Spanning forests and the golden ratio

For a graph G, let f_ij be the number of spanning rooted forests in which vertex j belongs to a tree rooted at i. In this paper, we show that for a path, the f_ij's can be expressed as the products of Fibonacci numbers; for a cycle, they are products of Fibonacci and Lucas numbers. The doubly stochastic graph matrix is the matrix F=(f_ij)_n×n/f, where f is the total number of spanning rooted forests of G and n is the number of vertices in G. F provides a proximity measure for graph vertices. By the matrix forest theorem, F^-1=I+L, where L is the Laplacian matrix of G. We show that for the paths and the so-called T-caterpillars, some diagonal entries of F (which provide a measure of the self-connectivity of vertices) converge to φ^-1 or to 1-φ^-1, where φ is the golden ratio, as the number of vertices goes to infinity. Thereby, in the asymptotic, the corresponding vertices can be metaphorically considered as “golden introverts” and “golden extroverts,” respectively. This metaphor is reinforced by a Markov chain interpretation of the doubly stochastic graph matrix, according to which F equals the overall transition matrix of a random walk with a random number of steps on G.     

Upper bounds on complexity of Frege proofs with limited use of certain schemata

The paper considers a commonly used axiomatization of the classical propositional logic and studies how different axiom schemata in this system contribute to proof complexity of the logic. The existence of a polynomial bound on proof complexity of every statement provable in this logic is a well-known open question. The axiomatization consists of three schemata. We show that any statement provable using unrestricted number of axioms from the first of the three schemata and polynomially-bounded in size set of axioms from the other schemata, has a polynomially-bounded proof complexity. In addition, it is also established, that any statement, provable using unrestricted number of axioms from the remaining two schemata and polynomially-bounded in size set of axioms from the first scheme, also has a polynomially-bounded proof complexity.     

High-efficiency longitudinally-pumped miniature Nd:YVO4 laser

We report on fundamental and intracavity frequency-doubled emission in a miniature Nd:YVO₄ (3×3×1 mm) laser. A maximum slope efficiency of η_s=58.6%, with optical efficiency of η₀=53.0% at 780 mW pump power was realized in a TEM₀₀ output beam. To obtain the optimum pump-beam focusing conditions we applied a new formalism in which the pump-beam propagation in the active medium was described by its M² factor. A good agreement between theoretical predictions and experimental results was observed. In second-harmonic regime, obtained by a KTP crystal, 230 mW green power that corresponds to 54% nonlinear conversion efficiency was reached.     

Bond polarizability calculations of hyper-Raman scattering in sodium, potassium and rubidium halides

Bond polarizability calculations extended for hyper-Raman scattering by the optical LO and TO zone-centre phonons in sodium, potassium and rubidium halides are presented. The effect of the crystal ionicity on both of the independent hyper-Raman tensor components is taken into consideration. Contributions of rotation of crystals bonds and their stretching to hyper-Raman scattering are determined. For LO phonons the electrooptic part of the hyper-Raman tensor is found. The results obtained are compared, when possible, with experimental and theoretical data reported in the literature.     

Optical forces acting on Rayleigh particle placed into interference field

We describe a general way how to calculate optical forces acting on Rayleigh particles or colloids placed into general interference field. In this paper, we focus on a configuration with three interfering beams laying in one plane and we present a comprehensive analysis of trap properties and particle behaviour. We found that this arrangement can be used for simultaneous 2D/3D confinement or sorting of particles having refractive index higher or lower comparing to the surrounding immersion medium.     

Development and design of a phase-locked loop in the subterahertz and terahertz ranges for a harmonic of the signal of a centimeter-wave synthesizer

We describe a phase-locked loop of a backward-wave oscillator in the range 667–857 GHz for the 34th–45th harmonic of the reference signal of a centimeter-wave synthesizer. Multiplication of the reference-synthesizer frequency in the phase-locked loop is performed by using a subharmonic mixer based on semiconductor superlattices. The mixer, operated at room temperature, has a nonlinear electron conductance.     

Experimental evidences for some unusual divalent cation complexes of valinomycin

From extraction experiments and γ-activity measurements, the exchange extraction constants corresponding to the general equilibrium M ²⁺ (aq) + 1 · Sr²⁺ (nb) ⇄ 1 · M ²⁺ (nb) + Sr²⁺ (aq) taking place in the two-phase water-nitrobenzene system (M ²⁺ = Cu²⁺, Zn²⁺, Cd²⁺, Pb²⁺, UO₂ ²⁺, Fe²⁺, Co²⁺, and Ni²⁺; 1 = valinomycin; aq = aqueous phase, nb = nitrobenzene phase) were evaluated. Moreover, the stability constants of the 1 · M ²⁺ complexes in water saturated nitrobenzene were calculated; they were found to increase in the order Fe²⁺ < Cd²⁺, Co²⁺ < Ni²⁺ < UO₂ ²⁺, Zn²⁺ < Cu²⁺ < Pb²⁺.