Continuous symmetry analysis of NMR chemical shielding anisotropy

Molecular symmetry is a key parameter which dictates the NMR chemical shielding anisotropy (CSA). Whereas correlations between specific geometrical features of molecules and the CSA are known, the quantitative correlation with symmetry--a global structural feature--has been unknown. Here we demonstrate a CSA/symmetry quantitative relation for the first time: We study how continuous deviation from exact symmetry around a nucleus affects its shielding. To achieve this we employed the continuous symmetry measures methodology, which allows one to quantify the degree of content of a given symmetry. The model case we use for this purpose is a population of distorted SiH(4) structures, for which we follow the (29)Si CSA as a function of the degree of tetrahedral symmetry and of square-planar symmetry. Quantitative correlations between the degree of these symmetries and the NMR shielding parameters emerge.     

On the Stark broadening of Sr+ and Ba+ resonance lines in ultracold neutral plasmas

We report results of the Stark broadening calculations for Sr⁺ and Ba⁺ resonance lines in ultracold plasmas using semiempirical formulas and numerical computer simulation technique. The simulation results show that strong collisions dominate Stark broadening at very low electron temperatures and weak collision approximation used recently by Vrinceanu et al. cannot be applied in this temperature region. Consequently, the temperature trend of Stark widths and shifts changes from successfully used at elevated temperatures to an increasing trend with temperature, which is characteristic for strong collisions at low temperature.     

Exchangeable measures for subshifts

Let Ω be a Borel subset of where S is countable. A measure is called exchangeable on Ω, if it is supported on Ω and is invariant under every Borel automorphism of Ω which permutes at most finitely many coordinates. De-Finetti's theorem characterizes these measures when . We apply the ergodic theory of equivalence relations to study the case , and obtain versions of this theorem when Ω is a countable state Markov shift, and when Ω is the collection of beta expansions of real numbers in [0,1] (a non-Markovian constraint).     

On the solvability for the mixed-type Lyapunov equation

^** Email: xsf{at}math.pku.edu.cn^*** Email: mscheng{at}math.pku.edu.cn In this paper, the linear matrix equation X = AXB^* + BXA^* +Q is considered, which is called the mixed-type Lyapunov equation.Some necessary and sufficient conditions for the existence ofa unique solution are presented. Since a Hermitian positivesemidefinite solution is important from the application pointof view, some sufficient conditions for the existence of a Hermitianpositive semidefinite solution are derived.     

Projections in operator ranges

If is a Hilbert space, is a positive bounded linear operator on and is a closed subspace of , the relative position between and establishes a notion of compatibility. We show that the compatibility of is equivalent to the existence of a convenient orthogonal projection in the operator range with its canonical Hilbertian structure.

     

Cuntz-Pimnser algebras, completely positive maps and Morita equivalence

Let be a completely positive map on and let be the associated GNS--correspondence. We prove a result that implies, in particular, that the Cuntz-Pimsner algebra of , , is strongly Morita equivalent to the Cuntz algebra , where is the index of .

     

A noncommutative extended de Finetti theorem

The extended de Finetti theorem characterizes exchangeable infinite sequences of random variables as conditionally i.i.d. and shows that the apparently weaker distributional symmetry of spreadability is equivalent to exchangeability. Our main result is a noncommutative version of this theorem.In contrast to the classical result of Ryll-Nardzewski, exchangeability turns out to be stronger than spreadability for infinite sequences of noncommutative random variables. Out of our investigations emerges noncommutative conditional independence in terms of a von Neumann algebraic structure closely related to Popa's notion of commuting squares and Kümmerer's generalized Bernoulli shifts. Our main result is applicable to classical probability, quantum probability, in particular free probability, braid group representations and Jones subfactors.     

Finite Morse index steady states of van der Waals force driven thin film equations

We study positive solutions of the equation

     

An almost periodic noncommutative Wiener's Lemma

We develop a theory of almost periodic elements in Banach algebras and present an abstract version of a noncommutative Wiener's Lemma. The theory can be used, for example, to derive some of the recently obtained results in time-frequency analysis such as the spectral properties of the finite linear combinations of time-frequency shifts.     

The King-Werner method for solving nonsymmetric algebraic Riccati equation

For the nonsymmetric algebraic Riccati equation arising from transport theory, we concern about solving its minimal positive solution. In [1], Lu transferred the equation into a vector form and pointed out that the minimal positive solution of the matrix equation could be obtained via computing that of the vector equation. In this paper, we use the King-Werner method to solve the minimal positive solution of the vector equation and give the convergence and error analysis of the method. Numerical tests show that the King-Werner method is feasible to determine the minimal positive solution of the vector equation.