Positive contraction mappings for classical and quantum Schrodinger systems

The classical Schrodinger bridge seeks the most likely probability law for a diffusion process, in path space, that matches marginals at two end points in time; the likelihood is quantified by the relative entropy between the sought law and a prior, and the law dictates a controlled path that abides by the specified marginals. Schrodinger proved that the optimal steering of the density between the two end points is effected by a multiplicative functional transformation of the prior; this transformation represents an automorphism on the space of probability measures and has since been studied by Fortet, Beurling and others. A similar question can be raised for processes evolving in a discrete time and space as well as for processes defined over non-commutative probability spaces. The present paper builds on earlier work by Pavon and Ticozzi and begins with the problem of steering a Markov chain between given marginals. Our approach is based on the Hilbert metric and leads to an alternative proof which, however, is constructive. More specifically, we show that the solution to the Schrodinger bridge is provided by the fixed point of a contractive map. We approach in a similar manner the steering of a quantum system across a quantum channel. We are able to establish existence of quantum transitions that are multiplicative functional transformations of a given Kraus map, but only for the case of uniform marginals. As in the Markov chain case, and for uniform density matrices, the solution of the quantum bridge can be constructed from the fixed point of a certain contractive map. For arbitrary marginal densities, extensive numerical simulations indicate that iteration of a similar map leads to fixed points from which we can construct a quantum bridge. For this general case, however, a proof of convergence remains elusive.Comment: 27 page

Typical support and Sanov large deviations of correlated states

Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov's theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states.Comment: 29 pages, no figures, references adde

Renormalized Coupled Cluster Approaches in the Cluster-in-Molecule Framework: Predicting Vertical Electron Binding Energies of the Anionic Water Clusters (H2O)n–

Anionic water clusters are generally considered to be extremely challenging to model using fragmentation approaches due to the diffuse nature of the excess electron distribution. The local correlation coupled cluster (CC) framework cluster-in-molecule (CIM) approach combined with the completely renormalized CR-CC(2,3) method [abbreviated CIM/CR-CC(2,3)] is shown to be a viable alternative for computing the vertical electron binding energies (VEBE). CIM/CR-CC(2,3) with the threshold parameter ζ set to 0.001, as a trade-off between accuracy and computational cost, demonstrates the reliability of predicting the VEBE, with an average percentage error of ∼15% compared to the full ab initio calculation at the same level of theory. The errors are predominantly from the electron correlation energy. The CIM/CR-CC(2,3) approach provides the ease of a black-box type calculation with few threshold parameters to manipulate. The cluster sizes that can be studied by high-level ab initio methods are significantly increased in comparison with full CC calculations. Therefore, the VEBE computed by the CIM/CR-CC(2,3) method can be used as benchmarks for testing model potential approaches in small-to-intermediate-sized water clusters

Development of technology for bakery products of functional purpose using non-traditional raw materials

Good and healthy nutrition is one of the most important and necessary conditions for conservation of life and health of the nation. In recent years in the science of nutrition a new direction - functional nutrition has developed. Functional nutrition products, when used systematically, should have a regulating effect on the macroorganism or certain organs and systems, providing a non-medicamentous correction of their function. Functional food products are intended for the systematic use in diet in all age- groups of healthy population. Consisting of physiologically functional food ingredients they reduce the risk of developing nutritional diseases, preserve and improve health. Physiologically functional food ingredients include biologically active and physiologically valuable ingredients , safe for health, having precise physicochemical characteristics. Their properties have been identified and scientifically justified, and daily intake of food products has been established. These are dietary fiber, vitamins, in particular vitamins-antioxidants, minerals, polyunsaturated fatty acids and their sources, probiotics, prebiotics, and synbiotics. Bakery products were and remain one of the main food products of the population of our country. Thanks to its daily consumption bread is one of the most important food products, the nutritional value of which is of primary importance. It provides more than 50% of the daily energy requirement and up to 75% of the demand for vegetable protein. Therefore, functional nutrition products are of great importance for improving the diet of the population

On groups of exponent 36.

Locally finiteness is proved for a group of exponent 36 containing an involution and no elements of order 6