An inequality for the trace of matrix products

In this paper we consider a product of n complex m×m matrices A_k (k=1,…,n) with singular values ∝^(k)_i ordered in decreasing magnitude. Using the spectral resolution for the operators $\text{A}{}^{\mathrm{dagger;}}{}_{\mathrm{k}}\text{A}{}_{\mathrm{k}}$, it is shown that $\text{|}\text{Tr}\text{A}{}_1\text{…A}{}_{\mathrm{n}}\text{|≤}\text{∑}\text{i=1}\text{m}\text{Φ}\text{i=1}\text{n}\text{α}{}^{\mathrm{(k)}}{}_{\mathrm{i}}\text{.}$This inequality is an extension of an inequality of von Neumann in the simple case that n=2. The necessary and sufficient condition for the equality sign to hold is established. Application of Hölder's inequality leads to further inequalities which can be useful in statistical mechanics.     

NMR study of guanidinium cation dynamics in C(NH2)3SbCl6

Proton NMR second moment and spin—lattice relaxation times T ₁ and T _1p have been studied for polycrystalline guanidinium hexachloroantimonate C(NH₂)₃SbCl₆ in a wide temperature range. A dynamic inequivalence of two cations has been detected in spite of their crystal-lographical equivalence. Activation parameters for C₃ reorientation and self-diffusion of the more mobile cation have been determined. It was shown that the para–ferroelastic phase transition at 351 K is connected with abrupt changes in the dynamics of the two cations. The weaker, second-order transition at 265 K is thought to be related to a change in the dynamics of one of the cations.     

Comparing BOLD fMRI signal changes in the awake and anesthetized rat during electrical forepaw stimulation

The difference between awake curarized and alpha-chloralose anesthetized animals was studied with respect to the BOLD signal response in an fMRI experiment. By studying the activation of the cortex upon electrical forepaw stimulation in the same rat, but following consecutively applied curarization and alpha-chloralose anesthesia protocols, it was possible to compare quantitatively the effect of both immobilization protocols on the fMRI data. The largest BOLD signal change as a result of forepaw stimulation was found in the awake condition, however the activated areas are less specific than those in the anesthetized state leaving it more difficult to interpret.     

On the free energy in systems with separable interactions. III

In this paper we extend the calculation of the free energy in systems with separable interactions of the ferromagnetic type as given in paper I of this series to a more general class of systems characterized by a hamiltonian, in which the one-particle operator Σ_kT(k) is replaced by a more general operator T which may contain interactions as well. This extension enables us to simplify a great deal the derivation given in paper II of the lower bound to the free energy for systems with both ferromagnetic and antiferromagnetic interactions. We also discuss in some detail under which conditions the upper and the lower bounds can be equal. (The conditions include in particular the hamiltonian treated in paper II.)     

Systems with separable many-particle interactions. I

In this paper we give an exact evaluation of the free energy per particle for systems with separable many-particle interactions described by a hamiltonian of the type ? = ∑_kT(k) + NP (N^-1 ∑_kV(k)), where P is an arbitrary polynomial. In the proof use is made of a fundamental theorem due to Bogoliubov Jr. for ferromagnetic quadratic operators. The free energy can be obtained from a trial hamiltonian, which is linear in the operators T and V.