Construction of solutions with exactly k blow-up points for the Schrödinger equation with critical nonlinearity

We consider the nonlinear Schrödinger equation: (1) $${{i\partial u} \mathord{\left/ {\vphantom {{i\partial u} {\partial t}}} \right. \kern-\nulldelimiterspace} {\partial t}} = - \Delta u - \left| u \right|^{{4 \mathord{\left/ {\vphantom {4 N}} \right. \kern-\nulldelimiterspace} N}} uandu\left( {0,.} \right) = \varphi \left( . \right),$$ whereu:[0,T)×? ^N →?. For any given pointsx ₁,x ₂,...,x _k in ? ^N , we construct a solution of Eq. (1),u(t), which blows up in a finite timeT at exactlyx ₁,x ₂,...,x _k . In addition, we describe the precise behavior of the solutionu(t) whent→T, at the blow-up points {x ₁,x ₂,...,x _k } and in ? ^N ?{x ₁,x ₂,...,x _k }.     

X-ray determination of the linear compressibilities of (SN)x

The linear compressibilities of (SN)_x have been determined up to 22 kbar using a hydrostatic-pressure X-ray cell: at 1 bar, k_a = 5.5;k_b = 2.2; k_c = 5.0; k_[201] = 3.8 —; all in units of 10^-12cm²/dyne. The results are compared with other chain-like solids and confirm the anisotropic three-dimensional bonding character of (SN)_x. Pressure dependence of the electrical conductivity is discussed.     

Preliminary results of the experiment on search for neutron nuclei on a nuclear reactor

An experiment on search for neutron nuclei in the reaction of neutron-induced fission of ²³⁵U nuclei has been performed on a nuclear reactor. The hypothetical reaction ¹²²Te(^xn, (x ? k)n)^{122 + k} Te → (β^?) → ^{122 + k} I (x = k ≥ 10) has been investigated. The radiochemical method for selecting iodine isotopes from tellurium was used. The upper limit on the probability of formation of neutron clusters has been obtained: P _k ≤ 10^?8 (fission)^?1 for ¹¹n clusters and P _k ≤ 10^?9 (fission)^?1 for ¹¹n clusters.     

Asymptotic inverse spectral problem for anharmonic oscillators

We study perturbationsL=A+B of the harmonic oscillatorA=1/2(??²+x ²?1) on ?, when potentialB(x) has a prescribed asymptotics at ∞,B(x)～|x|^?α V(x) with a trigonometric even functionV(x)=Σa _mcosω _m x. The eigenvalues ofL are shown to be λ _k =k+μ _k with small μ _k =O(k ^?γ), γ=1/2+1/4. The main result of the paper is an asymptotic formula for spectral fluctuations {μ _k }, $$\mu _k \sim k^{ - \gamma } \tilde V(\sqrt {2k} ) + c/\sqrt {2k} ask \to \infty ,$$ whose leading term \(\tilde V\) represents the so-called “Radon transform” ofV, $$\tilde V(x) = const\sum {\frac{{a_m }}{{\sqrt {\omega _m } }}\cos (\omega _m x - \pi /4)} .$$ as a consequence we are able to solve explicitly the inverse spectral problem, i.e., recover asymptotic part |x ^?α|V(x) ofB from asymptotics of {µ _k }. ₁ ^∞      

The evolutionary public goods game on scale-free networks with heterogeneous investment

The public goods game (PGG) is generally considered as a suitable paradigm to explain ubiquitous cooperative behavior. In this study, we investigated the evolutionary PGG on scale-free networks and studied the effect of individual heterogeneity by setting the cooperator x an investment value correlated to its degree as I_x=N⋅k_x^β/∑_jk_j^β, where k_x is the degree of x, j runs over all players and β is a tunable parameter. It is shown that the cooperation level is remarkably promoted by negative values of β whereas it is highly depressed by positive values of β. Moreover, the effect of environmental noise has also been investigated. Our result may sharpen the understanding of cooperation induced by the individual diversity.     

On-shell T-matrices in multiple scattering

The transition operator T for the scattering of a particle from N potentials V_j(x) can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in 〈k|T(k₀²±i0)|k′〉, |k| = |k′| = k₀, then each t_j is also on-shell.     

Anionic conductivity and thermal stability of single crystals of solid solutions based on calcium fluoride

The temperature and composition dependencies of the anionic conductivity in the temperature range from ambient to 1073 K were studied for single crystals of Ca_1−xGd_xF_2+x (x=1×10⁻⁴, 1×10⁻³, 1×10⁻² and 1×10⁻¹) and of Ca_0.8R_0.2F_2.2 (R=La, Ce, Pr, Nd, Gd, Dy, Er, Tm, Yb, Lu and Y), having the fluorite structure. The conductivity plots for the concentrated Ca_0.8R_0.2F_2.2 solid solutions display the low-temperature and high-temperature linear (Arrhenius) regions with the knee temperature T_k∼770 K. The values of the conductivity activation enthalpy obey the relation ΔH_HT(T>T_k)>ΔH_LT(T<T_k). The conductivity mechanism in heavily doped Ca_1−xR_xF_2+x crystals is associated with the clusters of the point defects which decrease the potential barriers for fluoride anions moving by hops over the structural sites of the anion sub-lattice. We studied the effect of the dimensional factor (doped cation radii) on the anionic transport in these crystals. A correlation between anionic transport and atomic structure of the studied crystals is discussed.     

Low-temperature property investigation of the lead indium-niobate-lead nickel-niobate solid solution

The complex perovskite solid solution (1−x) Pb(In_1/2Nb_1/2)O₃-(x) Pb(Ni_1/3Nb_2/3)O₃ has been successfully prepared by the Columbite precursor method. The temperature dependencies of the dielectric constant and pyroelectric coefficient were measured between −261 and 200 °C. Relaxor ferroelectric behavior has been noticed in all compositions across the solid solution. The room-temperature electrostrictive coefficient, Q₃₃, was 1.83×10⁻² C²/m⁴ for x=0.10. No room-temperature piezoelectric activity was detected; however, upon cooling to −261 °C the maximum coupling coefficients k_p=29%, k_t=11%, and k₃₃=31% were observed for the composition x=1.00.     

Improving the dielectric constant of Al2O3 by cerium substitution for high-k MIM applications

Process compatible high-k dielectric thin films are one of the key solutions to develop high performance metal–insulator–metal (MIM) structures for future microelectronic devices. Engineered cerium–aluminate (Ce_xAl_2–xO₃) thin films were deposited on titanium nitride metal electrodes by electron-beam co-evaporation of ceria and alumina in a molecular beam deposition chamber. X-ray photoelectron spectroscopy clearly reveals that Ce cations can be stabilized in the 3+ valence state in Ce_xAl_2–xO₃ up to x = 0.7 by accommodation in the alumina host matrix. Higher Ce content was observed to result in cerium dioxide segregation in cerium aluminate matrix, probably due to the chemical tendency of Ce cations to exist rather in the 4+ than in the 3+ state. Electrical characterization of the X-ray amorphous Ce_0.7Al_1.3O₃ films reveals a dielectric constant value of about 11 and leakage current lower than 10^?4 A/cm². No parasitic low-k interface formation between the high-k Ce_0.7Al_1.3O₃ film and the TiN metal electrode is detected.     

Results of search for heavy neutron clusters in nuclear fission

Experiments on search for heavy neutron clusters in the reaction of induced fission of ²³⁵U nuclei by neutrons have been carried out on the nuclear reactor, using the activation method. Two hypothetical reactions have been investigated: ⁹²Mo, ⁹⁴Mo, ⁹⁵Mo, ⁹⁶Mo, ⁹⁸Mo, ¹⁰⁰Mo(^xn, (x ? k)n)¹⁰⁵Mo → ¹⁰⁵Tc → ¹⁰⁵Ru → ¹⁰⁵Rh → ¹⁰⁵Pd, with a minimum transfer of 5 to 13 neutrons to activated Mo isotopes, and ¹²²Te(^xn, (x ? k)n)^{122 + k} Te → (β^?) → ^{122 + k} I, with a minimum transfer of 10 neutrons to activated ¹²²Te isotope. Radiochemical methods for selecting technetium isotopes from molybdenum and iodine from tellurium were used. For the first reaction, the upper limit on the probability of neutron cluster formation was found to be P _k ≤ 10^?8 per fission. For the second reaction, indications to the existence of heavy neutron clusters were obtained.     

Ground state energy of spin polarized hard core neutron matter

The expansion of the ground state energy of spin polarized hard core neutron matter in powers of x =k_Fc (k_F = Fermi momentum, c = hardcoreradius) is calculated up to terms ≈x⁸.     

High-resolution (e,e') study of isovector M1 and M2 transitions in the oxygen isotopes: (I). 16O

The M1 and M2 transition strength distribution for ¹⁶O in the excitation energy range from 16 to 20 MeV has been measured in a high-resolution electron scattering experiment. The M1 strength is concentrated in three sharp states at E_x = 16.22, 17.14 and 18.79 MeV (± 0.01 MeV) with B(M1, k)↑ = 0.20 ± 0.02, 0.32 ± 0.03 and 0.13 ± 0.03 μ_N², respectively. An additional strength of 0.35 ± 0.09 μ_N², distributed over eight weakly excited states with excitation energies E_x = 17.4 to 18.0 MeV, brings the total measured M1 strength to B(M1, k)↑ = 1.0 ± 0.1 μ_N². The experimental M2 strength is distributed over states at E_x = 16.82, 17.78, 18.50 and 19.0 MeV (± 0.01 MeV) with B(M2, k)↑ = 19 ± 2, 13 ± 2, 59 ± 7 and 341 ± 51 μ_N² · fm², respectively. Electric transitions were also measured to states at E_x = 16.45 MeV (2⁺, E2), 17.30 MeV (1⁺, E1) and 18.20 MeV (2⁺, E2). Calculations were performed using the modified surface delta interaction in a 2p-2h shell model for the M1 transitions and the random phase approximation for the M2 transitions. The results show the sensitivity of the M1 strength as a measure of ground-state correlations and compare well with results from the ¹⁵N($\text{p}$, γ) reaction.     

Temperature hysteresis of magnetic phase transitions in Tb1 − x Ce x Mn2O5 (x = 0, 0.20, 0.25)

The temperature evolution of the magnetic structure of multiferroics Tb_{1 ? x} Ce _x Mn₂O₅ (x = 0, 0.20, 0.25) has been investigated using the neutron scattering methods. It has been found that, despite the qualitative similarity of the magnetic states and the series of phase transitions for pure TbMn₂O₅ (TMO) and doped crystals Tb_{1 ? x} Ce _x Mn₂O₅ (TCMO, x = 0.20 and 0.25), there are significant differences in their properties. In contrast to TMO, where there are three magnetic phases, TCMO can include two magnetic phases that coexist in a wide temperature range and exhibit a rather wide temperature hysteresis. One of these phases with wave vector k ₁ = (0.5, 0, k _z1), k _z1 = 0.25, is commensurate and arises at temperatures below T _N ～ 39 K (for x = 0.2) and T _N ～ 38 K (x = 0.25). The second phase is incommensurate with wave vector k ₂ = (1/2, 0, k _z2), k _z2 = 0.256(2), and appears upon cooling at T = 21 K (x = 0.2) and T = 19 K (x = 0.25). Upon further cooling to 16 K, the component k _z2 increases to 0.292(2) and then remains constant. The component k _z1 increases to the value of 0.280(2) upon cooling in the range from 15 to 10 K and then remains constant down to 1.5 K. With an increase in the temperature, the components k _z1 and k _z2 undergo reverse changes to their initial values, but these changes occur at temperatures 7 K higher than those observed with a decrease in the temperature. For TMO, two phases also coexist, but the temperature hysteresis in this case is considerably smaller than for TCMO. This is explained by different densities of domain walls and different sizes of domains in pure and doped crystals.     

Exact solutions for a forced Burgers equation with a linear external force

We investigate the solutions of the Burgers equation , where F(x,t) is an external force and Φ(x,t) represents a forcing term. This equation is first analyzed in the absence of the forcing term by taking F(x,t)=k₁(t)−k₂(t)x into account. For this case, the solution obtained extends the usual one present in the Ornstein-Uhlenbeck process and depending on the choice of k₁(t) and k₂(t) it can present a stationary state or an anomalous spreading. Afterwards, the forcing terms Φ(x,t)=Φ₁(t)+Φ₂(t)x and Φ(x,t)=Φ₃x−Φ₄/x³ are incorporated in the previous analysis and exact solutions are obtained for both cases.     

Confined acoustic modes and spectral determination of network connectivity: Raman signatures of nanometric structure in g-GexSe1−x

Low-temperature Raman scattering results are presented on three glass compositions of Ge_xSe_1−x (15, 20, and 27 At. Ge%), and conciled with their reported structural data. The acoustic range is marked by the absence of Boson peak, and features discrete modes corresponding to nanometric dynamic aggregates. First sharp diffraction peak (FSDP) in structure factor S(k), due to Ge-Ge correlations brought about by shared Ge[Se_1/2]₄ tetrahedra medium-range structures, signifies 3D networking of Se-chains. A measure of volume-fraction of these entities determined from the FSDP area, and their number density ρ_MRS∝(k³Area)_FSDP is found to relate directly to the network connectivity, defined from optical Raman spectra in terms of the degree of cross-linking of Se-chains.     

Universal power law for the energy spectrum of breaking Riemann waves

The universal power law for the spectrum of one-dimensional breaking Riemann waves is justified for the simple wave equation. The spectrum of spatial amplitudes at the breaking time t = t _b has an asymptotic decay of k ^?4/3, with corresponding energy spectrum decaying as k ^?8/3. This spectrum is formed by the singularity of the form (x ? x _b )^1/3 in the wave shape at the breaking time. This result remains valid for arbitrary nonlinear wave speed. In addition, we demonstrate numerically that the universal power law is observed for long time in the range of small wavenumbers if small dissipation or dispersion is taken into account in the viscous Burgers or Korteweg-de Vries equations.     

Short-range order analysis and some physical properties of InxSe1−x glasses

Bulk In_xSe_1−x (with x=5-25 at%) glasses were prepared using the melt-quench technique. Short range order(SRO) was examined by the X-ray diffraction using Cu(k_α) radiation in the wave vector interval 0.28≤k≤6.5 A⁰⁻¹.The SRO parameters have been obtained from the radial distribution function. The inter-atomic distance obtained from the first and second peak are r₁=0.263 and r₂=0.460 nm, which is equivalent In-Se and Se-Se bond length. The fundamental structural unit for the studied glasses is In₂Se₃ pyramid. Using the differential scanning calorimetry (DSC), the crystallization mechanism of In_xSe_1−x chalcogenide glass has been studied. The glass transition activation energy (E_g) is 289±0.3 kj/mol.There is a correlation amongst the glass forming ability, bond strength and the number of lone pair electrons. The utility of the Gibbs-Di Marzio relation was achieved by estimating T_g theoretically.     

Phase transition behavior and piezoelectric properties of low-Li and high-Sb modified KNN based piezoceramics

(Na_0.5K_0.5)_0.975Li_0.025Nb_0.82−xSb_xTa_0.18O₃ lead-free piezoceramics were prepared by the conventional solid-state sintering method. All samples possess a pure perovskite phase, and no secondary phase could be certified. The crystal structure changes from tetragonal to pseudo-cubic with increasing amount of Sb. The ferroelectric Curie temperature (T_c) shifts to lower temperature while the tetragonal to orthorhombic phase transition temperature (T_o-t) shows no obvious change with increasing Sb⁵⁺. Enhanced piezoelectric and electromechanical properties are obtained with x=0.06: d₃₃=352 pC/N, k_p=47% and k_t=38%, showing that they could be promising candidates as lead-free piezoelectric materials.     

Magnetic structure of TmCu2Ge2

TmCu₂Ge₂ compound crystallizes in the tetragonal ThCr₂Si₂-type crystal structure. The neutron diffraction reveals the presence of an incommensurate antiferromagnetic order below T_N=2.5 K. The Tm magnetic moment of 5.0(1) μ_B at 0.47 K is parallel to the c-axis. The order is described by the propagation vector k=[k_x, k_x, 0], where k_x=0.117(3). The increase of the values of the components k_x near the Néel temperature is observed.