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Matrices induced by arithmetic functions acting on certain Krein spaces

100%
Cho I.
Special Matrices
|
2017
|
tom 5
|
nr 1
258-289
EN
In this paper, we study matrices induced by arithmetic functions under certain Krein-space representations induced by (multi-)primes less than or equal to fixed positive real numbers.
2
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Composition of Arithmetical functions with generalization of perfect and related numbers

100%
Shukla D. P., Yadav S.
Commentationes Mathematicae
|
2012
|
tom 52
|
nr 2
EN
In this paper we have studied the deficient and abundent numbers connected with the composition of \(\varphi\), \(\varphi^*\), \(\sigma\), \(\sigma^*\) and \(\psi\) arithmetical functions, where \(\varphi\) is Euler totient, \(\varphi^*\) is unitary totient, \(\sigma\) is sum of divisor, \(\sigma^*\) is unitary sum of divisor and \(\psi\) is Dedekind's function. In 1988, J. Sandor conjectured that \(\psi(\varphi(m)) \geq m\), for all \(m\), all odd \(m\) and proved that this conjecture is equivalent to \(\psi(\varphi(m)) \geq \frac{m}{2}\), we have studied this equivalent conjecture. Further, a necessary and sufficient conditions of primitivity for unitary r-deficient numbers and unitary totient r-deficient numbers have been obtained. We have discussed the generalization of perfect numbers for an arithmetical function \(E_\alpha\).
3
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Krein-space operators determined by free product algebras induced by primes and graphs

84%
Cho I., Jorgensen P. E. T.
Special Matrices
|
2017
|
tom 5
|
nr 1
1-35
EN
In this paper, we introduce certain Krein-space operators induced by free product algebras induced by both primes and directed graphs. We study operator-theoretic properties of such operators by computing free-probabilistic data containing number-theoretic data.
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