scholarly journals Proportional lumpability and proportional bisimilarity

2021 ◽  
Author(s):  
Andrea Marin ◽  
Carla Piazza ◽  
Sabina Rossi
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Upper And Lower Bounds ◽  
General Definition ◽  
Performance Indices ◽  
State Aggregation ◽  
Structural Regularity ◽  
Original Definition ◽  
Aggregation Technique ◽  
Definition Of

AbstractIn this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.

scholarly journals Limit theorems for asymptotic circular mth-order Markov chains indexed by an m-rooted homogeneous tree

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1817-1832
Author(s):  
Huilin Huang ◽  
Weiguo Yang
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Law Of Large Numbers ◽  
Homogeneous Tree ◽  
Strong Law ◽  
Nonhomogeneous Markov Chain ◽  
Asymptotic Equipartition Property ◽  
Large Numbers ◽  
Definition Of ◽  
Rooted Homogeneous Tree

In this paper, we give the definition of an asymptotic circularmth-order Markov chain indexed by an m rooted homogeneous tree. By applying the limit property for a sequence of multi-variables functions of a nonhomogeneous Markov chain indexed by such tree, we estabish the strong law of large numbers and the asymptotic equipartition property (AEP) for asymptotic circular mth-order finite Markov chains indexed by this homogeneous tree. As a corollary, we can obtain the strong law of large numbers and AEP about the mth-order finite nonhomogeneous Markov chain indexed by the m rooted homogeneous tree.

Classified Discrete-Time Markov Chains

2015 ◽  
pp. 87-115
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Formal Definition ◽  
Markov Chain Models ◽  
Definition Of ◽  
Markovian Systems

The main focus of this chapter is on the formalization of classified DTMCs. The chapter begins by presenting the formalization of some foundational notions of classified states, which are categorized based on reachability, periodicity, or absorbing features. Then, these results along with the formal definition of a DTMC, presented in the previous chapter, are used to formalize classified Markov chains, such as aperiodic and irreducible DTMCs. Based on these concepts, some long-term properties are verified for the purpose of formally checking the correctness of the functions of Markovian systems or analyzing the performance of Markov chain models.

scholarly journals Positive Dependence and Weak Convergence

2006 ◽  
Vol 43 (01) ◽  
pp. 48-59
Author(s):  
A. Colangelo ◽  
A. Müller ◽  
M. Scarsini
Keyword(s):  
Probability Measure ◽  
Weak Convergence ◽  
Markov Chains ◽  
Total Positivity ◽  
Precise Definition ◽  
General Definition ◽  
Actuarial Science ◽  
Stationary Distributions ◽  
Positive Dependence ◽  
Definition Of

A more general definition of MTP2 (multivariate total positivity of order 2) probability measure is given, without assuming the existence of a density. Under this definition the class of MTP2 measures is proved to be closed under weak convergence. Characterizations of the MTP2 property are proved under this more general definition. Then a precise definition of conditionally increasing measure is provided, and closure under weak convergence of the class of conditionally increasing measures is proved. As an application we investigate MTP2 properties of stationary distributions of Markov chains, which are of interest in actuarial science.

scholarly journals Dialectical and activity-based foundations for the modern original definition of science

2021 ◽  
Vol 21 (3-4) ◽  
pp. 22-30
Author(s):  
Roman V. Sorokin
Keyword(s):  
Social Life ◽  
Technological Development ◽  
Methodological Approach ◽  
Scientific Activity ◽  
General Definition ◽  
Modern Problem ◽  
Theoretical Understanding ◽  
Actual Problem ◽  
Original Definition ◽  
Definition Of

The article is devoted to the actual problem of uncertainty in science. The work done not only contributes to the removal/solution of this modern problem, but also contributes to the theoretical understanding of the need for qualitative changes in the sphere of scientific activity, the most organic combination of theory with practice to form a new form of science, guaranteed to remove the global dangers and uncertainties of scientific and technological development. The author presents the basis for the formation of a modern, general definition of science, the absence of which indicates the presence of the above problem. The aim of the research is to give a modern general definition of science, making a serious step, if not to the final solution, then to a generally meaningful solution of this problem. In the authors opinion, all substantive elements necessary for forming a modern general definition of science have already been identified, but they are not structured. The methodological approach has a high importance in solving this problem, and the emphasis should be placed on dialectics, rather than formal logic. In dialectical methodology, it is necessary to refer to the theory of dialectical development from the abstract to the concrete, as well as the general theory of activity. The specificity of science is defined in the consistent disclosure of the content of science as knowledge, as individual-collective, internalist-externalist activity and as a sphere of social life. As a result of the analysis of these aspects the author comes to the initial, general and actual definition of science.

scholarly journals A simple design rule for 1st order form-closure of underactuated hands

2011 ◽  
Vol 2 (1) ◽  
pp. 1-8 ◽  
Author(s):  
S. Krut ◽  
V. Bégoc
Keyword(s):  
International Workshop ◽  
Contact Forces ◽  
Design Rule ◽  
General Definition ◽  
Order Form ◽  
Contact Points ◽  
Original Definition ◽  
Form Closure ◽  
The Stability ◽  
Definition Of

Abstract. The property of form-closure of a grasp, as generally defined in the literature, is based on the assumption that contact points between the hand and the object are fixed in space. However, this assumption is false when considering a grasp exerted by an underactuated hand, since in this case, it is not possible to control the position of each phalanx independently. In spite of researchers' interest in studying form-closure, none of the available published work on this subject takes into consideration the particular kinematics of underactuated hands. Actually, there are few available tools to qualify or quantify the stability of a grasp exerted by an underactuated hand, thus the design of underactuated hands mostly results from an intuitive approach. This paper aims to reduce this gap. A classification of underactuated hands is proposed, based on the expression of contact forces. This highlights the influence of non-backdrivable mechanisms introduced in the transmission of the closing motion of the hand on the stability of the grasp. The way to extend the original definition of form-closure to underactuated grasps is illustrated. A more general definition is formulated, which checks the stability of the set "object + hand". Using this new definition, a simple rule is proposed for designing a hand capable of achieving 1st order form-closed grasps. This paper was presented at the IFToMM/ASME International Workshop on Underactuated Grasping (UG2010), 19 August 2010, Montréal, Canada.

scholarly journals Positive Dependence and Weak Convergence

2006 ◽  
Vol 43 (1) ◽  
pp. 48-59 ◽  
Author(s):  
A. Colangelo ◽  
A. Müller ◽  
M. Scarsini
Keyword(s):  
Probability Measure ◽  
Weak Convergence ◽  
Markov Chains ◽  
Total Positivity ◽  
Precise Definition ◽  
General Definition ◽  
Actuarial Science ◽  
Stationary Distributions ◽  
Positive Dependence ◽  
Definition Of

A more general definition of MTP2 (multivariate total positivity of order 2) probability measure is given, without assuming the existence of a density. Under this definition the class of MTP2 measures is proved to be closed under weak convergence. Characterizations of the MTP2 property are proved under this more general definition. Then a precise definition of conditionally increasing measure is provided, and closure under weak convergence of the class of conditionally increasing measures is proved. As an application we investigate MTP2 properties of stationary distributions of Markov chains, which are of interest in actuarial science.

scholarly journals Controlled Markov chains with non-exponential discounting and distribution-dependent costs

2021 ◽  
Author(s):  
Hongwei Mei ◽  
George Yin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
Existence And Uniqueness ◽  
Closed Loop ◽  
Mean Field ◽  
Time Inconsistency ◽  
Mean Field Game ◽  
Definition Of ◽  
Exponential Discounting

This paper deals with a controlled Markov chain in continuous time with a non-exponential discounting and distribution-dependent cost functional. A  definition of closed-loop  equilibrium  is given and its existence and uniqueness are established. Due to the time-inconsistency brought by the non-exponential discounting and distribution dependence, it is proved that the equilibrium is locally optimal in some appropriate sense. Moreover, it is shown that our problem is  equivalent to a mean-field game for infinite-many symmetric players with a non-exponential discounting cost.

scholarly journals IMPRECISE MARKOV CHAINS AND THEIR LIMIT BEHAVIOR

2009 ◽  
Vol 23 (4) ◽  
pp. 597-635 ◽  
Author(s):  
Gert de Cooman ◽  
Filip Hermans ◽  
Erik Quaeghebeur
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Transition Probabilities ◽  
Finite Markov Chain ◽  
Initial State ◽  
Frobenius Theorem ◽  
Credal Set ◽  
Credal Sets ◽  
Definition Of ◽  
Uncertainty Models

When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-calledcredal setsthat these probabilities are known or believed to belong to and by allowing the probabilities to vary over such sets. This leads to the definition of animprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-calledlowerandupper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at timenevolves asn→∞: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalization of the classical Perron–Frobenius theorem to imprecise Markov chains.

Populism and Liberal Democracy

2019 ◽  
Author(s):  
Takis S. Pappas
Keyword(s):  
United States ◽  
Liberal Democracy ◽  
The United States ◽  
Comprehensive Theory ◽  
The People ◽  
Original Definition ◽  
Key Issues ◽  
Great Relevance ◽  
Definition Of

Based on an original definition of modern populism as “democratic illiberalism” and many years of meticulous research, Takis Pappas marshals extraordinary empirical evidence from Argentina, Greece, Peru, Italy, Venezuela, Ecuador, Hungary, the United States, Spain, and Brazil to develop a comprehensive theory about populism. He addresses all key issues in the debate about populism and answers significant questions of great relevance for today’s liberal democracy, including: • What is modern populism and how can it be differentiated from comparable phenomena like nativism and autocracy? • Where in Latin America has populism become most successful? Where in Europe did it emerge first? Why did its rise to power in the United States come so late? • Is Trump a populist and, if so, could he be compared best with Venezuela’s Chávez, France’s Le Pens, or Turkey’s Erdoğan? • Why has populism thrived in post-authoritarian Greece but not in Spain? And why in Argentina and not in Brazil? • Can populism ever succeed without a charismatic leader? If not, what does leadership tell us about how to challenge populism? • Who are “the people” who vote for populist parties, how are these “made” into a group, and what is in their minds? • Is there a “populist blueprint” that all populists use when in power? And what are the long-term consequences of populist rule? • What does the expansion, and possibly solidification, of populism mean for the very nature and future of contemporary democracy? Populism and Liberal Democracy will change the ways the reader understands populism and imagines the prospects of liberal democracy.

scholarly journals A remark on local fractional calculus and ordinary derivatives

2016 ◽  
Vol 14 (1) ◽  
pp. 1122-1124 ◽  
Author(s):  
Ricardo Almeida ◽  
Małgorzata Guzowska ◽  
Tatiana Odzijewicz
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Short Note ◽  
General Definition ◽  
Local Fractional Derivative ◽  
Fundamental Properties ◽  
Local Fractional Calculus ◽  
Definition Of

AbstractIn this short note we present a new general definition of local fractional derivative, that depends on an unknown kernel. For some appropriate choices of the kernel we obtain some known cases. We establish a relation between this new concept and ordinary differentiation. Using such formula, most of the fundamental properties of the fractional derivative can be derived directly.

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