Narrow Results

Motivated by recent developments on large deviations in states of the spin chain, we reconsider the work of Petz, Raggio and Verbeure in 1989 on the variational expression of free energy density in the presence of a mean field type perturbation. We extend their results from the product state case to the Gibbs state case in the setting of translation-invariant interactions of finite range. In the special case of a locally faithful quantum Markov state, we clarify the relation between two different kinds of free energy densities (or pressure functions).

We show how algebraic identities, inequalities and constructions, which hold for numbers or matrices, often have analogs in the geometric classes of convex bodies or convex functions. By letting the polar body $K^{\circ }$ or the dual function ${\phi }^{\ast }$ play the role of the inverses “$K^{-1}$” and “${\phi }^{-1}$”, we are able to conjecture many new results, which often turn out to be correct. As one example, we prove that for every convex function $\phi$ one has

A formulation of singular classical theories (determined by degenerate Lagrangians) without constraints is presented. A partial Hamiltonian formalism in the phase space having an initially arbitrary number of momenta (which can be smaller than the number of velocities) is proposed. The equations of motion become first-order differential equations, and they coincide with those of multi-time dynamics, if a certain condition is imposed. In a singular theory, this condition is fulfilled in the case of the coincidence of the number of generalized momenta with the rank of the Hessian matrix. The noncanonical generalized velocities satisfy a system of linear algebraic equations, which allows an appropriate classification of singular theories (gauge and nongauge). A new antisymmetric bracket (similar to the Poisson bracket) is introduced, which describes the time evolution of physical quantities in a singular theory. The origin of constraints is shown to be a consequence of the (unneeded in our formulation) extension of the phase space, when the new bracket transforms into the Dirac bracket. Quantization is briefly discussed.

We discuss the proposal of a new method to transform an $f(R)$ metric gravity theory into a general relativistic theory including an auxiliary scalar field, recently introduced by Cotsakis et al. We argue that (i) the fact that the fourth-order equations of $f(R)$ metric gravity can be recast (via a Legendre transformation) into Einstein equations without any conformal rescaling has been thoroughly clarified in the previous literature, and (ii) the newly proposed method produces a set of equations that are not equivalent to the original theory. In the conclusion, a comment is added on another aspect of the Legendre transformation which seems to be generally overlooked.

Amphoteric hydrogels have been widely used in biologic and mechanic areas such as robot joints, biological cells, drug delivery systems. There are many factors that can influence the deformation of hydrogels, such as pH value, concentration of salt ions, solid constrains and external force, which all together makes the calculation very complicated. In this paper, an improved model for the calculation of amphoteric pH-sensitive hydrogels’ deformation is developed so that the behaviors of hydrogels in both acidic and alkaline solutions can be described quantitatively, and this model is implemented into finite element method (FEM) software ABAQUS in order to simulate hydrogel’s homogeneous and inhomogeneous deformations. The FEM results under free swelling state are compared with both analytical solution and FEM results existed before, and FEM results fit well with the experimental data. The buckling of amphoteric pH-sensitive hydrogel membrane is also analyzed, some natural phenomena, such as the wrinkling of human finger’s skin and the broad bean testa are successfully explained.

No abstract received.