scholarly journals Global Bounds for the Generalized Jensen Functional with Applications

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2105
Author(s):  
Slavko Simić ◽  
Bandar Bin-Mohsin
Keyword(s):  
Power Mean ◽  
Jensen Functional ◽  
Exact Bounds ◽  
Harmonic Means ◽  
Stolarsky Means ◽  
Generalized Arithmetic

In this article we give sharp global bounds for the generalized Jensen functional Jn(g,h;p,x). In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means.

scholarly journals Jensen Functional, Quasi-Arithmetic Mean and Sharp Converses of Hölder’s Inequalities

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3104
Author(s):  
Slavko Simić ◽  
Vesna Todorčević
Keyword(s):  
Generating Function ◽  
Hölder’S Inequality ◽  
Arithmetic Mean ◽  
Hölder's Inequality ◽  
Power Means ◽  
Jensen Functional ◽  
Exact Bounds ◽  
Stolarsky Means

In this article, we give sharp two-sided bounds for the generalized Jensen functional Jn(f,g,h;p,x). Assuming convexity/concavity of the generating function h, we give exact bounds for the generalized quasi-arithmetic mean An(h;p,x). In particular, exact bounds are determined for the generalized power means in terms from the class of Stolarsky means. As a consequence, some sharp converses of the famous Hölder’s inequality are obtained.

scholarly journals Sharp Power Mean Bounds for the One-Parameter Harmonic Mean

2015 ◽  
Vol 2015 ◽  
pp. 1-5
Author(s):  
Yu-Ming Chu ◽  
Li-Min Wu ◽  
Ying-Qing Song
Keyword(s):  
Harmonic Mean ◽  
Power Mean ◽  
Double Inequality ◽  
The One ◽  
Harmonic Means

We present the best possible parametersα=α(r)andβ=β(r)such that the double inequalityMα(a,b)<Hr(a,b)<Mβ(a,b)holds for allr∈(0, 1/2)anda, b>0witha≠b, whereMp(a, b)=[(ap+bp)/2]1/p  (p≠0)andM0(a, b)=abandHr(a, b)=2[ra+(1-r)b][rb+(1-r)a]/(a+b)are the power and one-parameter harmonic means ofaandb, respectively.

scholarly journals Schurm-Power Convexity of a Class of Multiplicatively Convex Functions and Applications

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Wen Wang ◽  
Shiguo Yang
Keyword(s):  
Convex Functions ◽  
Symmetric Functions ◽  
Power Mean ◽  
Harmonic Means

We investigate the conditions under which the symmetric functionsFn,k(x,r)=∏1≤i1<i2<⋯<ik≤n ‍f(∑j=1k‍xijr)1/r,  k=1,2,…,n,are Schurm-power convex forx∈R++nandr>0. As a consequence, we prove that these functions are Schur geometrically convex and Schur harmonically convex, which generalizes some known results. By applying the theory of majorization, several inequalities involving thepth power mean and the arithmetic, the geometric, or the harmonic means are presented.

Gaussian ϕ(ρz) curves: Comparison with other models

1990 ◽  
Vol 48 (2) ◽  
pp. 192-193
Author(s):  
Alberto Riveros ◽  
Gustavo Castellano
Keyword(s):  
Good Description ◽  
Bulk Sample ◽  
Incident Electron Energy ◽  
Power Mean ◽  
General Shape ◽  
Ionization Cross ◽  
X Ray ◽  
Mass Absorption ◽  
Absorption Correction ◽  
Image Position

X ray characteristic intensity Ii , emerging from element i in a bulk sample irradiated with an electron beam may be obtained throughwhere the function ϕi(ρz) is the distribution of ionizations for element i with the mass depth ρz, ψ is the take-off angle and μi the mass absorption coefficient to the radiation of element i.A number of models has been proposed for ϕ(ρz), involving several features concerning the interaction of electrons with matter, e.g. ionization cross section, stopping power, mean ionization potential, electron backscattering, mass absorption coefficients (MAC’s). Several expressions have been developed for these parameters, on which the accuracy of the correction procedures depends.A great number of experimental data and Monte Carlo simulations show that the general shape of ϕ(ρz) curves remains substantially the same when changing the incident electron energy or the sample material. These variables appear in the parameters involved in the expressions for ϕ(ρz). A good description of this function will produce an adequate combined atomic number and absorption correction.

scholarly journals A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenpeng Zhang ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Gauss Sums ◽  
Power Mean ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

Transforming Family Resemblance Concepts into Fuzzy Sets

2021 ◽  
pp. 004912412098619
Author(s):  
Francesco Veri
Keyword(s):  
Fuzzy Sets ◽  
Set Theory ◽  
Concept Formation ◽  
Family Resemblance ◽  
Prototype Theory ◽  
Power Mean

This article aims to clarify fundamental aspects of the process of assigning fuzzy scores to conditions based on family resemblance (FR) structures by considering prototype and set theories. Prototype theory and set theory consider FR structures from two different angles. Specifically, set theory links the conceptualization of FR to the idea of sufficient and INUS (Insufficient but Necessary part of a condition, which is itself Unnecessary but Sufficient for the result) sets. In contrast, concept membership in prototype theory is strictly linked to the notion of similarity (or resemblance) in relation to the prototype, which is the anchor of the ideational content of the concept. After an introductive section where I elucidate set-theoretic and prototypical aspects of concept formation, I individuate the axiomatic properties that identify the principles of transforming FR structures into fuzzy sets. Finally, I propose an algorithm based on the power mean that is able to operationalize FR structures considering both set-theoretic and prototype theory perspectives.

A Novel K-Harmonic Means Clustering Based on Multiple Initial Centers

2013 ◽  
Vol 321-324 ◽  
pp. 1947-1950
Author(s):  
Lei Gu ◽  
Xian Ling Lu
Keyword(s):  
New Method ◽  
Final Solution ◽  
Data Set ◽  
Number Of Clusters ◽  
Multiple Groups ◽  
Harmonic Means

In the initialization of the traditional k-harmonic means clustering, the initial centers are generated randomly and its number is equal to the number of clusters. Although the k-harmonic means clustering is insensitive to the initial centers, this initialization method cannot improve clustering performance. In this paper, a novel k-harmonic means clustering based on multiple initial centers is proposed. The number of the initial centers is more than the number of clusters in this new method. The new method with multiple initial centers can divide the whole data set into multiple groups and combine these groups into the final solution. Experiments show that the presented algorithm can increase the better clustering accuracies than the traditional k-means and k-harmonic methods.

Solubility prediction of supercritical carbon dioxide in 10 polymers using radial basis function artificial neural network based on chaotic self-adaptive particle swarm optimization and K-harmonic means

RSC Advances ◽  
2015 ◽  
Vol 5 (56) ◽  
pp. 45520-45527 ◽  
Author(s):  
Mengshan Li ◽  
Xingyuan Huang ◽  
Hesheng Liu ◽  
Bingxiang Liu ◽  
Yan Wu ◽  
...  
Keyword(s):  
Neural Network ◽  
Carbon Dioxide ◽  
Artificial Neural Network ◽  
Particle Swarm Optimization ◽  
Solubility Prediction ◽  
Prediction Modeling ◽  
Swarm Optimization ◽  
Hybrid Computation ◽  
Computation Algorithm ◽  
Harmonic Means

Excellent prediction modeling of CO2 solubility in polymers using hybrid computation algorithm.

scholarly journals Refinements of the Lower Bounds of the Jensen Functional

2011 ◽  
Vol 2011 ◽  
pp. 1-13 ◽  
Author(s):  
Iva Franjić ◽  
Sadia Khalid ◽  
Josip Pečarić
Keyword(s):  
Lower Bounds ◽  
Jensen Inequality ◽  
Left Hand ◽  
Right Hand ◽  
Special Cases ◽  
Jensen Functional ◽  
The Difference ◽  
The Right

The lower bounds of the functional defined as the difference of the right-hand and the left-hand side of the Jensen inequality are studied. Refinements of some previously known results are given by applying results from the theory of majorization. Furthermore, some interesting special cases are considered.

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