Sharpening of Cauchy Inequality

1998 ◽  
pp. 485-487
Author(s):  
Živojin Mijalković ◽  
Milan Mijalković
Keyword(s):  
Cauchy Inequality

scholarly journals Further Promotion of Cauchy Inequality

2009 ◽  
Vol 1 (2) ◽  
Author(s):  
Fengqing Sun
Keyword(s):  
Cauchy Inequality

Mesh Exponential Stability of Look-Ahead Interconnected System

2010 ◽  
Vol 29-32 ◽  
pp. 847-850 ◽  
Author(s):  
Zheng Li Cao ◽  
Tian Li ◽  
Ji Ye Zhang
Keyword(s):  
Exponential Stability ◽  
Function Theory ◽  
Cauchy Inequality ◽  
Sufficient Condition ◽  
Interconnected System ◽  
Vector Lyapunov Function ◽  
Look Ahead ◽  
Exponentially Stable ◽  
Inequality Technique ◽  
The Relationship

Mesh stability is the property of damping or eliminating disturbances as they travel in an interconnected system. In this paper, the mesh stability of a class of look-ahead vehicle following system is studied under the assumption that the all isolated subsystems are exponentially stable. Via idea of vector Lyapunov function theory and by using of Cauchy inequality technique, the sufficient condition to ensure the global mesh exponential stability is obtained. The obtained condition which shows the relationship and domain of the parameters is convenient to apply to design controller in practice.

scholarly journals On Hilbert's Inequality for Double Series and Its Applications

2008 ◽  
Vol 2008 ◽  
pp. 1-12
Author(s):  
Zhou Yu ◽  
Gao Mingzhe
Keyword(s):  
Real Function ◽  
Hardy Inequality ◽  
Double Series ◽  
Cauchy Inequality ◽  
Hilbert Inequality ◽  
Hilbert’S Inequality ◽  
Riesz Inequality

This study shows that a refinement of the Hilbert inequality for double series can be established by introducing a real functionu(x)and a parameterλ. In particular, some sharp results of the classical Hilbert inequality are obtained by means of a sharpening of the Cauchy inequality. As applications, some refinements of both the Fejer-Riesz inequality and Hardy inequality inHpfunction are given.

On a converse Cauchy inequality of D. Zagier

1992 ◽  
Vol 58 (2) ◽  
pp. 157-159 ◽  
Author(s):  
Horst Alzer
Keyword(s):  
Cauchy Inequality

scholarly journals Generalized reverse Cauchy inequality and applications to operator means

2018 ◽  
pp. 1029-1039 ◽  
Author(s):  
Hiroyuki Osaka ◽  
Yukihiro Tsurumi ◽  
Shuhei Wada
Keyword(s):  
Cauchy Inequality ◽  
Operator Means

scholarly journals The probability of the correct majority made decision

2019 ◽  
Vol 5 (1) ◽  
pp. 42-53
Author(s):  
Vladimir M. Strepetov ◽  
Konstantin E. Voevodskii
Keyword(s):  
Asymptotic Representation ◽  
Power Series Expansion ◽  
Magnetic Suspension ◽  
Transport Systems ◽  
Cauchy Inequality ◽  
Individual Decision ◽  
Geometric Average ◽  
Individual Probability ◽  
The Individual ◽  
Bilateral Estimates

Aim: the probability of correctness of the collegial decision, which is made by a majority vote of some collective (board), consisting of an odd number of members is investigated, if the probability of correctness of the individual decision of each member of Board is known. Мaterials and methods: Bernoulli scheme, asymptotic representation, estimation via geometric series, power series expansion, the formula of Wallis, a power scale of averages, average of Kolmogorov. Result: it is established, that if for each member of the board the probability of correctness of the individual decision is more than ½, then with an unlimited increase in the number of members of the Board the probability of correctness of the collegial decision tends to 1. The asymptotic representation and a number of bilateral estimates characterizing the speed of this aspiration are obtained. For heterogeneous Board (that is a Board, whose members make the right individual decision with different probability) introduced the concept of collegial average as an average characteristics, which can replaced the individual probability of each member of the board with the preservation of the probability of a collegial decision. The existence and uniqueness of the collegial average are proved. We derive a collegial inequality showing that the collegial average of some a set of numbers is not less than the geometric average of the same numbers with the equality takes place in the case and only if all the numbers are equal to each other. The collegial inequality serves as an analogue and complement to known set of inequalities establishing a connection between different averages (for example, Cauchy inequality for arithmetic average and geometric average). Conclusion: thus, the results of the study fully meet the aim of determining the probability of correctness of collegial decision taken by a majority of votes under the assumptions. As a result we obtain an asymptotic representation and bilateral estimates characterizing the rate of striving for the correct solution. For a heterogeneous board, the existence uniqueness of the concept of collegial average as an average characteristics is introduced and strictly proved, which can be replaced by an individual probability of each with preserving the probability of correctness of the collegial decision. It is established that the collegial average is not less than the geometric average. Possible applications of the results obtained can be the quantitative evaluation of election procedures and the solution of problems associated with improving the reliability of recognition of weak signals of control sensors of various transport systems, including high-speed transport systems on magnetic suspension.

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