Congruent families and invariant tensors

Classical results of Chentsov and Campbell state that -- up to constant multiples -- the only $2$-tensor field of a statistical model which is invariant under congruent Markov morphisms is the Fisher metric and the only invariant $3$-tensor field is the Amari-Chentsov tensor. We generalize this result for arbitrary degree $n$, showing that any family of $n$-tensors which is invariant under congruent Markov morphisms is algebraically generated by the canonical tensor fields defined in an earlier paper

Quantum and Fisher Information from the Husimi and Related Distributions

The two principal/immediate influences -- which we seek to interrelate here -- upon the undertaking of this study are papers of Zyczkowski and Slomczy\'nski (J. Phys. A 34, 6689 [2001]) and of Petz and Sudar (J. Math. Phys. 37, 2262 [1996]). In the former work, a metric (the Monge one, specifically) over generalized Husimi distributions was employed to define a distance between two arbitrary density matrices. In the Petz-Sudar work (completing a program of Chentsov), the quantum analogue of the (classically unique) Fisher information (montone) metric of a probability simplex was extended to define an uncountable infinitude of Riemannian (also monotone) metrics on the set of positive definite density matrices. We pose here the questions of what is the specific/unique Fisher information metric for the (classically-defined) Husimi distributions and how does it relate to the infinitude of (quantum) metrics over the density matrices of Petz and Sudar? We find a highly proximate (small relative entropy) relationship between the probability distribution (the quantum Jeffreys' prior) that yields quantum universal data compression, and that which (following Clarke and Barron) gives its classical counterpart. We also investigate the Fisher information metrics corresponding to the escort Husimi, positive-P and certain Gaussian probability distributions, as well as, in some sense, the discrete Wigner pseudoprobability. The comparative noninformativity of prior probability distributions -- recently studied by Srednicki (Phys. Rev. A 71, 052107 [2005]) -- formed by normalizing the volume elements of the various information metrics, is also discussed in our context.Comment: 27 pages, 10 figures, slight revisions, to appear in J. Math. Phy

Optimal routing in a problem with constraints and cost functions depending on the task list

A routing problem with precedence conditions and complex cost functions is considered. In this problem, we must choose a starting point, a route (permutation of indices) and a specific trajectory for our process. This trajectory must be consistent with the route. In addition, this route or index permutation defines the sequence of tasks. In addition, the selection of the above route must satisfy the precedence conditions defined by the system of ordered index pairs. These ordered pairs are called address pairs. We consider the additive criterion routing problem. This criterion is natural for the problem of dismantling the system of radiation sources. In this article, we will focus on this engineering problem. In this problem very naturally cost functions arise with a dependency on the list of tasks. Namely, each time the performer touches those and only those sources that were not dismantled at that time. The solution uses widely understood dynamic programming. We build the optimal algorithm for the PC; information about the computational experiment is given. © Published under licence by IOP Publishing Ltd.The research was supported by Russian Foundation for Basic Research, project no. 19-01-00573

Generalized a bottleneck routing problem: dynamic programming and the start point optimization

One routing problem with constraints is considered. These constraints are reduced to precedence conditions which be to visiting sequence of megalopolises. This sequence is selected together with concrete trajectory and initial state for minimization of nonadditive criterion. These criterion is some generalization of known criterion for the bottleneck routing problem. The basis singularity of the used solving method consists of using of unique dynamic programming procedure for all initial states. The used criterion includes a controlled parameter influences on significance of different fragments of solution. © 201

О задаче последовательного обхода мегаполисов с условиями предшествования и функциями стоимости с зависимостью от списка заданий

A constrained routing problem with complicated cost functions is studied. The construction of the cost functions can be difficult, and therefore the stages of this construction are elements of the solution of the problem. This situation arises, in particular, in studying the engineering problem of dismantling radiation hazardous elements, where, in the framework of a problem statement traditional for discrete optimization, it takes an unacceptably long time to construct a cost matrix whose entries characterize the radiation doses received by performers at the stage of displacement and dismantling. It is assumed that, at the stage of the computational implementation of the resulting optimal algorithm, the corresponding “parts” of the matrix may be not fed to the computer’s memory but calculated as needed. Possible applications of the developed methods may be related to the problem of dismantling a decommissioned generator unit of an NPP. © Krasovskii Institute of Mathematics and Mechanics.Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00573) and is a part of the research carried out at the Ural Mathematical Center

Dynamic programming in the generalized bottleneck problem and the start point optimization

We consider one non-additive routing problem, which is a generalization of the well-known “bottleneck problem”. The parameter is assumed to be a positive number, the degree of which determines the weight of the corresponding stage of the displacement system. By varying the parameter, it is possible to make the initial or, on the contrary, the final stages of displacement dominant. The variant of aggregation of values with the above-mentioned weights corresponds to the ideological formulation of the “bottleneck problem”, but opens the possibility of investigating new versions of routing problems with constraints. It is assumed, however, that the statement of the problem is complicated by the dependence of values on the list of tasks and includes restrictions in the form of precedence conditions. In addition, in the interest of optimization, an arbitrary choice of the initial state from a given a priori set is allowed. For the construction, the apparatus of widely understood dynamic programming is used. The possibility of realizing a global extremum with any degree of accuracy under conditions when the set of possible initial states is not finite is investigated. © 2018 Udmurt State University. All rights reserved.Russian Academy of Sciences, RASFunding. The work was supported by the Presidium of the Russian Academy of Sciences, project no. 30 “Theory and Technology of Multilevel Decentralized Group Control in Conditions of Conflict and Cooperation”

One task of routing jobs in high radiation conditions

The problem of sequential bypass of megalopolises is investigated, focused on the problem of dismantling a system of radiation hazardous objects under constraints in the form of precedence conditions. The radiation impact on the performers is assessed by the doses received during movements and during the performance of dismantling works. The route problem of minimizing the dose load of workers carrying out dismantling in one or another sequence of operations is considered. The procedure for constructing an optimal solution using a variant of dynamic programming is investigated. On this basis, an algorithm is built, implemented on a PC. Examples of the numerical solution of a model problem for the minimum dose load are given. © 2021 by the authors.Ministry of Education and Science of the Russian Federation, Minobrnauka: 075-02-2021-1383Funding. The work was performed as a part of the research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2021-1383)

On the Problem of Sequential Traversal of Megalopolises with Precedence Conditions and Cost Functions Depending on a List of Tasks

A constrained routing problem with complicated cost functions is studied. The construction of the cost functions can be difficult,and therefore the stages of this construction are elements of the solution of the problem. This situation arises, in particular, in studyingthe engineering problem of dismantling radiation hazardous elements, where, in the framework of a problem statement traditional fordiscrete optimization, it takes an unacceptably long time to construct a cost matrix whose entries characterize the radiation dosesreceived by performers at the stage of displacement and dismantling. It is assumed that, at the stage of the computational implementationof the resulting optimal algorithm, the corresponding “parts” of the matrix may be not fed to the computer’s memory but calculated as needed.Possible applications of the developed methods may be related to the problem of dismantling a decommissioned generator unit of a nuclear power plant. © 2021, Pleiades Publishing, Ltd.Russian Foundation for Basic Research, РФФИ: 19-01-00573This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00573) and within the research carried out at the Ural Mathematical Center