Two Formalisms of Stochastization of One-Step Models (original) (raw)
Implementing a Method for Stochastization of One-Step Processes in a Computer Algebra System
Programming and Computer Software
When modeling such phenomena as population dynamics, controllable flows, etc., a problem arises of adapting the existing models to a phenomenon under study. For this purpose, we propose to derive new models from the first principles by stochastization of one-step processes. Research can be represented as an iterative process that consists in obtaining a model and its further refinement. The number of such iterations can be extremely large. This work is aimed at software implementation (by means of computer algebra) of a method for stochastization of one-step processes. As a basis of the software implementation, we use the SymPy computer algebra system. Based on a developed algorithm, we derive stochastic differential equations and their interaction schemes. The operation of the program is demonstrated on the Verhulst and Lotka-Volterra models.
One-Step Stochastic Processes Simulation Software Package
Background. It is assumed that the introduction of stochastic in mathematical model makes it more adequate. But there is virtually no methods of coordinated (depended on structure of the system) stochastic introduction into deterministic models. Authors have improved the method of stochastic models construction for the class of one-step processes and illustrated by models of population dynamics. Population dynamics was chosen for study because its deterministic models were sufficiently well explored that allows to compare the results with already known ones. Purpose. To optimize the models creation as much as possible some routine operations should be automated. In this case, the process of drawing up the model equations can be algorithmized and implemented in the computer algebra system. Furthermore, on the basis of these results a set of programs for numerical experiment can be obtained. Method. The computer algebra system Axiom is used for analytical calculations implementation. ...
Platonic Investigations / Платоновские исследования, 2022
The article deals with the logical structure of action in Stoicism. Both Early and Imperial Stoics looked up to the same sequence: impression — assent — judgment — action. However, Socratic identity of virtue and knowledge developed in two different ways. Early Stoics supposed that the right judgment follows to ethics, which follows to physics and logic; therefore, good action is consistent with a long chain of premises and conclusions. The advantage of this approach is a strict logical necessity of the way of life, but there is deficiency too: if some philosophical opponents will criticize the intermediate premises of the logical consistency, the whole reasoning will be ruined. Imperial Stoics changed this situation: each action is a consequence of principles (τὰ δόγματα, decreta); then action passes same intermediate premises and goes back to the first principles. However, the formal structure remains the same: we recognize a thing by the general concepts, which directly sends us to the first principles, not to the ethical precepts. Although Early Stoics mention dogma in a similar sense three times, we have to admit: this structure was inherent in Late Stoa par excellence. At the same time, we cannot assert some key difference between two approaches, because both of them provide consistency of theory and practice.
On stochastic equations in filtering of Markov processes
Lithuanian Mathematical Journal
The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: Д. Сургайлис. О стохастических уравнениях фильтрации марковских процессов D. Surgailis. Apie Markovo procesų filtracijos stochastines lygtis
Modeling of the Immobilization Process by the Random Walk Method
ВЕСТНИК ПЕРМСКОГО УНИВЕРСИТЕТА. ФИЗИКА, 2016
Работа посвящена разработке макроскопической модели, описывающей процессы осаждения примеси и закупорку пор для произвольных значений концентрации примеси. Основной и самой распространенной причиной засорения фильтров является сорбция частиц примеси стенками пор или «физическая сорбция». В настоящей работе исследована задача о дрейфе твердых не взаимодействующих между собой частиц в капилляре. Между входом и выходом из капилляра задан постоянный перепад давления. В начальный момент времени внутри канала возникает течение Пуазейля. Расположение частиц на входе в капилляр задается случайным образом по времени и пространству. Учет взаимодействия частиц с потоком производится в приближении Стокса. Кроме этого, в модели учтены случайные столкновения, вызванные диффузией. Задача решена численно в рамках модели случайных блужданий. Получена эволюция течения жидкости в поре при ее закупорке: поля функции тока, давления и завихренности. Определены зависимости скорости оседания частиц от скорости потока и начальной концентрации частиц в потоке. Исследована зависимость расхода через поперечное сечение поры от концентрации осевших частиц. Произведены оценки времени закупорки канала.
On the three-stage version of stable dynamic model
An attempt to merge into a single model, which reduces to the solution of non-smooth convex optimization problem: calculation model of OD-matrix (entropy model), the mode split model and the model of the equilibrium distribution of flows (Stable dynamic model, Nesterov - de Palma, 2003). To best of our knowledge, this is the first attempt to combine this three models. Previously such attempts were done for other types of equlibrium models, mainly with the BMW-model (1955), the calibration of which is significantly more difficult. We also remark, that our model much better then traditional from computational point of view.
Stochastic model of business process decomposition
arXiv (Cornell University), 2019
Decomposition is the basis of works dedicated to business process modelling at the stage of information and management systems' analysis and design. The article shows that the business process decomposition can be represented as a Galton-Watson branching stochastic process. This representation allows estimating the decomposition tree depth and the total amount of its elements, as well as explaining the empirical requirement for the business function decomposition (not more than 7 elements). The problem is deemed relevant as the obtained results allow objectively estimating the labor input in business process modelling.
Partial Hamiltonian formalism, multi-time dynamics and singular theories
We formulate singular (with degenerate Lagrangians) classical theories (for clarity, in local coordinates) without involving constraints. First, we recall the standard action principle (for pedagogical reasons and in order to establish notation). Then, applying it to the action (27), we develop a partial (in the sense that not all velocities are transformed to momenta) Hamiltonian formalism in an initially reduced phase space (with canonical coordinates q i , p i , where the number n p ≤ n of momenta p i , i = 1, . . . , n p , see (17), is arbitrary, where n is the dimension of the configuration space) in terms of a partial Hamiltonian H 0 (q i , p i , q α ,q α ), see (18), and (n − n p ) additional Hamiltonians H α (q i , p i , q α ,q α ), α = n p + 1, . . . , n, see (20) (instead of the remaining momenta p α defined in the standard full Hamiltonian formalism (6)). In this way we obtain (n − n p + 1) Hamilton-Jacobi equations (25)-(26) which fully determine the dynamics. The equations of motion are first-order differential equations (33)-(34) with respect to the canonical coordinates q i , p i and second-order differential equations (35) in the noncanonical coordinates q α (which have no corresponding momenta). In the partial Hamiltonian formalism (which describes the same dynamics as the Lagrange equations of motion ), the number of momenta n p ≤ n is arbitrary. The limit cases n p = n and n p = 0 correspond to the standard Hamiltonian and Lagrangian dynamics (discussed in ), respectively.