? Bc}=235.9^+352.8_-122.0\varGamma _{Z^{0}\to B_{c}}=235.9^{+352.8}_{-122.0} KeV, where the errors are caused by the main sources of uncertainty.     

Low Temperature Properties for Correlation Functions in Classical N-Vector Spin Models

We obtain convergent multi-scale expansions for the one-and two-point correlation functions of the low temperature lattice classical N - vector spin model in d S 3 dimensions, N S 2. The Gibbs factor is taken as exp[-b(1/2 ||?f||² +l/8 || |f|² - 1 ||² + v/2||f- h||²)], \exp [-\beta (1/2 ||\partial \phi||^2 +\lambda/8 ||\, |\phi|^2 - 1 ||^2 + v/2||\phi - h||^2)], where f(x), h ? R^N\phi(x), h \in R^N, x ? Z^dx \in Z^d, |h|=1, b < ￥|h|=1, \beta < \infty, l 3 ￥\lambda \geq \infty are large and 0 < v h 1. In the thermodynamic and v ˉ 0v \downarrow 0 limits, with h = e₁, and j L ‘^½ ‘, the expansion gives áf₁(x)? = 1+0(1/b^1/2)\langle \phi_1(x)\rangle = 1+0(1/\beta^{1/2}) (spontaneous magnetization), áf₁(x)f_i(y)? = 0\langle \phi_1(x)\phi_i(y)\rangle=0, áf_i (x)f_i (y)? = c₀ D^-1(x,y)+R(x,y)\langle \phi_i (x)\phi_i (y)\rangle = c_0 \Delta^{-1}(x,y)+R(x,y) (Goldstone Bosons), i = 2, 3, ?, Ni= 2, 3,\,\ldots, N, and áf₁(x)f₁(y)?^T=R￠(x,y)\langle \phi_1(x)\phi_1(y)\rangle^T=R'(x,y), where |R(x,y)||R(x,y)|, |R￠(x,y)| < 0(1)(1+|x-y|)^d-2+r|R'(x,y)|< 0(1)(1+|x-y|)^{d-2+\rho} for some „ > 0, and c₀ is aprecisely determined constant.     

A Certain Necessary Condition of Potential Blow up for Navier-Stokes Equations

We show that a necessary condition for T to be a potential blow up time is lim_{t- T} ||v(·,t)||_L3=￥{\lim\nolimits_{t\uparrow T} \|v(\cdot,t)\|_{L_3}=\infty}.     

Combined constraints on modified Chaplygin gas model from cosmological observed data: Markov Chain Monte Carlo approach

We use the Markov Chain Monte Carlo method to investigate a global constraints on the modified Chaplygin gas (MCG) model as the unification of dark matter and dark energy from the latest observational data: the Union2 dataset of type supernovae Ia (SNIa), the observational Hubble data (OHD), the cluster X-ray gas mass fraction, the baryon acoustic oscillation (BAO), and the cosmic microwave background (CMB) data. In a flat universe, the constraint results for MCG model are, W_bh² = 0.02263^+0.00184_-0.00162 (1s)^+0.00213_-0.00195 (2s){\Omega_{b}h^{2}\,{=}\,0.02263^{+0.00184}_{-0.00162} (1\sigma)^{+0.00213}_{-0.00195} (2\sigma)}, B_s = 0.7788^+0.0736_-0.0723(1s)^+0.0918_-0.0904 (2s){B_{s}\,{=}\,0.7788^{+0.0736}_{-0.0723}(1\sigma)^{+0.0918}_{-0.0904} (2\sigma)}, a = 0.1079^+0.3397_-0.2539 (1s)^+0.4678_-0.2911 (2s){\alpha\,{=}\,0.1079^{+0.3397}_{-0.2539} (1\sigma)^{+0.4678}_{-0.2911} (2\sigma)}, B = 0.00189^+0.00583_-0.00756(1s)^+0.00660_-0.00915 (2s){B\,{=}\,0.00189^{+0.00583}_{-0.00756}(1\sigma)^{+0.00660}_{-0.00915} (2\sigma)}, and H₀=70.711^+4.188_-3.142 (1s)^+5.281_-4.149(2s){H_{0}=70.711^{+4.188}_{-3.142} (1\sigma)^{+5.281}_{-4.149}(2\sigma)}.     

Upper Bounds for Coarsening for the Deep Quench Obstacle Problem

The deep quench obstacle problem models phase separation at low temperatures. During phase separation, domains of high and low concentration are formed, then coarsen or grow in average size. Of interest is the time dependence of the dominant length scales of the system. Relying on recent results by Novick-Cohen and Shishkov (Discrete Contin. Dyn. Syst. B 25:251–272, 2009), we demonstrate upper bounds for coarsening for the deep quench obstacle problem, with either constant or degenerate mobility. For the case of constant mobility, we obtain upper bounds of the form t ^1/3 at early times as well as at times t for which E(t) ￡ \frac(1-[`(u)]<sup>2</sup>)4E(t)\le\frac{(1-\overline{u}^{2})}{4}, where E(t) denotes the free energy. For the case of degenerate mobility, we get upper bounds of the form t <sup>1/3</sup> or t <sup>1/4</sup> at early times, depending on the value of E(0), as well as bounds of the form t <sup>1/4</sup> whenever E(t) ￡ \frac(1-[`(u)]²)4E(t)\le\frac{(1-\overline{u}^{2})}{4}.     

Dielectric spectroscopy study of the new compound [C<Subscript>12</Subscript>H<Subscript>17</Subscript>N<Subscript>2</Subscript>]<Subscript>2</Subscript>CdCl<Subscript>4</Subscript>

The temperature dependence of the electrical conductivity of the compound 2,4,4-trimethyl-4,5-dihydro-3H-benzo[b] [1,4] diazepin-1-ium tetrachlorocadmiate in the different phases follows the Arrhenius law. The imaginary part of the permittivity constant is analyzed with the Cole–Cole formalism. In the temperature range 348–394 K, the activation energy of conductivity obtained from complex permittivity in regions I and II are, respectively, 1.03 and 0.33 eV, and E _m (in regions I and II are, respectively, 0.97 and 0.36 eV) obtained from the modulus spectra is close, suggesting that the ion transport is probably due to a hopping mechanism. The Kohlrausch–Williams–Watts function, j(t) = exp( - ( \fractt_\textKWW )^b ) \varphi (t) = \exp \left( { - {{\left( {\frac{t}{{{\tau_{\text{KWW}}}}}} \right)}^\beta }} \right) , and the coupling model are utilized for analyzing electric modulus at various temperatures. The decreasing of β at 373 K is due to approaching the temperatures of change in the conduction mechanism of the sample.