On Extracting Maximum Stable Sets in Perfect Graphs Using Lovász's Theta Function

We study the maximum stable set problem. For a given graph, we establish several transformations among feasible solutions of different formulations of Lovász's theta function. We propose reductions from feasible solutions corresponding to a graph to those corresponding to its induced subgraphs. We develop an efficient, polynomial-time algorithm to extract a maximum stable set in a perfect graph using the theta function. Our algorithm iteratively transforms an approximate solution of the semidefinite formulation of the theta function into an approximate solution of another formulation, which is then used to identify a vertex that belongs to a maximum stable set. The subgraph induced by that vertex and its neighbors is removed and the same procedure is repeated on successively smaller graphs. We establish that solving the theta problem up to an adaptively chosen, fairly rough accuracy suffices in order for the algorithm to work properly. Furthermore, our algorithm successfully employs a warm-start strategy to recompute the theta function on smaller subgraphs. Computational results demonstrate that our algorithm can efficiently extract maximum stable sets in comparable time it takes to solve the theta problem on the original graph to optimality. This work was supported in part by NSF through CAREER Grant DMI-0237415. Part of this work was performed while the first author was at the Department of Applied Mathematics and Statisticsat Stony Brook University, Stony Brook, NY, USA.     

Spectral manifestations of the splitting of the lowest triplet levels in Pd complexes of nonsymmetric porphyrins

Based on the study of the phosphorescence and phosphorescence excitation spectra of Pd(II) tetramethylporphyrin (PdTMP) and Pd(II) tetramethyldiethylporphyrin (PdTMDEP) in solutions in 2-methyltetrahydrofuran and dodecane in the temperature range 77–283 K, the occurrence of the splitting of the lowest degenerate singlet (S ₁) and triplet (T ₁, T ₂) levels of porphyrin molecules is established. In the absorption of molecules of the compounds studied, two components, S ₁ and S ₂, are revealed in the range of allowed long-wavelength Q(0,0) transitions (530–550 nm) and four components, T ₁–T ₄, are observed in the range of spin-forbidden intersystem crossing transitions S ₀ → T _n (560–670 nm), with all the triplet levels T ₁-T ₄ being located below the S ₁ level on the energy scale. It is shown that an increase in the degree of deformation of the porphyrin macrocycle caused by steric interactions between β alkyls and hydrogen meso atoms is accompanied by an increase in the splitting in the system of singlet levels δE(S ₂–S ₁) from 120 cm^?1 for PdTMDEP to 215 cm^?1 for PdTMP, as well as by an increase in the splitting in the system of triplet levels δE(T ₂–T ₁) from 190 cm^?1 for PdTMDEP to 250 cm^?1 for PdTMP.     

The Fermi to Gamow-Teller mixing ratio of theβ + decay of52Mn and time-reversal invariance

To date, ten experimental measurements of the asymmetry parameter have been made for the positron decay of⁵²Mn. The Fermi to Gamow-Teller mixing ratioy can be deduced from such measurements and this quantity is important because the time-reversal violating amplitude is proportional toy/(1+y²). Our theoretical calculation using the Collective Model yieldsy=? 3.5×10^?4 for the deformation parameterβ=0.1 andy=? 4.6×10^?4 forβ=0.2. Such small values ofy are consistent with time-reversal invarisance.     

Nonlinear dynamics of a spinor Bose-Einstein condensate in a double-well potential

We examine the nonlinear dynamical behavior of a spinor Bose-Einstein condensate in a double-well potential. Considering a condensate with large number of atoms, such that it can be described using the mean field theory, we separate the spinor dynamics from the spatial dynamics under the single-mode approximation. We limit ourselves to certain initial conditions under which the spatial mode is frozen so that we can focus on the spinor dynamics only. Identifying collective spin variables of our system, we derive the corresponding nonlinear equations of motion for them. Employing standard stability analysis, we find and characterize fixed points of the system. For a wide range of physical parameters such as tunneling strength and non-linear interactions, as well as for various initial preparations of the system, we identify qualitatively different dynamical regimes possible in the system. In particular, complete and incomplete oscillations of spin variables between quantum wells are found. We also show that by bringing some fixed points close to each other in the phase space of the system, it is possible to induce amplitude modulation to those otherwise regular tunneling oscillations.     

Low-frequency Raman scattering in a polycrystalline C60 film: The role of orientational disorder

Low-frequency Raman scattering in the orientationally disordered phase of a polycrystalline C₆₀ film is investigated. By analogy with disordered media (glasses), the low-frequency Raman spectra are interpreted in terms of light scattering by localized vibrational states.     

Probing antiferromagnetic order with heat capacity in exchange-biased structures

We explore the magnetic heat capacity in exchange-biased ferromagnet/antiferromagnet bilayers theoretically. We show that changes in the antiferromagnetic structure due to the reversal of the ferromagnet layer can be detected by distinct features in the heat capacity. This offers a method for probing antiferromagnetic domains in exchange-biased systems.     

Reconstructing electron concentrations with small horizontal gradients from oblique ionospheric sounding

Translated from Chislennye Metody Resheniya Obratnykh Zadach Matematicheskoi Fiziki, pp. 147–153.