Automorphisms and symmetries of quantum logics

Given an amalgam of groups then every quantum logicQ ₀ = (L ₀,M ₀) (L ₀ is aσ-orthomodular poset,M ₀ is a full set of states on it) satisfying some reasonable conditions can be embedded in a quantum logicQ = (L, M), in which (1) all the automorphisms ofL form a group ∼-G ₁, (2) all the automorphisms ofM form a group ∼-G ₂, and (3) all the symmetries ofQ form a group ∼-G ₀. The quantum logic of all closed subspaces of a Hilbert spaceH and all its measures satisfies the conditions required fromQ ₀; hence, enlarging it, one can obtain “anything.”     

On the trace and the relativistic scalar products involving Dirac matricesγ i and the 4 × 4 unit matrix

In this paper we have dealt with the relativistic scalar products likeγ _i uγ _i,Tr(γ _i u) Tr(γ _i v) and Tr(γ _i γ _j v). Here any string likeu andv involve Dirac matrices in the manneru=Π _n ^r=1 a _r where the element \(a_r = a_{r_i } \gamma _i + ia_{r_5 }\) i.e.,a _r in general involves a term \(ia_{r_5 }\) which is multiplied by a 4 × 4 unit matrix. We have further evaluated Tr (γ _b u) Tr (γ _b v) where the suffix ‘b’, unlike the dummy suffixesi andj, does not imply any summation and can assume any specific value from 1 to 5. Some reduction formulae for the evaluation of Tr (γ₅ u) and Tr(u) have been obtained in this paper.     

Logistic equation of arbitrary order

The paper is concerned with the new logistic equation of arbitrary order which describes the performance of complex executive systems X vs. number of tasks N, operating at limited resources K, at non-extensive, heterogeneous self-organization processes characterized by parameter f. In contrast to the classical logistic equation which exclusively relates to the special case of sub-extensive homogeneous self-organization processes at f=1, the proposed model concerns both homogeneous and heterogeneous processes in sub-extensive and super-extensive areas. The parameter of arbitrary order f, where −∞<f<+∞, depends on both the coefficient of external resource utilization u=N/K, where 0<u<1, and the internal microscopic character of realized processes related to the depth of feedback β. The coefficient β directly influences self-organization of processes by the change of microscopic parameters V_i, S_i, i and Z, where V_i is the number of references (visit) to the ith component of the system during the service of each task, S_i is the time of serving the task by the ith component, and Z is the think time of a given process. In the general case of complex system, parameters V_i, S_i, i and Z can have values in the range from 0 to +∞. In this way the new equation includes all possible cases of a complex executive system’s operation. Furthermore, it allows us to define the optimal matching point between X and N with f as the parameter. It also helps to balance the load in complex systems and to equip artificial systems with self-optimization mechanisms similar to those observed in natural systems.     

Inverse problem for the Grad-Shafranov equation with affine right-hand side

For a given domain ω ⋐ ℝ² with boundary γ = ∂ω, we study the cardinality of the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) of pairs of numbers (a, b) for which there is a function u = u _(a,b): ω → ℝ such that ∇² u(x) = au(x) + b ⩾ 0 for x ∈ ω, u| _γ = 0, and ||∇u(s)| − Φ(s) ⩽ η for s ∈ γ. Here η ⩾ 0 stands for a very small number, Φ(s) = |∇(s)| / ∫ _γ |∇v| d γ, and v is the solution of the problem ∇² v = a ₀ v + 1 ⩾ 0 on ω with v| _γ = 0, where a ₀ is a given number. The fundamental difference between the case η = 0 and the physically meaningful case η > 0 is proved. Namely, for η = 0, the set $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) contains only one element (a, b) for a broad class of domains ω, and a = a ₀. On the contrary, for an arbitrarily small η > 0, there is a sequence of pairs (a _j , b _j ) ∈ $ \mathfrak{A}_\eta \left( \Phi \right) $ \mathfrak{A}_\eta \left( \Phi \right) and the corresponding functions u _j such that ‖f _u ^j+1‖ − ‖f _u ^j ‖ > 1, where ‖f _u ^j = max _x∈ω |f _u ^j (x)| and f _u ^j (x) = a _j u _j (x) + b _j . Here the mappings f _u ^j : ω → ℝ necessarily tend as j → ∞ to the δ-function concentrated on γ.     

On asymptotic behaviour of four spin correlation functions in a linear chain of spins

Second and fourth moments of the four spin frequency correlation functions S^Z_i(t) S^Z_i+1 (t) S^Z_j+1 〉_k,ω and 〈S^x_i(t) S^z_i+1 (t) S^x_jS^z_j+1 〉_kω have been calculated in the high temperature limit for an arbitrary spin with an isotropic Heisenberg Hamiltonian.     

Li2分子自旋一致激发态a3Σ+u和b3Πu的分析势能函数的从头算

使用对称性匹配簇-组态相互作用方法首次计算了Li₂分子自旋一致激发态a³Σ⁺_u和b³Π_u的离解能、平衡几何及其谐振频率。使用最小二乘法、利用Murrell-Sorbie函数形式拟合出了Li₂分子三重态的第一激发态a³Σ⁺_u 和第二激发态b³Π_u的完整势能函数，并计算了这两个态的光谱常数 (B_e, α_e, ω_e 和 ω_eχ_e) 和力常数 (f₂, f₃和f₄)。得到了Murrell-Sorbie函数形式既适用于基态、又适用于激发态的结论。将计算得到的激发态（a³Σ⁺_u和b³Π_u）的离解能、平衡几何及其谐振频率与实验结果及其它理论计算结果进行了比较。从比较的结果中可以清楚地看出，本文的计算结果在计算精度方面有很大的改进。     

Higher-order fine structure of the a4Πu state of O2+

It is shown that the quality of the fit of frequencies for the (4, 4) band of the b⁴Σ_g^?-a⁴Π_u transition of O₂⁺, obtained in an ion-beam experiment, can be significantly improved by the inclusion of a fine structure term of the form L_zS_z³ in the effective Hamiltonian for the a⁴Π_u state, where L_z and S_z are respectively the components of the total orbital and spin angular momenta along the internuclear axis. The various contributions to this term in the Hamiltonian are assessed from a general standpoint by the use of a new order-of-magnitude scheme. The separate contributions to the parameter involved, η, are then considered in detail by means of a perturbation treatment of the spin-orbit and spin-spin interactions through third order. Attempts to interpret the parameter values determined for the a⁴Π_u state of O₂⁺ are severely limited by the present lack of detailed knowledge of the properties of the various electronic states of O₂⁺.     

A stochastic derivation of the Sivashinsky equation for the self-turbulent motion of a free particle

Within the framework of the Kershaw approach and of a hypothesis on spatial stochasticity, the relativistic equations of Lehr and Park, Guerra and Ruggiero, and Vigier for stochastic Nelson mechanics are obtained. In our model there is another set of equations of the hydrodynamical type for the drift velocityv _i(x _j,t) and stochastic velocityu _i(x _j,t) of a particle. Taking into account quadratic terms in l, the universal length, we obtain from these equations the Sivashinsky equations forv _i(x _j,t) in the caseu _i →0. In the limit l →0, these equations acquire the Newtonian form.     

Fragmentation of the high spin discrete neutron and proton states of209Pb and209Bi

The distribution of the discrete 1h _9/2, 2f _7/2, 1i _13/2 and 2f _5/2 proton states of²⁰⁹Bi and 2g _9/2, 1i _11/2, 1j _15/2 and 2g_7/2 neutron states of²⁰⁹Pb have been obtained within the particle-vibration coupling model calculation and compared with the experimental datas baising on the most recent high resolution stripping reaction on²⁰⁸Pb using 480 MeV¹²C projectile. The optimised shell model energies arising from the core-polarisation effect have profound influence for both the study of the structures of the high spin continuum shell model states of²⁰⁸Pb and stability of superheavy nuclei.     

Gibbs Distributions for Random Partitions Generated by a Fragmentation Process

In this paper we study random partitions of {1,…,n} where every cluster of size j can be in any of w _j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w _j , the time-reversed process is the discrete Marcus–Lushnikov coalescent process with affine collision rate K _i,j = a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton–Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions. Research supported in part by N.S.F. Grant DMS-0405779.     

Conversion ratio to F-H center pairs from self-trapped excitons in alkali chloride crystals

Abstract

The time delayed double excitation spectroscopy has been utilized to determine the conversion ratio to F-H center pairs from self-trapped excitons(STE_L) at the lowest state (1s[sgrave]_g;a_1g). The final conversion ratios, η_F/(η_F+η_X), were 0.86, 0.49 and 0.20 for NaCl, KCl and RbCl at 14K, respectively. The conversion efficiency (η =η_F+η_X) from STE_L to F-H center pairs(η_F) and to unknown states(η_X) were 0.25, 0.90 and 0.76 for the hole excitation to π_g, while 0.03, 0.01 and 0.01 for the electron excitation to b_1u, b_2u or b_3u, in NaCl, KCl and RbCl, respectively.     

An analytic approach to the measurement of nestedness in bipartite networks

We present an index that measures the nestedness pattern of bipartite networks, a problem that arises in theoretical ecology. Our measure is derived using the sum of distances of the occupied elements in the incidence matrix of the network. This index quantifies directly the deviation of a given matrix from the nested pattern. In the simplest case the distance of the matrix element a_i,j is d_i,j=i+j, the Manhattan distance. A generic distance is obtained as d_i,j=(i^χ+j^χ)^1/χ. The nestedness index is defined by ν=1−τ, where τ is the “temperature” of the matrix. We construct the temperature index using two benchmarks: the distance of the complete nested matrix that corresponds to zero temperature and the distance of the average random matrix where the temperature is defined as one. We discuss an important feature of the problem: matrix occupancy ρ. We address this question using a metric index χ that adjusts for matrix occupancy.     

Axiomatization of quantum logics

Starting with a quantum logic (a -orthomodular poset)L, a set of probabilistically motivated axioms is suggested to identifyL with a standard quantum logicL(H) of all closed linear subspaces of a complex, separable, infinite-dimensional Hilbert space. Attention is paid to recent results in this field.     

A simple Fourier photopolarimeter with rotating polarizer and analyzer for measuring Jones and Mueller matrices

The classic and simplest polarimetric scheme of examining a “sample” by placing it between a pair of linear polarizers and observing the intensity of the transmitted light can be transformed into a powerful photopolarimeter if the two polarizers are synchronously rotated at different speeds and the transmitted flux is linearly detected and its periodic waveform Fourier analyzed. In particular, if the angular speed of rotation of one polarizer is ω and that of the other is 3ω, the detected signal has the waveform, i=a₀ + Σ⁴_n=1a_n cos nω_ft + b_n sin nω_ft, where ω_f=2ω is the fundamental frequency. The nine Fourier amplitudes a₀ and (a_n, b_n), n=1,2,3,4, to be derived by performing a discrete Fourier transform (DFT) of the signal i, determine all nine elements of the 3×3 submatrix M_3×3 obtained by deleting the fourth row and fourth column of the Mueller matrix M. If the sample is nondepolarizing, the absolute values of all the four elements of the equivalent Jones matrix $\text{J}\text{=(J}{}_{\mathrm{ij}}\text{)}$, i,j=1,2 and their angle differences θ_ij?θ₂₂ (where θ_ij=argJ_ij) can be determined.     

Photoelectron spectra of pyromellitic dianhydride,dithioanhydride and diimide and of trimellitic anhydride

The photoelectron (He(I)) spectra of the tricyclic tetracarbonyl compounds pyromellitic dianhydride, dithioanhydride and diimide and of the tetracyclic hexacarbonyl compound trimellitic anhydride have been investigated. To aid the interpretation of the main features of the spectra, i.e. the ordering and splitting of the ⁿCO ionisations and the behaviour of the ‘benzenic’ and heteroatom π ionisations, MO calculations based on a ZDO pragmatic model and semiempirical SCF-PP calculations have been carried out. The evolution of the ⁿCO and π_X ionisations upon progressive fusion of anhydride moieties with a benzene nucleus is analysed in detail. The proposed orbital sequences for the n orbitals are: a_g(S⁺) $\text{\$}\text{?}$b_1u(AS⁺) $\text{\$}\text{?}$b_2u(S^?) $\text{\$}\text{?}$b_3g(AS^?) for the tetracarbonyls and a′₁(S⁺) $\text{\$}\text{?}$e′(AS⁺) $\text{\$}\text{?}$a′₂ (AS^?) ≈ e′(S^?) for the hexacarbonyl.     

Evidence for low temperature phase transition in Rb2ZnCl4 from dielectric constant and birefringence measurements

Temperature dependences of the dielectric constants ?_i and of the birefringences Δn_i for light propagation directions along all the three crystallographic axes (i = a, b, c) have been measured between 5 and 350 K. The optical measurements clearly reveal a new phase transition at T_L = 75 K, below which the crystal structure is transformed from the ferroelectric Pna2₁ phase into an unknown low temperature phase, possibly the monoclinic space group P112₁. Small peaks in the dielectric constants ?_a and ?_b have been observed at this transition temperature.     

Ab initio calculation of transition dipole moments for transitions between valence states in oxygen molecules

Results of ab initio calculations of potential-energy curves for 20 singlet and 20 triplet valence states of oxygen with configuration interaction taken into account in the 6-31G basis are presented. Transition dipole moments of triplet-triplet (1³Π_g ← B ³Σ _u ^? , 1³Π_g ← A ³Σ _u ⁺ , 1³Π_g ← A′ ³Δ_u, B ³Σ _u ^? ← X ³Σ _g ^? , 2³Π_u ← 1³Π _g, 2³Σ _g ^? ← B ³Σ _u ^? , 1³Π_u ← X ³Σ _g ^? , 2³Π_u ← X ³Σ _g ^? , 2³Π _g ← A′³Δ_u, 3³Π_g ← A′ ³Δ _u, 2³Δ_u ← 2³Π_g, 3³Π_g ← B ³Σ _u ^? , and 2³Π_g ← A ³Σ _u ⁺ ) and singlet-singlet (2¹Σ _g ⁺ ← 2¹Π_u, 2¹Π_u ← 1¹Π ^g, ¹Π_u ← 2¹Δ_g, 1¹Π_g ← c ¹Σ _u ^? , ¹Π_u ← b ¹Σ _g ⁺ , 1¹Δ _u ← a ¹Δ_g, 2¹Π_u ← a ¹Δ_g, 2¹Δ_g ← 1¹Δ_u, ¹Π _u ← a ¹Δ _g, 1¹Π_u ← b ¹Σ _g ⁺ , 2¹Π_g ← 1¹Π_u, 2¹Π _g ← c ¹Σ _u ^? , 1¹Δ _u ← 1¹Π _g, f′Σ _u ⁺ ← b ¹Σ _g ⁺ , 2¹Σ _g ⁺ ← f ′¹Σ _u ⁺ , 3¹Π_g ← 1¹Δ_u) radiative transitions are calculated as functions of internuclear separation. The possibility of observing these transitions under experimental conditions is discussed.     

Relative partial photoionization cross-sections and photoelectron branching ratios of ethylene

Relative partial photoionization cross-sections and photoelectron branching ratios of the valence bands (~10–25 eV binding energy) of ethylene are reported over the photon energy ranges 18–100 eV and 21–100 eV, respectively. The four lowest ionization energy bands (1b⁻¹_1u, 1b⁻¹_1g, 3a⁻¹_g, and 1b⁻¹_2u) show monotonic cross-section decreases with photon energy from 33 eV, the 1b⁻¹_1u CC π band showing the least rapid decline. In contrast, the 2b⁻¹_3u and 2a⁻¹_g bands show almost constant cross-sections up to ~50 eV photon energy, followed by similar, although slower, monotonic decreases. This is attributed to the substantial carbon 2s character of the 2b_3u and 2a_g orbitais. The cross-section behaviour of all bands is interpreted with the aid of SCF-LCAO-MO calculations on the neutral molecule using the Gaussian-80 series of programs.