Entropic dimension for completely positive maps

We extend the concept of quantum dynamical entropyh _φ (γ) to cover the case of a completely positive map γ. Forh _φ (γ) = 0 we examine the limit $$h_\phi (N,\gamma ,\beta ) = \mathop {\lim }\limits_n (1/n^\beta )H_\phi (N,\gamma {\rm N},...,\gamma ^{n -- 1} N)$$ calling the turning point β₀ between zero and infiniteh _φ (N, γ, β) the “entropic dimension”D _N (γ). The application of this theory to a solvable irreversible quantum dynamical semigroup on a one-dimensional fermion lattice provides any value ofD _N (γ) between 0 and 1.     

ΛN and ΣN Interactions from Lattice QCD

We present our recent study on ΛN and ΣN (isospin I = 3/2) interactions by measuring Nambu–Bethe–Salpeter wave functions on the Lattice QCD. The lattice QCD calculation is performed by using the N _f  = 2 + 1 gauge configurations generated by PACS-CS collaboration together with employing an improved method to obtain potentials in lattice QCD simulations. For the ¹ S ₀ channel, the central ΣN (I = 3/2) potential and the central ΛN (¹ S ₀) potential are found to be very similar. For the spin triplet (³ S ₁?³ D ₁) channels, the central ΛN(³ S ₁?³ D ₁) potential is attractive while the central ΣN(I = 3/2, ³ S ₁?³ D ₁) potentials is repulsive. Tensor potentials, on the other hand, are rather weak in both ΛN and ΣN(I = 3/2) systems.     

Calculation of the Eigenstates and Eigenvalues for the Power and Inverse-Power Hypercentral Potentials

The bound-state solutions to the hyperradial Schr?dinger equation is constructed for any general case comprising any hypercentral power and inverse-power potentials. The hypercentral potential depends only on the hyperradius which itself is a function of Jacobi relative coordinates that are functions of particle positions (r ₁,r ₂, … , and r _N ). This paper is mainly devoted to the demonstration of the fact that any ψ of the form ψ = power series × exp(polynomial) = [f(x) exp (g(x))] is potentially a solution of the Schr?dinger equation, where the polynomial g(x) is an ansatz depending on the interaction potential.     

Selective effects of homogeneous and heterogeneous reactions in the system N21-CO

In the system N₂¹-CO, effects arising only from the reactions in the homogeneous phase have been isolated in a reactor with inert walls. After an induction period of 0.13 s, they involve a stationary rate of enhancement of N₂(B³Π_g) and N(⁴S) concentrations according to the reactions N₂(X¹Σ_g⁺)_v¹+CO→ CO(X¹Σ⁺)_v1 + N₂(X¹Σ_g⁺ and CO(X¹Σ⁺)_v1 + N₂(A³Σ_u⁺) → N (⁴S) + N(⁴S) + CO(X¹Σ⁺).In a reactor with active walls, both the above reactions in the homogeneous phase and heterogeneous reactions due to CO adsorbed on the walls are involved according to CO_ads + N₂(X¹Σ_g⁺)_{v1 ads or not} → CO(X¹Σ⁺)_v1 + N₂(X¹Σ_g⁺ and N(⁴S) + N(⁴S) + CO_ads → N₂(X¹Σ_g⁺) + CO(X¹Σ⁺)N(⁴S) + CO_ads → N₂(X¹Σ_g⁺) + CO(X¹Σ⁺). In this case, the rate of enhancement is not stationary. Furthermore, for cylindrical reactors with large diameters, the two types of reaction do not interact and their effects are additive.     

The localization of local and gap modes by the sign count method

In the theory of local states it is usual to bring the linear problem ¦glI _N ?D¦=0 into a non-linear one ¦R(λ)¦=0, where the ordern of the condensed stiffness matrixR(λ) is much smaller than the orderN of the matrixD. Powerful methods exist for determining the number of the eigenvalues ofD in some interval without solving the equation ¦glI _N ?D¦=0 explicitly. It is shown, that analogous methods can be applied to the condensed matrix R(λ), when computing the eigen-valuesλ outside the frequency spectrum of the ideal lattice.     

The Waring formula and fusion rings

The potential that generates the cohomology ring of the Grassmannian is given in terms of the elementary symmetric functions using the Waring formula that computes the power sum of roots of an algebraic equation in terms of its coefficients. As a consequence, the fusion potential for su(N)_K is obtained. This potential is the explicit Chebyshev polynomial in several variables of the first kind. We also derive the fusion potential for sp(N)_K from a reciprocal algebraic equation. This potential is identified with another Chebyshev polynomial in several variables. We display a connection between these fusion potentials and generalized Fibonacci and Lucas numbers. In the case of su(N)_K the generating function for the generalized Fibonacci numbers obtained are in agreement with Lascoux using the theory of symmetric functions. For sp(N)_K, however, the generalized Fibonacci numbers are obtained from new sequences.     

Effect of surface orientation and thickness on the magnetization of anisotropic FCC ferromagnetic films

The dependence on temperature of the layer magnetization of a Heisenberg ferromagnetic ultrathin film in presence of magnetocrystalline single-ion anisotropy was theoretically investigated in the framework of a Green's function approach using the random phase approximation (RPA). The effect of surface orientation and of film thickness N on the Curie temperature T_C was carefully investigated in the case of face centered cubic (FCC) films: the steepest increase of T_C(N) was found in the case of the FCC(1 1 1) orientation and the smoothest in the FCC(1 1 0) one. Our results for T_C(N) were successfully fitted by a finite-size scaling relation [T_C(∞)−T_C(N)]/T_C(N)=(N/N₀)^−λ, giving a shift exponent λ≃1.5, irrespectively of the surface orientation. Finally, the temperature evolution of the magnetization profile was analyzed, as well as its limiting shape at T_C.     

n-Dimensional complex riemann-einstein spaces with O(n−1, C) as the symmetry group

Working with a complex Riemannian space V_n, we determine the most general form of the metric which admits O(n?1, C) as a symmetry subgroup. The conformal curvature tensor, which is of type D for the real cross-section of signature (+---) with n = 4, has a very similar structure for n > 4 and complex V_n. Solutions of the empty space Einstein equations with the λ-term are then studied. There are two different static solutions which, with n = 4 and real cross-section of signature (+---), reduce to Schwarzschild-de Sitter metric, or to the Nariai's solution. A proof of a slight generalization of Birkhoff's theorem that every time-dependent solution of G_μv = λg_μv is reducible, by a coordinate transformation, to one of these two solutions, is given. The complete symmetry groups of the solutions obtained are examined. The techniques of embedding in a higher-dimensional flat space and of Kruskal's variables which exhibit the spurious nature of Schwarzschild's singularity are generalized to the case of (complex) n dimensions. General properties of geodesic lines for both solutions are examined and various real cross-sections of Nariai's solution for n = 4 are discussed. The last section studies some contractions of the two solutions.     

Relativistic comparison theorems for the Klein-Gordon equation with scalar and vector potentials in d-dimensions

Relativistic comparison theorems are established for discrete eigenvalues of Klein-Gordon equation with vector and scalar potentials in d-dimensions. Theorem 1: If V(λ) and S(λ) depend on a parameter λ, ∂S/∂λ?0, S?0, ∂V/∂λ?0, V?0, E>0, then it follows that ∂En/∂λ?0. Theorem 2: If S₂?S₁?0, 0?V₂?V₁, E>0, then the corresponding eigenvalues are ordered as En(2)?En(1). Theorem 3: If 0?V₂?V₁, S₂?|S₁|, En(1)>0, En(2)>0, then En(2)?En(1). Some illustrative examples are given.     

Investigation of the optical constants of arsenic chalcogenide films in the wavelength range 0.5–2.5 μm

Thin films of chalcogenide glasses deposited on quartz glass substrates by thermal evaporation in vacuum have been investigated. The dependences n(λ) and k(λ) for films of different composition have been determined from the transmission spectra. Expressions of the n = A + BL + CL ² + Dλ ² + Eλ ⁴ type (L = (λ ² ? 0.028)^?1 and A, B, C, D, and E are constants) for calculating the refractive indices of As₂Se₃, AsSe₄, AsS₄, and AsS_16.2Se_16.2 films in the wavelength range from 0.5 to 2.5 μm are reported.     

Influence of the dimension of a polycrystalline film and the optical anisotropy of crystallites on the effective dielectric constant of the film

The dimension D of a polycrystalline film and the optical anisotropy m = ε_z/ε_x of uniaxial crystallites with the principal components ε_x = ε_y and ε_z of the tensor of the dielectric constant have been shown to produce a strong influence on the effective dielectric constant ε_D^* and the effective refractive index n_D^* = (ε_D^*)^1/2 of the film in the optical transparency region, as well as on the boundaries of the intervals B_Dl ≤ ε_D^* ≤ B_Du. The intervals Δ₂(m) = B_2l–B_2u and Δ₃(m) = B_3l–B_3u are separated by a gap for m in the range 1 < m < 2, whereas the theoretical dependence ε₂^*(m) is separated by a gap from the interval Δ₃(m) for m in the range 1 < m < 4. This is confirmed by a comparison of the experimental (n_oP) and theoretical (n_D^*) ordinary refractive indices for uniaxial polycrystalline films of the conjugated polymer poly(p-phenylene vinylene) (PPV) with uniaxial crystallites and appropriate values of m. In the visible transparency region of the PPV films with a change in m(λ) in the range 2 < m(λ) < 3 due to the dependence of the components ε_x,z(λ) on the light wavelength λ, the refractive indices n_oP²(λ) = ε_oP(λ) are consistent with the theoretical values of ε₂^*(λ) and lie outside the interval Δ₃(m). For m(λ) > 3 near the electronic absorption band of the crystallites, the values of ε_oP(λ) lie in the region of the overlap of the intervals Δ₂(m) and Δ₃(m). The boundaries mc of the range 1 < m < m_c are determined, for which the interval Δ₂(m) is separated by a gap from the dependences ε₃^*(m) corresponding to the effective medium theory with spherical crystallites and hierarchical models of a polycrystal, as well as from the proposed new dependence ε₃^*(m).     

The refractive index dispersion of Hg1−xCdxTe by infrared spectroscopic ellipsometry

The refractive indices of Hg_1−xCd_xTe (x=0.276, 0.309, and 0.378) bulk samples in the region below, in, and above the fundamental band gap have been measured by infrared spectroscopic ellipsometry at room temperature. A refractive index peak, in which the corresponding energy equals approximately the band gap energy, is observed for each refractive index spectrum with different compositions. Above the band gap, the refractive index drops quickly near the gap, then decreases slowly as photon energy increases. The refractive index n above the band gap is found to follow the Sellmeier dispersion relationship n²(λ)=A+B/λ²+C/λ⁴+D/λ⁶ as a function of the wavelength of light λ.     

Assortativity of complementary graphs

Newman's measure for (dis)assortativity, the linear degree correlationρ _D , is widely studied although analytic insight into the assortativity of an arbitrary network remains far from well understood. In this paper, we derive the general relation (2), (3) and Theorem 1 between the assortativity ρ _D (G) of a graph G and the assortativityρ _D (G ^c) of its complement G ^c. Both ρ _D (G) and ρ _D (G ^c) are linearly related by the degree distribution in G. When the graph G(N,p) possesses a binomial degree distribution as in the Erd?s-Rényi random graphs G _p (N), its complementary graph G _p ^c (N) = G _{1- p} (N) follows a binomial degree distribution as in the Erd?s-Rényi random graphs G _{1- p} (N). We prove that the maximum and minimum assortativity of a class of graphs with a binomial distribution are asymptotically antisymmetric: ρ _max(N,p) = -ρ _min(N,p) for N → ∞. The general relation (3) nicely leads to (a) the relation (10) and (16) between the assortativity range ρ _max(G)–ρ _min(G) of a graph with a given degree distribution and the range ρ _max(G ^c)–ρ _min(G ^c) of its complementary graph and (b) new bounds (6) and (15) of the assortativity. These results together with our numerical experiments in over 30 real-world complex networks illustrate that the assortativity range ρ _max–ρ _min is generally large in sparse networks, which underlines the importance of assortativity as a network characterizer.     

The exponential interaction in Rn

We prove that the Schwinger functions for the ultraviolet cut-off exponential interaction with euclidean measure exp {;?λ∫Λ:e^αξk(x):dx} dμ₀(ξ/ ∫ exp{?λ∫Λ:e^αξk(x):dx} dμ₀(ξ), λ > 0, converge as the ultraviolet cut-off is removed. The limits are the free Schwinger functions in the case of space-time dimension n ? 3. In the case n = 2 this holds for |α| sufficiently big, whereas for |α| < 2 √π, one has the well-known nontrivial Schwinger functions of the exponential interaction.     

Nuclear effects in 800 GeV proton interactions

We discuss here the nuclear effects in multiparticle production processes in 800 GeV proton-emulsion nuclei interactions. The multiplicity of medium energy particles (N _g ) was used as a measure of the average number of projectile collisions (<v(N _g )>). To study nuclear effects, we determined <v(N _g )> and ratio of normalized rapidity density of all produced charged particles in hadron-nucleus to hadron-hadron interaction,R(N _η ). It was found that the average multiplicity of shower particles increases with <v(N _g )>. From theR(N _η ) distribution, we find; a strong maxima in the target fragmentation region, a plateau in the central region and depletion of particle density in the projectile fragmentation region.     

States on the current algebra

We introduce the field algebra Σ$\text{D}$(M;ⁿ?ⁿ$\text{g}$) associated with the current algebra $\text{D}$_r(M;$\text{g}$) for the Lie algebra $\text{g}$ over physical space M. The Heisenberg magnet model is generalized to this continuum. It is shown that the Hamiltonian can be given meaning as implementing a derivation of the field algebra in certain representations.We introduce new representations of the current algebra. For example, if G = SU(2), a representation in L²(R³)?³ is [σ(?)F]^j = ε^jkl?_kψ_l for (?_k) = ? in $\text{D}$_r(M;$\text{g}$)(ψ_l = F. This has cyclic subrepresentations with prime parts.