scholarly journals Relative Entropy in Determining Degressive Proportional Allocations

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 903
Author(s):  
Katarzyna Cegiełka ◽  
Piotr Dniestrzański ◽  
Janusz Łyko ◽  
Arkadiusz Maciuk ◽  
Maciej Szczeciński
Keyword(s):  
Power Function ◽  
Relative Entropy ◽  
Significant Relationship ◽  
Power Functions ◽  
Allocation Rules

The principle of degressively proportional apportionment of goods, being a compromise between equality and proportionality, facilitates the application of many different allocation rules. Agents with smaller entitlements are more interested in an allocation that is as close to equality as possible, while those with greater entitlements prefer an allocation as close to proportionality as possible. Using relative entropy to quantify the inequity of allocation, this paper indicates an allocation that neutralizes these two contradictory approaches by symmetrizing the inequities perceived by the smallest and largest agents participating in the apportionment. First, based on some selected properties, the set of potential allocation rules was reduced to those generated by power functions. Then, the existence of the power function whose exponent is determined so as to generate the allocation that symmetrizes the relative entropy with respect to equal and proportional allocations was shown. As a result, all agents of the apportionment are more inclined to accept the proposed allocation regardless of the size of their entitlements. The exponent found in this way shows the significant relationship between the problem under study and the well-known Theil indices of inequality. The problem may also be seen from this viewpoint.

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

The Wind-Sand Flux Structure of Grassland on the Northern Foot of Yinshan Mountain

2014 ◽  
Vol 668-669 ◽  
pp. 1530-1537
Author(s):  
Hong Tao Jiang ◽  
Chun Rong Guo ◽  
Chun Xing Hai ◽  
Shan Shan Sun ◽  
Yun Hu Xie ◽  
...  
Keyword(s):  
Vertical Distribution ◽  
Power Function ◽  
Exponential Function ◽  
Vertical Direction ◽  
Power Functions ◽  
Height Range ◽  
Sand Flux ◽  
Soil Flux ◽  
The Vertical Distribution ◽  
Better Than

Sand samplers were layed out in the grassland located in the northern foot of Yinshan Mountain for collecting soil flux samples from 0 to 1.5m height above the surface from Mar., 1, 2008 to Feb., 29, 2009.Exponential and Power functions were both used for describing vertical distribution of sand flux in the grassland, the results indicated that determination coefficient of Power function varied from 0.898 to 0.992 while 0.432 to 0.661 for exponential function. Power function is better than exponential function in describing the vertical distribution of both annual and seasonal soil flux, summer excluded. Annual cumulative percentage of each height was determined indirectly according to the power function mentioned above, the result indicated that up to 2m height,15-25% of soil flux concentrated with in 10cm above the surface,25-35% of soil flux concentrated within 20cm above the surface,30-40% of soil flux concentrated within 30 cm above the surface, 43-54% of soil flux concentrated within 50 cm above the surface,85-90% of soil flux concentrated within 150 cm above the surface, respectively. No significant differences of soil flux structures in spring, autumn, winter and in the whole year were found. The research on wind erosion of grassland in the vertical direction more dispersed, in the height range of sediment accumulated percentage was lower than that of the previous research.

Three Models of Temporal Summation Evaluated Using Normal-Hearing and Hearing-Impaired Subjects

1983 ◽  
Vol 26 (2) ◽  
pp. 256-262 ◽  
Author(s):  
George M. Gerken ◽  
Adele D. Gunnarson ◽  
Craig M. Allen
Keyword(s):  
Power Function ◽  
Time Constant ◽  
Temporal Summation ◽  
Exponential Model ◽  
Normal Hearing ◽  
Hearing Impaired ◽  
Power Functions ◽  
Critical Test ◽  
Combined Model ◽  
Level Of Significance

Temporal summation effects were measured in normal-hearing and hearing impaired subjects using stimuli of different durations and temporal patterns. Threshold decreased with increasing stimulus duration for either single- or multiple-burst stimuli, but the hearing-impaired group showed smaller threshold shifts, which differed from those obtained with the normal-hearing group at the .0001 level of significance. Three models of temporal summation were evaluated: One model employed a time constant in an exponential function, one used a power function characterized by an exponent, and the last combined the properties of the exponential and power functions and was also characterized by an exponent. Estimates of the parameters that best described the data were obtained for each model. Data from the hearing-impaired subjects provided the most critical test of the models. The power function model and the combined model were both satisfactory with the range of stimulus durations used. but the exponential model failed to describe the data from the hearing-impaired subjects. It is suggested that there may not be a decrease in the time constant for temporal summation for subjects with sensorineural hearing-loss but that a factor related to the utilization of sensory input is altered.

Approximation of the Temperature and Time Dependence of the Stress versus Rupture Life Curves by Power Functions

2010 ◽  
Vol 659 ◽  
pp. 385-391
Author(s):  
Zoltán Dudás
Keyword(s):  
Time Dependence ◽  
Power Function ◽  
Approximation Method ◽  
Function Approximation ◽  
Rupture Life ◽  
Measured Data ◽  
Time Dependent ◽  
Power Functions ◽  
Temperature Dependent

This paper explains and demonstrates an approximation method for the calculation of the stress values depending on the temperature and time, or time values depending on the stress and temperature using time dependent and temperature dependent power functions. The approximation is based on measured data and the paper shows the verification of the power function approximation.

Scaling Apparent Distance in a Large Open Field: Some New Data

1983 ◽  
Vol 56 (1) ◽  
pp. 135-138 ◽  
Author(s):  
José Aparecido Da Silva ◽  
Cleuza Beatriz Da Silva
Keyword(s):  
Standard Deviation ◽  
Open Field ◽  
Power Function ◽  
Magnitude Estimation ◽  
Apparent Distance ◽  
Physical Distance ◽  
Power Functions ◽  
The Mean

Judged distance in a large open field, scaled by the method of magnitude estimation, is related to physical distance by a power function with an exponent smaller than unity. The exponents obtained with two ranges of distance were not affected by the availability of a standard. The mean exponent for all 80 individual power functions was 0.86, with a standard deviation of 0.11.

Delving Deeper: Derivatives of Generalized Power Functions

2009 ◽  
Vol 102 (7) ◽  
pp. 554-557
Author(s):  
John M. Johnson
Keyword(s):  
Power Function ◽  
Exponential Function ◽  
Power Functions ◽  
Exponential Functions ◽  
First Semester ◽  
Derivatives Of

After several years of teaching multiple sections of first-semester calculus, it was easy for me to think that I had nothing new to learn. But every year and every class bring a new group of students with their unique gifts and insights. In a recent class, after covering the derivative rules for power functions and exponential functions, I asked the class about the derivative of a function like y = xsinx, which is neither a power function (the power is not constant) nor an exponential function (the base is not constant).

scholarly journals On an integral transform

1988 ◽  
Vol 11 (4) ◽  
pp. 635-642
Author(s):  
D. Naylor
Keyword(s):  
Bessel Function ◽  
Power Function ◽  
Inversion Formula ◽  
Bessel Functions ◽  
Integral Transform ◽  
Finite Interval ◽  
Power Functions ◽  
Inversion Theorem ◽  
Eigenvalue Parameter

A formula of inversion is established for an integral transform whose kernel is the Bessel functionJu(kr)wherervaries over the finite interval(0,a)and the orderuis taken to be the eigenvalue parameter. When this parameter is large the Bessel function behaves for varyingrlike the power functionruand by relating the Bessel functions to their corresponding power functions the proof of the inversion formula can be reduced to one depending on the Mellin inversion theorem.

scholarly journals Sample size determinations for stepped-wedge clinical trials from a three-level data hierarchy perspective

2016 ◽  
Vol 27 (2) ◽  
pp. 480-489 ◽  
Author(s):  
Moonseong Heo ◽  
Namhee Kim ◽  
Michael L Rinke ◽  
Judith Wylie-Rosett
Keyword(s):  
Statistical Model ◽  
Sample Size ◽  
Power Function ◽  
Data Structures ◽  
Statistical Models ◽  
Simulation Studies ◽  
Stepped Wedge ◽  
Power Functions ◽  
Intervention Effect ◽  
Level Data

Stepped-wedge (SW) designs have been steadily implemented in a variety of trials. A SW design typically assumes a three-level hierarchical data structure where participants are nested within times or periods which are in turn nested within clusters. Therefore, statistical models for analysis of SW trial data need to consider two correlations, the first and second level correlations. Existing power functions and sample size determination formulas had been derived based on statistical models for two-level data structures. Consequently, the second-level correlation has not been incorporated in conventional power analyses. In this paper, we derived a closed-form explicit power function based on a statistical model for three-level continuous outcome data. The power function is based on a pooled overall estimate of stratified cluster-specific estimates of an intervention effect. The sampling distribution of the pooled estimate is derived by applying a fixed-effect meta-analytic approach. Simulation studies verified that the derived power function is unbiased and can be applicable to varying number of participants per period per cluster. In addition, when data structures are assumed to have two levels, we compare three types of power functions by conducting additional simulation studies under a two-level statistical model. In this case, the power function based on a sampling distribution of a marginal, as opposed to pooled, estimate of the intervention effect performed the best. Extensions of power functions to binary outcomes are also suggested.

The Construction and Prediction of Psychophysical Power Functions for the Sweetness of Equiratio Sugar Mixtures

Perception ◽  
1983 ◽  
Vol 12 (6) ◽  
pp. 753-767 ◽  
Author(s):  
Jan E R Frijters ◽  
Peter A M Oude Ophuis
Keyword(s):  
Mixture Models ◽  
Power Function ◽  
Alternative Model ◽  
Power Functions ◽  
Great Precision ◽  
Sugar Mixtures ◽  
Simple Alternative ◽  
Single Compound ◽  
The Relationship ◽  
Interaction Phenomena

Psychophysical taste mixture models describe the relationship between the perceived intensities of the unmixed components and the intensity of the mixture. Three of these models are discussed. As all of these appear either to be internally inconsistent or lack sufficient generality, a simple alternative model has been developed especially for the prediction of the intensity of equiratio mixtures. This model was experimentally tested with glucose—fructose mixtures. On the basis of the data obtained it is shown that a psychophysical equiratio mixture function can be constructed in the same way as a power function for a single compound. The results show that the new mixture model can predict the functions for equiratio mixtures with great precision. Implications for mixture interaction phenomena are discussed.

scholarly journals The Confidence Interval of the Estimator of the Periodic Intensity Function in the Presence of Power Function Trend on the Nonhomogeneous Poisson Process

CAUCHY ◽  
2021 ◽  
Vol 7 (1) ◽  
pp. 73-83
Author(s):  
Ikhsan Maulidi ◽  
Bonno Andri Wibowo ◽  
Nina Valentika ◽  
Muhammad Syazali ◽  
Vina Apriliani
Keyword(s):  
Confidence Interval ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Power Function ◽  
Intensity Function ◽  
Nonhomogeneous Poisson Process ◽  
Constructive Proof ◽  
Power Functions ◽  
Periodic Intensity ◽  
Intensity Estimator

The nonhomogeneous Poisson process is one of the most widely applied stochastic processes. In this article, we provide a confidence interval of the intensity estimator in the presence of a periodic multiplied by trend power function. This estimator's confidence interval is an application of the formulation of the estimator asymptotic distribution that has been given in previous studies. In addition, constructive proof of the convergent in probability has been provided for all power functions.

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