scholarly journals A foundation for probabilistic beliefs with or without atoms

2019 ◽  
Vol 14 (2) ◽  
pp. 709-778 ◽  
Author(s):  
Andrew Mackenzie
Keyword(s):  
Probability Measure ◽  
Pairwise Disjoint ◽  
Third Order ◽  
Qualitative Probability ◽  
Measure Representation ◽  
Probabilistic Beliefs ◽  
Additive Probability ◽  
Probability Spaces ◽  
The Given

We propose two novel axioms for qualitative probability spaces: (i) unlikely atoms, which requires that there is an event containing no atoms that is at least as likely as its complement; and (ii) third‐order atom‐swarming, which requires that for each atom, there is a countable pairwise‐disjoint collection of less‐likely events that can be partitioned into three groups, each with union at least as likely as the given atom. We prove that under monotone continuity, each of these axioms is sufficient to guarantee a unique countably‐additive probability measure representation, generalizing work by Villegas to allow atoms. Unlike previous contributions that allow atoms, we impose no cancellation or solvability axiom.

scholarly journals On atom-swarming and Luce’s theorem for probabilistic beliefs

2020 ◽  
Author(s):  
Andrew Mackenzie
Keyword(s):  
New York ◽  
Measurement Theory ◽  
Academic Press ◽  
Third Order ◽  
Qualitative Probability ◽  
Measurement Volume ◽  
Measure Representation ◽  
Polynomial Representations ◽  
Probabilistic Beliefs ◽  
Probability Spaces

Abstract For qualitative probability spaces, monotone continuity and third-order atom-swarming are together sufficient for a unique countably additive probability measure representation that may have atoms (Mackenzie in Theor Econ 14:709–778, 2019). We provide a new proof by appealing to a theorem of Luce (Ann Math Stat 38:780–786, 1967), highlighting the usefulness of extensive measurement theory (Krantz et al. in Foundations of Measurement Volume I: Additive and Polynomial Representations. Academic Press, New York, 1971) for economists.

DYNAMICS AND CHAOS CONTROL OF NONLINEAR SYSTEMS WITH ATTRACTION/REPULSION FUNCTION

2008 ◽  
Vol 18 (08) ◽  
pp. 2345-2371 ◽  
Author(s):  
XIAN LIU ◽  
JINZHI WANG ◽  
ZHISHENG DUAN ◽  
LIN HUANG
Keyword(s):  
Absolute Stability ◽  
Chaos Control ◽  
Linear Matrix ◽  
Chaotic Attractors ◽  
Matrix Inequality ◽  
Third Order ◽  
Simple Method ◽  
Stabilizing Controller ◽  
The Given ◽  
Absolute Stability Theory

In this paper, a more general third-order chaotic system with attraction/repulsion function is introduced on the basis of [Duan et al., 2005]. A gallery of chaotic attractors, bifurcation diagrams and Lyapunov exponent spectra are presented to show the interesting phenomena of the given system. Based on the absolute stability theory and linear matrix inequality (LMI), a simple method of chaos control for the system is proposed and a stabilizing controller is derived such that chaos oscillations of the system disappear and all chaotic trajectories of it are led to certain equilibrium. Numerical simulations are provided to illustrate the efficiency of the proposed method.

scholarly journals Remarks on continuously distributed sequences

2021 ◽  
Vol 13 (1) ◽  
pp. 89-97
Author(s):  
M. Paštéka
Keyword(s):  
Distribution Function ◽  
Probability Measure ◽  
Finitely Additive Probability ◽  
Additive Probability

In the first part of the paper we define the notion of the density as certain type of finitely additive probability measure and the distribution function of sequences with respect to the density. Then we derive some simple criterions providing the continuity of the distribution function of given sequence. These criterions we apply to the van der Corput's sequences. The Weyl's type criterions of continuity of the distribution function are proven.

scholarly journals g-FUNCTION IN PERTURBATION THEORY

2004 ◽  
Vol 19 (15) ◽  
pp. 2545-2559
Author(s):  
ANATOLY KONECHNY
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Perturbation Theory ◽  
Beta Function ◽  
Open String ◽  
Quantum Field ◽  
Two Dimensional ◽  
Third Order ◽  
The Third ◽  
The Given

We present some explicit computations checking a particular form of gradient formula for a boundary beta function in two-dimensional quantum field theory on a disk. The form of the potential function and metric that we consider were introduced in Refs. 16 and 18 in the context of background independent open string field theory. We check the gradient formula to the third order in perturbation theory around a fixed point. Special consideration is given to situations when resonant terms are present exhibiting logarithmic divergences and universal nonlinearities in beta functions. The gradient formula is found to work to the given order.

scholarly journals Global Attractivity Results on Complete Ordered Metric Spaces for Third-Order Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Mujahid Abbas ◽  
Maher Berzig
Keyword(s):  
Metric Spaces ◽  
Global Attractivity ◽  
Iterative Algorithms ◽  
Monotone Mappings ◽  
Contractive Condition ◽  
Third Order ◽  
Ordered Metric Spaces ◽  
All Solutions ◽  
The Difference ◽  
The Given

We establish fixed-point theorems for mixed monotone mappings in the setting of ordered metric spaces which satisfy a contractive condition for all points that are related by a given ordering. We also give a global attractivity result for all solutions of the difference equation where satisfies certain monotonicity conditions with respect to the given ordering. As an application of our obtained results, we present some iterative algorithms to solve a class of matrix equations. A numerical example is also presented to test the validity of the algorithms.

The Transformation Between the Eulerian and Lagrangian Solutions for Irrotational Standing Gravity Waves

2009 ◽  
Author(s):  
Yang-Yih Chen ◽  
Hung-Chu Hsu
Keyword(s):  
Gravity Waves ◽  
Particle Motion ◽  
Mass Conservation ◽  
Perturbation Parameter ◽  
Third Order ◽  
The Third ◽  
Asymptotic Expressions ◽  
Lagrangian Form ◽  
Standing Gravity Waves ◽  
The Given

This study reports the transformations between the third-order Eulerian and Lagrangian solutions for the standing gravity waves on the uniform depth. Regarding the motion of a marked fluid particle, the instantaneous velocity, the mass conservation and the free surface must be the same for either Eulerian or Lagrangian methods. We impose the assumption that the Lagrangian wave frequency is a function of wave steepness. Expanding the unknown function in a small perturbation parameter and using a successive expansion in a Taylor series for the water particle path and the period of a particle motion, the third order asymptotic expressions for the particle trajectories and the period of particle motion can be derived directly in Lagrangian form. It shows that the given Eulerian solutions are capable of being transformed into the completely unknown Lagrangian solutions and the reversible process is also identified.

PROBABILITY MEASURES ON PROJECTIONS IN VON NEUMANN ALGEBRAS

1989 ◽  
Vol 01 (02n03) ◽  
pp. 235-290 ◽  
Author(s):  
SHUICHIRO MAEDA
Keyword(s):  
Probability Measure ◽  
Linear Functional ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Probability Measures ◽  
Type I ◽  
Von Neumann ◽  
Positive Linear Functional ◽  
Finitely Additive Probability ◽  
Additive Probability

A state ϕ on a von Neumann algebra A is a positive linear functional on A with ϕ(1) = 1, and the restriction of ϕ to the set of projections in A is a finitely additive probability measure. Recently it was proved that if A has no type I 2 summand then every finitely additive probability measure on projections can be extended to a state on A. Here we give precise and complete arguments for proving this result.

scholarly journals On the complete integrability and linearization of nonlinear ordinary differential equations. II. Third-order equations

2006 ◽  
Vol 462 (2070) ◽  
pp. 1831-1852 ◽  
Author(s):  
V.K Chandrasekar ◽  
M Senthilvelan ◽  
M Lakshmanan
Keyword(s):  
General Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Complete Integrability ◽  
Nonlinear Ordinary Differential Equations ◽  
Powerful Method ◽  
Integrals Of Motion ◽  
Third Order ◽  
Non Local ◽  
The Given

We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle–Singer method. We describe a procedure to deduce all the integrals of motion associated with the given equation, so that the general solution follows straightforwardly from these integrals. The method is illustrated with several examples. Further, we propose a powerful method of identifying linearizing transformations. The proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation. In addition to the above, we provide an algorithm to invert the non-local linearizing transformation. Through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation.

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