Stability of Asymptotics of Christoffel–Darboux Kernels

We study the stability of convergence of the Christoffel–Darboux kernel, associated with a compactly supported measure, to the sine kernel, under perturbations of the Jacobi coefficients of the measure. We prove stability under variations of the boundary conditions and stability in a weak sense under ? ¹ and random ? ² diagonal perturbations. We also show that convergence to the sine kernel at x implies that μ({x}) = 0.     

Non-Intersecting Squared Bessel Paths at a Hard-Edge Tacnode

The squared Bessel process is a 1-dimensional diffusion process related to the squared norm of a higher dimensional Brownian motion. We study a model of n non-intersecting squared Bessel paths, with all paths starting at the same point a > 0 at time t = 0 and ending at the same point b > 0 at time t = 1. Our interest lies in the critical regime ab = 1/4, for which the paths are tangent to the hard edge at the origin at a critical time ${t^*\in (0,1)}$ . The critical behavior of the paths for n → ∞ is studied in a scaling limit with time t = t ^* + O(n ^?1/3) and temperature T = 1 + O(n ^?2/3). This leads to a critical correlation kernel that is defined via a new Riemann-Hilbert problem of size 4 × 4. The Riemann-Hilbert problem gives rise to a new Lax pair representation for the Hastings-McLeod solution to the inhomogeneous Painlevé II equation q′′(x) = xq(x) + 2q ³(x) ? ν, where ν = α + 1/2 with α > ?1 the parameter of the squared Bessel process. These results extend our recent work with Kuijlaars and Zhang (Comm Pure Appl Math 64:1305–1383, 2011) for the homogeneous case ν = 0.     

Rise of proton proton total cross section and inclusive spectra

The origin of the rise of p-p total cross section (σ_t(pp)) is investigated by making use of the inclusive sum rule. The contribution to the rise of σ_t(pp) from the development of sharp peak near the kinematical boundary in the process p+p→p+ “anything” is almost completely canceled out by the decrease of the cross section in other x-region between s=45 (GeV/c)² and s=2820 (GeV/c)². Then the net contribution to the rise of σ_t(pp) from the process p+p→p+ “anything” is very likely to be zero. The true origin of the rise of σ_t(pp) is found to be the increase of inclusive cross sections at x ? 0 between PS and ISR energies. The contribution from the process p+p→π+ “anything” at x ? 0 to the rise of σ_t(pp) is estimated to be 3.2 mb, that from the process p+p→K + “anything”, 0.71 mb and that from the process p+p→$\text{p}$ “anything”, 0.62 mb. According to our conclusion we expect that all total cross sections will rise with energy at high energy.     

Quasiparticle band structure and optical spectrum of LiBr

The electronic structures of the Fe-doped perovskite ruthenates BaRu_1?x Fe _x O₃ with x = 0, 0.25, 0.5, 0.625, 0.75, and 1 are investigated through density-functional calculations. Large exchange splitting and small crystal field splitting are found in BaFeO₃, and a contrary scenario can take place on BaRuO₃ as expected since the Ru atom has a highly extended 4d orbital. The small exchange splitting and extended 4d states are the reasons why the obtained spin magnetic moment (0.628μ _B ) is significantly lower than the spin only value (2μ _B ) for the t _2g 3↑ t _2g 1↓ electronic configuration for Ru⁴⁺ ion. Further investigations suggest that Fe substitution at the Ru sites can suppress the bandwidths of Ru 4d orbital, leading to the half-metallic behaviour in BaRu_1?x Fe _x O₃ with x = 0.625 and 0.75. The different orbital feature of the Ru⁴⁺ ions in BaRu_0.375Fe_0.625O₃ is presented, which reflects the influence of Fe dopant on Ru 4d orbitals.     

Structure and optical properties of Cd-substituted ZnO (Zn1−x Cd x O) thin films synthesized by the dc reactive magnetron sputtering

The ternary Zn_1?x Cd _x O (x = 0, 0.2) thin films with wurtzite structure and highly (002)-preferred orientations were deposited on glass substrates by the direct current (dc) reactive magnetron sputtering method. The X-ray diffraction, X-ray photoelectron spectroscopy (XPS), transmission electron microscopy (TEM), optical absorption spectra and photoluminescence (PL) were employed to investigate the structural and the optical properties in detail. The results indicated that as x varied from x = 0–0.2, the diffraction angle of the (002) peaks decreased from ~34.36° to ~33.38° and the lattice spacing increased from 0.260 to 0.268 nm. Moreover, the optical band-gap of the Zn_1?x Cd _x O thin films with the wurtzite structure decreased from 3.20 eV at x = 0–2.70 eV at x = 0.2. Correspondingly, the near-band-edge PL was tuned in a wide visible region from ~393 to 467 nm. The chemical bonding states of Cd in Zn_1?x Cd _x O alloy thin films were examined by XPS analysis.     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

The low temperature magnetic phase diagram of EuxSr1−xTiO3

Specific heat and magnetization measurements demonstrate that the antiferromagnetic (AFM) phase transition at T _N  = 5.7 K of EuTiO₃ is rapidly suppressed with Sr doping in Eu _x Sr_1?x TiO₃. Close to x = 0.25, T _N  = 0 K and AFM order vanishes. Above this critical concentration a finite transition temperature to an AFM phase is observed. The exchange couplings are derived as a function of x and the corresponding low temperature phase diagram is presented.     

Kinetics of phase transitions with singular multiplicative noise

The behavior of the most probable values of the order parameter x and the amplitude p of conjugate force fluctuations is studied for a stochastic system with a noise amplitude depending on x as |x|^a. It is shown that the phase half-plane x>0 for the canonical pair x, p is divided into isolated regions of large, intermediate, and small values of x. In the first region, the trajectories converge to values of x, p → ∞ as the time t → ∞, and the probability of their realization is negligibly small. In the intermediate region, the configuration point tends to the attraction center corresponding to a stationary ordered state. In the region , the trajectories converge to the point x=p=0 for 0<a<1/2 and to x=0, p → ∞ for 1/2<a≤1. In the former case, the probability of realization of trajectories is finite, while, in the latter case, it is negligibly small, and an absorbing state can be formed.     

Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

Limit of Fluctuations of Solutions of Wigner Equation

We consider fluctuations of the solution W _ε (t, x, k) of the Wigner equation which describes energy evolution of a solution of the Schrödinger equation with a random white noise in time potential. The expectation of W _ε (t, x, k) converges as ε → 0 to \({\bar{W}(t,x,k)}\) which satisfies the radiative transport equation. We prove that when the initial data is singular in the x variable, that is, W _ε (0, x, k) = δ(x)f(k) and \({f\in {\mathcal{S}}(\mathbb{R}^d)}\), then the laws of the rescaled fluctuation \({Z_\varepsilon(t):=\varepsilon^{-1/2}[W_\varepsilon(t,x,k)-\bar{W}(t,x,k)]}\) converge, as ε → 0+, to the solution of the same radiative transport equation but with a random initial data. This complements the result of [6], where the limit of the covariance function has been considered.     

Evolution of structural,magnetic and magnetocaloric properties in Sn-doped manganites La0.57Nd0.1Sr0.33Mn1−x Sn x O3 (x = 0.05–0.3)

Structural and magnetic properties of manganites series La_0.57Nd_0.1Sr_0.33Mn_1?x Sn _x O₃ with (0.05 ≤ x ≤ 0.30) have been investigated, and the critical exponents and magnetocaloric effect are studied around the room temperature, to shed light on Sn substitution influence. A solid-state reaction method was used in the preparation. A structural study using Rietveld refinement of XRD patterns indicates rhombohedral structure with R \( \overline{3} \) c space group for (0.05 ≤ x ≤ 0.20) and shows the existence of a secondary phase attributed to the neodymium tin oxide (Nd₂Sn₂O₇) pyrochlore for x = 0.3. The variation of the magnetization (M) vs. temperature (T), under an applied magnetic field of 0.05 T, reveals a ferromagnetic–paramagnetic transition at the Curie temperature T _C. In addition, it was discovered that increasing the tin content leads to a reduction in magnetization and a lowering of T _C from 282 K (x = 0.05) to 158 K (x = 0.20) with increasing Sn substitution. The samples exhibit the characteristics of spin/cluster-glass state which is evident from (zero-field-cooled and field-cooled) magnetization vs. temperature curves. Indeed, the thermal evolution of magnetization in the ferromagnetic phase at low temperature varies as T ^3/2, in accordance with Bloch’s law. The spin-stiffness constant D obtained from the Bloch constant was determined. A large magnetocaloric effect has been observed in both samples (x = 0.05 and x = 0.10): the maximum entropy change, \( \left| {\varDelta S_{\text{M} }^{\text{peak}} } \right| \) , reaches the highest value of 3.22 J/kg K under a magnetic field change of 5 T with a RCP value of 56 J/kg for x = 0.10 composition. This opens an interesting opportunity to this compound to compete with materials which work as magnetic refrigerants near room temperature. Besides, we show that the samples follow the conventional behavior of a second-order ferromagnetic transition. This was possible by investigating the critical behavior at the transition region by adopting the modified Arrott plot method. The values of the critical exponents (β, γ, δ and n) are determined and they are between those predicted by the three-dimensional Heisenberg model.     

On Optimal Separation of Eigenvalues for a Quasiperiodic Jacobi Matrix

We consider quasiperiodic Jacobi matrices of size N with analytic coefficients. We show that, in the positive Lyapunov exponent regime, after removing some small sets of energies and frequencies, any eigenvalue is separated from the rest of the spectrum by N ^?1(log N)^?p , with p > 15.     

Magnetization and Mössbauer study of double layer perovskites La1.5Ca1.5Mn2 − x Fe x O7 prepared by sol–gel method

Magnetization and Mössbauer studies have been made for understanding magnetic behavior of three double perovskite systems La_1.5Ca_1.5Mn_2???x Fe _x O₇ corresponding to x = 0.05, 0.10 and 0.50. These have been prepared following sol–gel route. Substitution of Fe does not lead to any major change in the tetragonal cell but increased iron leads to greater distortion in octahedral site. The three samples undergo paramagnetic–ferromagnetic transition. Curie temperature (T _c) for the system with 0.05 Fe is ～150 K which is lower than (190 K) for the system without iron; with 0.50 Fe T _c goes below 50 K. Iron goes as Fe^3?+? and replaces Mn in ab plane. With increasing Fe the valence states of Mn get re-distributed in a way that number of the Jahn–Teller ions Mn^3?+? increases and that of the pairs of Mn^3?+?–O–Mn^4?+? experiencing double exchange decreases.     

Magnetic properties and Mössbauer studies in Y1−x Gd x Fe2

Alloys of Y_1???x Gd _x Fe₂B _y (x = 0, 0.25, 0.5, 0.75 and 1; y = 0, 0.1, 0.15 and 0.2) have been prepared and investigated for structural and magnetic properties. The compounds with x = 0 and 1 are found to form in single phase with C15-type cubic Laves phase structure, while those with x = 0.25, 0.5 and 0.75 are observed to form with small quantities of secondary (Y,Gd)Fe₃ phase. The lattice parameters, Curie temperature and the average Fe hyperfine field are found to increase with increasing x. The Gd–Gd and Gd–Fe interactions are attributed to be the main reason for the enhancement of magnetic properties. Boron was found to stabilize the (Y,Gd)Fe₂ phase without affecting the magnetic properties.     

A study of the microstructure,thermal properties and wetting kinetics of Sn–3Ag–<Emphasis Type="Italic">x</Emphasis>Zn lead-free solders

Microstructure, thermal properties and wetting kinetics of Sn–3Ag–xZn solders (x = 0.4, 0.6, 0.8, 1, 2 and 4 wt%) were systematically investigated. The results indicate that a small amount of Zn (Zn wt% ≤ 1 wt%) has a rather moderate effect on the microstructure morphology of the Sn–3Ag–xZn solders. The microstructures are composed of a β-Sn phase and the mixture of Ag₃Sn and ζ-AgZn particles. However, the β-Sn phase reduces its volume fraction in the entire microstructure and the intermetallic compounds population increases with the increasing of Zn content. The microstructure is dramatically changed with a further increase in the Zn content. The γ-AgZn phase is formed in a Sn–3Ag–2Zn solder. The ε-AgZn phase is formed in a Sn–3Ag–4Zn solder. The melting temperature and the undercooling of the Sn–3Ag–xZn solder alloys decrease with the increase in Zn content, reach to a minimum value when the content of Zn is 1 wt%, and then increase with further increase in Zn content. The Sn–3Ag–1Zn demonstrates the minimum value of 228.13 °C in the melting temperature and 13.87 °C in undercooling. The wetting kinetics of the main spreading stage features the power law of R ⁿ  ~ t (n = 1), which is controlled by chemical reactions at the triple line.     

Substitution effects on the bulk and surface properties of (Li,Ni)Mn<Subscript>2</Subscript>O<Subscript>4</Subscript>

Manganese oxides of spinel structure, LiMn₂O₄, Li_1-x Ni _x Mn₂O₄ (0.25 ≤ x≤ 0.75), and NiMn₂O₄, were studied by EDS, XRD, SEM, magnetic (M-H, M-T), and XPS measurements. The samples were synthesized by an ultrasound-assisted sol-gel method. EDS analysis showed good agreement with the formulations of the oxides. XRD and Rietveld refinement of X-ray data indicate that all samples crystallize in the Fd3m space group characteristic of the cubic spinel structure. The a-cell parameter ranges from a = 8.2276 Å (x = 0) to a = 8.3980 Å (x = 1). SEM results showed particle agglomerates ranging in size from 2.3 μm (x = 0) down to 0.8 μm (x = 1). Hysteresis magnetization vs. applied field curves in the 5–300K range was recorded. ZFC-FC measurements indicate the presence of two magnetic paramagnetic-ferrimagnetic transitions. The experimental Curie constant was found to vary from 5 to 7.1 cm³ K mol^?1 for the range of compositions studied (0 ≤ x ≤ 1). XPS studies of these oxides revealed the presence of Ni²⁺, Mn³⁺, and Mn⁴⁺. The experimental Ni/Mn atomic ratios obtained by XPS were in good agreement with the nominal values. A linear relationship of the average oxidation state of Mn with Ni content was observed. The oxide’s cation distributions as a function of Ni content from x = 0 ?Li⁺[Mn³⁺Mn⁴⁺]O₄ to x = 1 \( {\mathrm{Ni}}_{0.35}^{2+}{\mathrm{Mn}}_{0.65}^{3+}\left[{\mathrm{Ni}}_{0.65}^{2+}\right.\left.{\mathrm{Mn}}_{1.35}^{3+}\right]{\mathrm{O}}_4 \) were proposed.     

On Conjectures of A. Eremenko and A. Gabrielov for Quasi-Exactly Solvable Quartic

We study polynomials p(x) satisfying a differential equation of the form p′′?h′p′ + Hp = 0, where h = x ³/3 + ax and H is a polynomial. We prove a conjecture of A. Eremenko and A. Gabrielov.     

Correlation between structural and giant magnetoresistance properties of Fe–Ag nanogranular films

Fe _x Ag_1?x granular thin-films, with the atomic Fe concentration, x, ranging from 0 up to 0.5, were deposited by dc magnetron co-sputtering. The giant magnetoresistance (GMR) intensity is maximum at x _I  = 0.32, while the maximum of GMR efficiency, γ, i.e., the change of GMR intensity for a unit change of reduced squared magnetization, is observed at x _γ = 0.26. Owing to the spin-dependent scattering features, the GMR intensity and γ depend on both the concentration and the arrangement of the magnetic material. Therefore, to explain the difference between x _I and x _γ and to understand how the structural properties affect the magnetoresistive behaviour, we performed magnetization, Mössbauer and X-ray diffraction measurements as a function of x. X-ray data indicate that the granular films exhibit three different regimes: for x < 0.2, they can be described as a Fe–Ag solid solution; for 0.2 < x < 0.32 the Fe–Ag solid solution is still observed and very small Fe precipitates are found; finally, for x > 0.32, a Fe–Ag saturated solid solution is detected, containing bcc Fe clusters whose size is about 10 nm. Differently, for all the concentrations, magnetization data show the presence of Fe precipitates, whose size increases with x, and the Mössbauer investigation confirms this picture. We find that the samples grown at x = x _γ display the finest Fe dispersion within the Ag matrix, as the Fe–Ag solid solution is nearly at saturation and the Fe cluster size is of the order of a few nanometers; this arrangement possibly maximizes the magnetic/non-magnetic interface extension thus enhancing the GMR efficiency. If x is slightly increased, the increase in total Fe content compensates the GMR efficiency reduction, so the GMR intensity maximum is observed.     

Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field

We investigate solutions to the equation ? _t ?? $\mathcal{D}$ Δ?=λS ²?, where S(x, t) is a Gaussian stochastic field with covariance C(x?x′, t, t′), and x∈ $\mathbb{R}$ ^d . It is shown that the coupling λ _cN (t) at which the N-th moment <? ^N (x, t)> diverges at time t, is always less or equal for $\mathcal{D}$ >0 than for $\mathcal{D}$ =0. Equality holds under some reasonable assumptions on C and, in this case, λ _cN (t)=Nλ _c (t) where λ _c (t) is the value of λ at which <exp[λ ∫ ^t ₀ S ²(0, s) ds]> diverges. The $\mathcal{D}$ =0 case is solved for a class of S. The dependence of λ _cN (t) on d is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, $\mathcal{D}$ →i $\mathcal{D}$ , the case of interest for backscattering instabilities in laser-plasma interaction.     

Asymptotics of Block Toeplitz Determinants and the Classical Dimer Model

We compute the asymptotics of a block Toeplitz determinant which arises in the classical dimer model for the triangular lattice when considering the monomer-monomer correlation function. The model depends on a parameter interpolating between the square lattice (t = 0) and the triangular lattice (t = 1), and we obtain the asymptotics for 0 < t ≤ 1. For 0 < t < 1 we apply the Szegö Limit Theorem for block Toeplitz determinants. The main difficulty is to evaluate the constant term in the asymptotics, which is generally given only in a rather abstract form.