A Markovian Growth Dynamics on Rooted Binary Trees Evolving According to the Gompertz Curve

Inspired by biological dynamics, we consider a growth Markov process taking values on the space of rooted binary trees, similar to the Aldous-Shields (Probab. Theory Relat. Fields 79(4):509?C542, 1988) model. Fix n??1 and ??>0. We start at time 0 with the tree composed of a root only. At any time, each node with no descendants, independently from the other nodes, produces two successors at rate ??(n?k)/n, where k is the distance from the node to the root. Denote by Z _n (t) the number of nodes with no descendants at time t and let T _n =?? ^?1 nln(n/ln4)+(ln2)/(2??). We prove that 2^?n Z _n (T _n +n??), ?????, converges to the Gompertz curve exp(?(ln2)?e ^??|? ). We also prove a central limit theorem for the martingale associated to Z _n (t).     

The Excited Triplet State of Azoalkanes: Electron Spin Polarization and Magnetic Field Effects During Triplet-Sensitized Photolysis of trans-Azocumene in Solution

The triplet-sensitized photodecomposition of azocumene into nitrogen and cumyl radicals is investigated by time-resolved electron paramagnetic resonance and absorption spectroscopy. The radicals are found to be created spin polarized with a yield depending on the strength of the applied magnetic field. The phenomenon arises because in triplet azocumene, the decay into radicals competes with a fast triplet-sublevel selective intersystem crossing back to the azocumene ground state. The size of the initial spin polarization of the radicals and the magnetic field effect on their yield are determined in solvents of different viscosities. Data analysis yields rate constants for the intersystem crossing and the cleavage reaction of triplet azocumene as well as its zero-field splitting D _ZFS. At room temperature in nonpolar solvents, the most probable values are: k _x ?=?k _y ?=?1.2?×?10¹¹?s^?? and k _z ?=?1.9?×?10¹⁰?s^?? for the intersystem crossing from the energetically lower and upper triplet substates, respectively, k _p ?=?1.6?×?10⁹?s^?? for the cleavage reaction and for the zero-field splitting D _ZFS?=???.4?×?10^10?s^?? (0.18?cm^??).     

Knight shifts and magnetic susceptibilities of the intermetallic compounds CeCu4 and CeCu5

The magnetic susceptibility and Knight shift of the compounds CeCu₄ and CeCu₅ have been measured over the temperature ranges 80–800 and 140–400 K, respectively. The most important contributions to the magnetic susceptibility are the Curie-Weiss term, expressing the paramagnetism of the localized ?-electrons, and a temperature independent term, which have both been determined. The phenomenological exchange integral F_s? between the 4?-electron spins and conduction electron spins was found to be ?10.43× 10^?3 eV for CeCu₄ and 3.9 × 10^?3 eV for CeCu₅. A reversal in the sign of the s?? coupling for CeCu₅ is noted.     

The statistics of Curie-Weiss models

LetS _n denote the random total magnetization of ann-site Curie-Weiss model, a collection ofn (spin) random variables with an equal interaction of strength 1/n between each pair of spins. The asymptotic behavior for largen of the probability distribution ofS _n is analyzed and related to the well-known (mean-field) thermodynamic properties of these models. One particular result is that at a type-k critical point (S _n-nm)/n^1–1/2k has a limiting distribution with density proportional to exp[-_s ^2k/(2k)!], wherem is the mean magnetization per site and A is a positive critical parameter with a universal upper bound. Another result describes the asymptotic behavior relevant to metastability.Research supported in part by National Science Foundation Grants MPS 76-06644 (to RSE) and MPS 74-04870 A01 (to CMN).     

Some tips on the decomposition of tensor product representations of compact connected lie groups

Let ?₁, ?₂, ?₃, ?₄ be irreducible representations of a compact connected semisimple Lie group G with highest weights Λ₁, Λ₂, Λ₃, Λ₄, respectively. Let ?₁?₂ (resp. ?₃?₄) be irreducible representations of G with highest weights Λ₁·Λ₂ (resp. Λ₃·Λ₄). It is assumed that one knows the Clebsch-Gordan series of ?₁??₃ and ?₂??₄ (resp. of S²?₁, S²?₂, A²?₁ and A²?₂). Then we formulate a result (Theorem 2) which gives information on the decomposition of ?₁?₂??₃?₄ (resp. of S²(?₁?₂) and A²(?₁?₂)). Though this result is not complete, it is useful because it delivers the information very quickly and in a very simple manner.     

Critical behaviour of a uniaxial ferromagnet,ErCl3·6H2O with predominant dipolar interactions

The parallel magnetic susceptibility χ of a uniaxial ferromagnet ErCl₃·6H₂O has been measured between 0.3 and 4.2K and specially near T_c = 0.353 K. The predominant contribution to the Curie-Weiss temperature is due to the dipolar interactions. χ is proportional to ?^?γ with $\text{? =}\text{T}\text{T}{}_{\mathrm{c}}\text{?1}$ in the range 10^?3 < ? < 5 × 10^?2. The γ value, γ = 1.01 ±0.03 is consistent with the theoretical prediction for a uniaxial dipolar ferromagnet.     

Glauber Dynamics for the Mean-Field Potts Model

We study Glauber dynamics for the mean-field (Curie-Weiss) Potts model with q??3 states and show that it undergoes a critical slowdown at an inverse-temperature ?? _s (q) strictly lower than the critical ?? _c (q) for uniqueness of the thermodynamic limit. The dynamical critical ?? _s (q) is the spinodal point marking the onset of metastability. We prove that when ??<?? _s (q) the mixing time is asymptotically C(??,q)nlogn and the dynamics exhibits the cutoff phenomena, a sharp transition in mixing, with a window of order n. At ??=?? _s (q) the dynamics no longer exhibits cutoff and its mixing obeys a power-law of order n ^4/3. For ??>?? _s (q) the mixing time is exponentially large in n. Furthermore, as ?????? _s with n, the mixing time interpolates smoothly from subcritical to critical behavior, with the latter reached at a scaling window of O(n ^?2/3) around ?? _s . These results form the first complete analysis of mixing around the critical dynamical temperature??including the critical power law??for a model with a first order phase transition.     

Classical stability of direct products of spheres in gravitational systems

Classical stability of Einstein spaces S^d₁ ×?×S^d_n(d_j ? 2) against all fluctuations is investigated in euclidean gravity with a cosmological constant. It is shown that S^d is classically stable, while S^d₁ ×?× S^d_n(n ? 2) is classically unstable. As a generalization of this analysis it is proved that a compact Einstein space B₁ ×?× B_n(n ? 2) which is a direct product of each Einstein space is classically unstable. Non-Einstein spaces M² × S⁴ (M² × S² × S²) are also considered in six- dimensional Einstein-Maxwell theory and are shown to be classically stable (unstable).     

Effective magnetic moments of the europium-monochalcogenides

We have measured the paramagnetic susceptibility and the saturation magnetic moment of a number of polycrystalline samples of all four europium monochalcogenides. Many of them exhibit a negative susceptibility at 2 K and magnetic fields of 70 kOe, rather than the expected saturation behaviour. These anomalies are believed to be due to lattice defects. Reproducible saturation moments have been obtained after subtraction of this linear high field magnetization. Since all materials were nonstoichiometric to within a few percent, the ratio between the Curie-Weiss constantC and the saturation momentM ₀ has been evaluated in order to eliminate the unreliably known numberN of the Eu^{+ +} ions. The quantityR = g _J (J+1) extracted in this way should have the valueR ₀ = 9.00 forg = 2 andJ ≡S = 7/2. The measured values, 9.72, 9.33, 9.32 and 9.50 for EuO, EuS, EuSe and EuTe, respectively, can be correlated directly with the magnetic hyperfine fields, which are ? 305, ? 330, ? 328 and ? 315 kOe in the ferromagnetic aligned states. Both results together lead to the assumption that the magnetic moment per Eu-ion is enhanced over the⁸ S _7/2-value by spin contributions of outer 5d or 6s electrons.     

Generalizations of CPn−1 with non-abelian gauge symmetry in 2 + ϵ dimensions

This paper considers non-linear σ models having global U(n) and local U(k) symmetries (1 ? k < n) in space-time dimensions d > 2. The special case k = 1 is the CP^n?1 model. The renormalizability to all orders of these models (in the presence of gauge invariant sources) is demonstrated in 2 + ? dimensions. A second order phase transition is shown to occur at a coupling strength of order ?, analogous to that in the O(n) model. Certain critical exponents associated with this transition are evaluated at two loop order. We are able to compare the two phases only in the limit n → ∞, k fixed.     

A Level 1 Large-Deviation Principle for the Autocovariances of Uniquely Ergodic Transformations with Additive Noise

A large-deviation principle (LDP) at level 1 for random means of the type $$M_n \equiv \frac{1}{n}\sum\limits_{j = 0}^{n - 1} {Z_j Z_{j + 1} ,{\text{ }}n = 1,2,...}$$ is established. The random process {Z _n} _n≥0 is given by Z _n = Φ(X _n) + ξ _n , n = 0, 1, 2,..., where {X _n} _n≥0 and {ξ _n} _n≥0 are independent random sequences: the former is a stationary process defined by X _n = T ⁿ(X ₀), X ₀ is uniformly distributed on the circle S ¹, T: S ¹ → S ¹ is a continuous, uniquely ergodic transformation preserving the Lebesgue measure on S ¹, and {ξ_n} _n≥0 is a random sequence of independent and identically distributed random variables on S ¹; Φ is a continuous real function. The LDP at level 1 for the means M _n is obtained by using the level 2 LDP for the Markov process {V _n = (X _n, ξ _n , ξ _n+1)} _n≥0 and the contraction principle. For establishing this level 2 LDP, one can consider a more general setting: T: [0, 1) → [0, 1) is a measure-preserving Lebesgue measure, $\Phi :\left[ {0,\left. 1 \right)} \right. \to \mathbb{R}$ is a real measurable function, and ξ _n are independent and identically distributed random variables on $\mathbb{R}$ (for instance, they could have a Gaussian distribution with mean zero and variance σ²). The analogous result for the case of autocovariance of order k is also true.     

Nuclear Spin relaxation mediated by Fermi-edge electrons in n-type GaAs

A method based on the optical orientation technique was developed to measure the nuclear-spin lattice relaxation time T ₁ in semiconductors. It was applied to bulk n-type GaAs, where T ₁ was measured after switching off the optical excitation in magnetic fields from 400 to 1200 G at low (< 30 K) temperatures. The spin-lattice relaxation of nuclei in the studied sample with n _D = 9 × 10¹⁶ cm^?3 was found to be determined by hyperfine scattering of itinerant electrons (Korringa mechanism) which predicts invariability of T ₁ with the change in magnetic field and linear dependence of the relaxation rate on temperature. This result extends the experimentally verified applicability of the Korringa relaxation law in degenerate semiconductors, previously studied in strong magnetic fields (several Tesla), to the moderate field range.     

Stimulated Raman scattering in natural crystals of SrSO4, BaSO4 and PbSO4: High-order Stokes and anti-Stokes generation with?single-wavelength UV, visible, and near-IR excitation, as?well?as cascaded up-conversion nonlinear �� (3)?�� (3) lasing effects under dual-wavelength picosecond collinear coherent pumping

Orthorhombic crystals of SrSO₄, BaSO₄, and PbSO₄, known as natural crystals celestine, barite, and anglesite, were found to be attractive ?? ⁽³⁾-active nonlinear optical materials. High-order Stokes and anti-Stokes picosecond generation that spans almost two octaves has been recorded with single-wavelength laser excitation in the UV, visible, and near-IR ranges. All recorded Raman induced lasing components were identified and attributed to the SRS-promoting vibration modes of the studied crystals (?? _SRS??999?cm^?1 for SrSO₄,?? _SRS??985?cm^?1 for BaSO₄ and ?? _SRS??977?cm^?1 for PbSO₄). Under dual-wavelength (?? _f1=1.06415???m + ?? _f2=0.53207???m) collinear coherent picosecond pumping several new manifestations of cascaded ?? ⁽³⁾??? ⁽³⁾ nonlinear up-conversion lasing effects were observed in BaSO₄ and SrSO₄ crystals. We classify all three studied sulfate crystals as promising SRS-active materials for Raman laser frequency converters and as efficient ?? ⁽³⁾-crystals that efficiently generate Stokes and anti-Stokes frequency combs, which can enable experiments of ultra-short pulse syntheses.     

Longitudinal magnetoresistance in zero-gap CdxHg1−xTe at low temperatures

The longitudinal magnetoresistance Δ?₆(H)/?₀ is studied experimentally in gapless solid solutions Cd_xHg_1?xTe (0 < x < 0.15) for temperatures 1.3–15 K and the electron concentrations n ～ 10¹⁵ cm^?3. The temperature and the magnetic field dependences of the observed negative longitudinal magnetoresistance are explained by the resonant nature of electron scattering by an acceptor level. The quantitative analysis of the $\text{Δ?6(H)}\text{?}{}_0$ field dependence for weak magnetic fields under strong carrier degeneracy makes it possible to evaluate parameters of the acceptor level involved.     

Theoretical prediction of cosmological constant Λ in Veneziano ghost theory of QCD

Based on the Veneziano ghost theory of QCD, we predict the cosmological constant ??, which is related to energy density of cosmological vacuum by $ \Lambda = \frac{{8\pi G}} {3}\rho _\Lambda $ . In the Veneziano ghost theory, the vacuum energy density ?? _?? is expressed by absolute value of the product of quark vacuum condensate and quark current mass: $ \rho _\Lambda = \frac{{2N_f H}} {{m_{\eta '} }}c|m_q < 0|:\bar qq:|0 > | $ . We calculate the quark local vacuum condensates ??0|: $ \bar q $ q: |0?? by solving Dyson-Schwinger Equations for a fully dressed confining quark propagator S _f (p) with an effective gluon propagator G _???? ^ab (q). The quark current mass m _q is predicted by use of chiral perturbation theory. Our theoretical result of ??, with the resulting ??0|: 471-4 q: |0?? = ?(235 MeV)³ and light quark current mass m _q ? 3.29?C6.15 MeV, is in a good agreement with the observable of the ?? ?? 10^?52 m^?2 used widely in a great amount of literatures.     

Effects of complex parameters on classical trajectories of Hamiltonian systems

Anderson et al have shown that for complex energies, the classical trajectories of real quartic potentials are closed and periodic only on a discrete set of eigencurves. Moreover, recently it was revealed that when time is complex t \((t=t_{r}\mathrm {e}^{i\theta _{\tau }}),\) certain real Hermitian systems possess close periodic trajectories only for a discrete set of values of ?? _τ . On the other hand, it is generally true that even for real energies, classical trajectories of non-PT symmetric Hamiltonians with complex parameters are mostly non-periodic and open. In this paper, we show that for given real energy, the classical trajectories of complex quartic Hamiltonians H=p ²+a x ⁴+b x ^k (where a is real, b is complex and k=1 or 2) are closed and periodic only for a discrete set of parameter curves in the complex b-plane. It was further found that the given complex parameter b, the classical trajectories are periodic for a discrete set of real energies (i.e., classical energy gets discretized or quantized by imposing the condition that trajectories are periodic and closed). Moreover, we show that for real and positive energies (continuous), the classical trajectories of complex Hamiltonian H = p ² + μx ⁴, μ = μ _r e ^i?? ) are periodic when ??=4 tan^?1[(n/(2m+n))] for ? n and \(m\in \mathbb {Z}\) .     

Fluorescence resolue dans le temps des niveax de singulet et d du zinc

The time resolved spectroscopy technique is applied to investigate the (4s, nd) ¹D₂ series of Zn I using a selective stepwise excitation via the resonant 4¹P₁ state. Radiative lifetimes of n¹S₀ (n = 6?8) and n¹D₂ (n = 4,5) levels are measured. Upper limits for transition probabilities 4¹P₁?n¹S₀ (n = 9?12) and 4¹P₁?n¹ (n = 6?10 ) are obtained. Strong configuration interaction effects are observed in the ¹D₂ series and compared with those in Mg I and Cd I.     

Magnetic monopoles in neutron stars

The effects of monopole-antimonopole annihilation on a previously reported bound [1] on the product of the galactic flux of grand unified magnetic monopoles and the cross section for monopole catalyzed nucleon decay: (F_M/cm^?2s^?1sr^?1)(σ_ΔB/10^?2 cm²) ? 10^?22 are examined for several models of neutron star interiors. For neutron stars with superconducting interiors or large internal magnetic fields this bound is unaltered. In the unlikely event that old neutron stars are not superconducting and have internal magnetic fields B_int ? 10⁸ Gauss the effects of monopole-antimonopole annihilation relax the bound to (F_M/cm^?2 s^?1 sr^?1)(σ_ΔB/10^?27 cm^?2)² ? 10^?18. Magnetic monopoles may also have a significant effect on the structure of the interior magnetic field in neutron stars.     

Isotope-selective ionization utilizing molecular alignment and non-resonant multiphoton ionization

We demonstrate laser nitrogen isotope separation, which is based on field-free alignment and angular-dependent ionization of ¹⁴N₂ and ¹⁵N₂ isotopologues. A linearly polarized short laser pulse (???~?795?nm, ?????~?60?fs) creates rotational wave packets in the isotopologues, which periodically revive with different revival times as a result of different moments of inertia. Another linearly polarized short laser pulse (???~?795?nm, ?????~?60?fs) ionizes one of the isotopologues selectively as a result of their different angular distributions. In the present experiments, the ion yield ratio R [=I(¹⁵N₂ ⁺)/I(¹⁴N₂ ⁺)] can be changed in the range from 0.85 to 1.22, depending on the time delay between the two laser pulses.     

NMR and magnetic susceptibility of CeCu6

NMR and magnetic susceptibility of CeCu₆ intermetallic compound were investigated. The most important contributions to the magnetic susceptibility are the Curie-Weiss term, expressing the paramagnetism of the localized f-electrons, and a temperature independent term for which all the contributions were estimated. The phenomenological exchange constantJ _sf between 4f — electron spins and conduction — electron spins is derived to be ?0·012 eV. The Knight shift and Curie temperature are discussed in terms of the Ruderman-Kittel-Kasuya-Yosida theory and in the Rocher's virtual bound states model. The s — f exchange integralΓ and the Fermi wavevectork _F are derived to be — 0·8 eV and 1·32 Å^?1, respectively.