Mindful Application of Standards for Avionics - An Intentional, Systematic, and Measurable Transformation

2018 ◽  
Author(s):  
Uma D. Ferrell
Keyword(s):  
Measurable Transformation

scholarly journals Some Remarks on a Recent Paper by Borch

1961 ◽  
Vol 1 (5) ◽  
pp. 265-272 ◽  
Author(s):  
Paul Markham Kahn
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Fixed Number ◽  
Point Of View ◽  
Optimum Amount ◽  
Measurable Transformation ◽  
The Difference ◽  
Image Position ◽  
Stop Loss ◽  
Condition B

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

scholarly journals On the almost everywhere convergence of the ergodic averages

1990 ◽  
Vol 10 (1) ◽  
pp. 141-149
Author(s):  
F. J. Martín-Reyes ◽  
A. De La Torre
Keyword(s):  
Measure Space ◽  
Uniform Boundedness ◽  
Finite Measure ◽  
Necessary And Sufficient Condition ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

AbstractLet (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

The Pointwise Ergodic Theorem for Transformations whose Orbits contain or are contained in the Orbits of a Measure-Preserving Transformation

1982 ◽  
Vol 34 (6) ◽  
pp. 1303-1318 ◽  
Author(s):  
John C. Kieffer ◽  
Maurice Rahe
Keyword(s):  
Information Theory ◽  
Ergodic Theory ◽  
Probability Space ◽  
Sufficient Conditions ◽  
Measurable Transformation ◽  
Input And Output ◽  
Measure Preserving Transformation ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Measure Preserving Transformations

1. Introduction. Let be a probability space with standard. Let T be a bimeasurable one-to-one map of Ω onto itself. Let U: Ω → Ω be another measurable transformation whose orbits are contained in the T-orbits; that is,where Z denotes the set of integers. (This is equivalent to saying that there is a measurable mapping L: Ω → Z such that U(ω) = TL(ω)(ω), ω ∈ Ω.) Such pairs (T, U) arise quite naturally in ergodic theory and information theory. (For example, in ergodic theory, one can see such pairs in the study of the full group of a transformation [1]; in information theory, such a pair can be associated with the input and output of a variable-length source encoder [2] [3].) Neveu [4] obtained necessary and sufficient conditions for U to be measure-preserving if T is measure-preserving. However, if U fails to be measure-preserving, U might still possess many of the features of measure-preserving transformations.

scholarly journals Dynamics via measurability

2006 ◽  
Vol 2006 ◽  
pp. 1-15
Author(s):  
Svetlana Butler ◽  
Joseph Rosenblatt
Keyword(s):  
Quantitative Methods ◽  
Ergodic Averages ◽  
Measurable Transformation

Generatorsfforσ-algebras can be used to view the dynamics of an invertible measurable transformationTin terms of the range values off∘T. Such generators are the norm rather than the exception. Related measurable and quantitative methods of estimating a function from the behavior of ergodic averages are also discussed.

Weighted inequalities for Cesàro maximal operators in Orlicz spaces

2005 ◽  
Vol 135 (6) ◽  
pp. 1287-1308 ◽  
Author(s):  
Pedro Ortega Salvador ◽  
Consuelo Ramírez Torreblanca
Keyword(s):  
Orlicz Space ◽  
Integral Inequality ◽  
Orlicz Spaces ◽  
Sufficient Conditions ◽  
Maximal Operators ◽  
Weighted Inequalities ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Integrable Functions ◽  
Image Position

Let 0 < α ≤ 1 and let M+α be the Cesàro maximal operator of order α defined by In this work we characterize the pairs of measurable, positive and locally integrable functions (u, v) for which there exists a constant C > 0 such that the inequality holds for all λ > 0 and every f in the Orlicz space LΦ(v). We also characterize the measurable, positive and locally integrable functions w such that the integral inequality holds for every f ∈ LΦ(w). The discrete versions of this results allow us, by techniques of transference, to prove weighted inequalities for the Cesàro maximal ergodic operator associated with an invertible measurable transformation, T, which preserves the measure.Finally, we give sufficient conditions on w for the convergence of the sequence of Cesàro-α ergodic averages for all functions in the weighted Orlicz space LΦ(w).

The Investigation of a Novel Large-Strain Sensor Based on a Dual Capacitive Structure

2014 ◽  
Vol 530-531 ◽  
pp. 66-70
Author(s):  
Cheng Li Duan ◽  
Ya Dong Jiang ◽  
Hui Ling Tai ◽  
Li Jie Chen ◽  
Qi Dong Li ◽  
...  
Keyword(s):  
Strain Measurement ◽  
Large Strain ◽  
Strain Sensor ◽  
The Other ◽  
Excellent Performance ◽  
Relative Shift ◽  
Measurable Transformation ◽  
Voltage Signal ◽  
Capacitor Structure ◽  
Overlap Area

In this paper, a novel large-strain sensor based on a dual planar capacitive structure has been developed. It has the capacity of large-strain measurement up to 200,000 με (0.2 ε). The change in strain causes a measurable transformation in the capacitance of the sensor by relative shift of the overlap area between two capacitive plates, one fixed (i.e. fixed plate) and the other one movable (i.e. movable plate), and is thus converted into a voltage signal by a read-out circuit module. The dual capacitor structure was designed for increasing the initial capacitance and improving the resolution of sensors compared with a single capacitor structure. The experimental results showed that the sensor had a linearity of 2.29% full scale (FS), a hysteresis error of 1.146%FS, repeatability of 0.226%FS and a resolution of 0.5%FS, suggesting excellent performance of the sensor.

scholarly journals An Approach of Randomness of a Sample Based on Its Weak Ergodic Limit

2017 ◽  
Vol 2017 ◽  
pp. 1-5
Author(s):  
Jaime A. Londoño
Keyword(s):  
Probability Measure ◽  
Equivalence Relation ◽  
Sample Space ◽  
Schnorr Randomness ◽  
Measurable Transformation ◽  
Ergodic Limit ◽  
Probability Spaces ◽  
Sample Points ◽  
Time Invariant ◽  
Uniquely Ergodic

For a Polish Sample Space with a Borel σ-field with a surjective measurable transformation, we define an equivalence relation on sample points according to their ergodic limiting averages. We show that this equivalence relation partitions the subset of sample points on measurable invariant subsets, where each limiting distribution is the unique ergodic probability measure defined on each set. The results obtained suggest some natural objects for the model of a probabilistic time-invariant phenomenon are uniquely ergodic probability spaces. As a consequence of the results gained in this paper, we propose a notion of randomness that is weaker than recent approaches to Schnorr randomness.

scholarly journals Injective composition operators on Lorentz–Bochner spaces

2013 ◽  
Vol 46 (4) ◽  
Author(s):  
Gopal Datt ◽  
S. C. Arora
Keyword(s):  
Composition Operators ◽  
Weighted Composition Operators ◽  
Measurable Transformation ◽  
Weighted Composition ◽  
Vector Valued Functions ◽  
Vector Valued ◽  
Equivalent Conditions

AbstractIn this paper, we extend the notion of essential range to vector-valued functions and present various equivalent conditions for the injectiveness of the composition operators alongwith a characterisation for measurable transformations inducing composition operators between Lorentz–Bochner spaces. Some aspects of the weighted composition operators on Lorentz–Bochner spaces, induced by a measurable transformation and an operator valued map, are also discussed.

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