scholarly journals A Geometric Mean of Parameterized Arithmetic and Harmonic Means of Convex Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Sangho Kum ◽  
Yongdo Lim
Keyword(s):  
Convex Functions ◽  
Convex Analysis ◽  
Geometric Mean ◽  
Positive Semidefinite ◽  
Dual Operator ◽  
Positive Semidefinite Matrices ◽  
Essential Properties ◽  
Harmonic Means ◽  
Further Development ◽  
Interesting Generalization

The notion of the geometric mean of two positive reals is extended by Ando (1978) to the case of positive semidefinite matricesAandB. Moreover, an interesting generalization of the geometric meanA # BofAandBto convex functions was introduced by Atteia and Raïssouli (2001) with a different viewpoint of convex analysis. The present work aims at providing a further development of the geometric mean of convex functions due to Atteia and Raïssouli (2001). A new algorithmic self-dual operator for convex functions named “the geometric mean of parameterized arithmetic and harmonic means of convex functions” is proposed, and its essential properties are investigated.

scholarly journals A trace bound for integer-diagonal positive semidefinite matrices

2020 ◽  
Vol 8 (1) ◽  
pp. 14-16
Author(s):  
Lon Mitchell
Keyword(s):  
Positive Semidefinite ◽  
Positive Semidefinite Matrix ◽  
Positive Semidefinite Matrices ◽  
Semidefinite Matrix

AbstractWe prove that an n-by-n complex positive semidefinite matrix of rank r whose graph is connected, whose diagonal entries are integers, and whose non-zero off-diagonal entries have modulus at least one, has trace at least n + r − 1.

scholarly journals Examination of Future Psychologists Empathy Level

2019 ◽  
Vol 39 (1) ◽  
pp. 13-17
Author(s):  
S. Dmitrieva ◽  
O. Machushnyk
Keyword(s):  
Moral Development ◽  
Interpersonal Relations ◽  
Value Orientations ◽  
Successful Implementation ◽  
Prosocial Behaviour ◽  
Essential Properties ◽  
The Individual ◽  
Further Development ◽  
Different Levels

In current conditions, the priority of young people's preparation for life and work is especially important. Consequently, the requirements for the training of a future psychologist are also changing. One of the essential properties of a psychologist, necessary for the successful implementation of their activities, is empathy. The problem of empathy is one of the most difficult psychological sciences. The implacability of this phenomenon for researchers confirms the diversity in the definitions of its essence, mechanisms, functions, the role of empathy in the personality moral development, prosocial behaviour, altruism, and others like that. The presence of the appropriate level of empathic properties of students-psychologists acts as a condition for the formation of their professional compliance. Subjective factors of empathy formation: value sphere of personality, type of interpersonal relations, level of self-centeredness, type of accentuation of character, types of attitudes to different spheres of life, level of subjective control. Therefore, in the article, empathy development is studied in students who get a psychologist's degree. It is determined that in general subjects have average empathy level. By dividing students into groups, according to their level of empathy, it has been established that different value orientations characterize boys with different levels of empathy. It is determined that the overwhelming majority of respondents have a mean self-centeredness level. It was found that the obtained data provide an opportunity for further development of empathy among students. As a result of our research, we are convinced that the objective factors for the formation of empathy are: the perception of other people, the maturity of the individual. Our research is not exhaustive; our further development will concern the deepening of the ideas about the empathy component of the personality of the future psychologist and the methods of its development.  

scholarly journals Fischer Type Log-Majorization of Singular Values on Partitioned Positive Semidefinite Matrices

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Benju Wang ◽  
Yun Zhang
Keyword(s):  
Singular Values ◽  
Positive Semidefinite ◽  
Positive Semidefinite Matrices ◽  
Fischer’S Inequality ◽  
New Inequalities

In this paper, we establish a Fischer type log-majorization of singular values on partitioned positive semidefinite matrices, which generalizes the classical Fischer's inequality. Meanwhile, some related and new inequalities are also obtained.

scholarly journals A Characterization of Polynomial Density on Curves via Matrix Algebra

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1231
Author(s):  
Carmen Escribano ◽  
Raquel Gonzalo ◽  
Emilio Torrano
Keyword(s):  
Matrix Algebra ◽  
Optimization Problems ◽  
Positive Semidefinite ◽  
Point Of View ◽  
Alternative Proof ◽  
General Context ◽  
Positive Semidefinite Matrices ◽  
Jordan Curves ◽  
Point Evaluations

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

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