scholarly journals On Positive Partial Transpose Matrices

2020 ◽  
Vol 36 (36) ◽  
pp. 256-264
Author(s):  
Mehmet Gumus ◽  
Jianzhen Liu ◽  
Samir Raouafi ◽  
Tin-Yau Tam
Keyword(s):  
Sufficient Conditions ◽  
Norm Inequalities ◽  
Positive Partial Transpose ◽  
Partial Transpose ◽  
The Relationship

This paper mainly focuses on the class of $2 \times 2$ block PPT matrices. The relationship between PPT matrices and the norm inequalities is further explored. Some properties of a non-PPT matrix in terms of its eigenvalues are investigated. Moreover, a number of useful sufficient conditions for a matrix to be PPT are provided.

scholarly journals Pairwise Completely Positive Matrices and Conjugate Local Diagonal Unitary Invariant Quantum States

2019 ◽  
Vol 35 ◽  
pp. 156-180 ◽  
Author(s):  
Nathaniel Johnston ◽  
Olivia MacLean
Keyword(s):  
Sufficient Conditions ◽  
Positive Semidefinite ◽  
Quantum States ◽  
Completely Positive ◽  
Positive Partial Transpose ◽  
Completely Positive Matrices ◽  
Partial Transposition ◽  
Partial Transpose ◽  
Positive Matrices ◽  
Necessary And Sufficient

A generalization of the set of completely positive matrices called pairwise completely positive (PCP) matrices is introduced. These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite while the other one is necessarily entrywise non-negative. Basic properties of these matrix pairs are explored and several testable necessary and sufficient conditions are developed to help determine whether or not a pair is PCP. A connection with quantum entanglement is established by showing that determining whether or not a pair of matrices is pairwise completely positive is equivalent to determining whether or not a certain type of quantum state, called a conjugate local diagonal unitary invariant state, is separable. Many of the most important quantum states in entanglement theory are of this type, including isotropic states, mixed Dicke states (up to partial transposition), and maximally correlated states. As a specific application of these results, a wide family of states that have absolutely positive partial transpose are shown to in fact be separable.

scholarly journals Some Modifications of Pairwise Soft Sets and Some of Their Related Concepts

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1781
Author(s):  
Samer Al Ghour
Keyword(s):  
Topological Space ◽  
Sufficient Conditions ◽  
Soft Sets ◽  
New Concepts ◽  
Bitopological Spaces ◽  
The Relationship ◽  
Soft Topological Space

In this paper, we first define soft u-open sets and soft s-open as two new classes of soft sets on soft bitopological spaces. We show that the class of soft p-open sets lies strictly between these classes, and we give several sufficient conditions for the equivalence between soft p-open sets and each of the soft u-open sets and soft s-open sets, respectively. In addition to these, we introduce the soft u-ω-open, soft p-ω-open, and soft s-ω-open sets as three new classes of soft sets in soft bitopological spaces, which contain soft u-open sets, soft p-open sets, and soft s-open sets, respectively. Via soft u-open sets, we define two notions of Lindelöfeness in SBTSs. We discuss the relationship between these two notions, and we characterize them via other types of soft sets. We define several types of soft local countability in soft bitopological spaces. We discuss relationships between them, and via some of them, we give two results related to the discrete soft topological space. According to our new concepts, the study deals with the correspondence between soft bitopological spaces and their generated bitopological spaces.

The Maxslope Taxometric Procedure: Mathematical Derivation, Parameter Estimation, Consistency Tests

2004 ◽  
Vol 95 (2) ◽  
pp. 517-550 ◽  
Author(s):  
William M. Grove
Keyword(s):  
Parameter Estimation ◽  
Nonlinear Regression ◽  
Classification Accuracy ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Indicator Variable ◽  
Regression Slope ◽  
The Relationship ◽  
Mathematical Derivation ◽  
Made In

This article first explains concepts in taxometrics, including the meaning of “taxon” in relation to taxometric procedures. It then mathematically develops the MAXSLOPE procedure of Grove and Meehl which relies on nonlinear regression of one taxometric indicator variable on another. Sufficient conditions for MAXSLOPE's validity are set forth. The relationship between the point of maximum regression slope (MAXSLOPE point) and the HITMAX cut, i.e., the point on a variable which, if used as a diagnostic cut-off score, yields maximum classification accuracy, is analyzed. A sufficient condition is given for the MAXSLOPE point to equal the HITMAX cut; however, most distributions have different MAXSLOPE and HITMAX points. Equations and an algorithm are spelled out for making a graphical test for the existence of a taxon, estimating taxometric parameters, and conducting consistency tests; the latter serve as stringent checks on the validity of a taxonic conjecture. The plausibility of assumptions made, in deriving MAXSLOPE equations, is discussed, and the qualitative effects of violations of these assumptions are explained.

scholarly journals Property $(w)$ of upper triangular operator Matrices

2020 ◽  
Vol 51 (2) ◽  
pp. 81-99
Author(s):  
Mohammad M.H Rashid
Keyword(s):  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Operator Matrices ◽  
Single Valued Extension Property ◽  
Triangular Operator ◽  
Upper Triangular Operator Matrices ◽  
Necessary And Sufficient ◽  
The Relationship ◽  
Number Of Authors

Let $M_C=\begin{pmatrix} A & C \\ 0 & B \\ \end{pmatrix}\in\LB(\x,\y)$ be be an upper triangulate Banach spaceoperator. The relationship between the spectra of $M_C$ and $M_0,$ and theirvarious distinguished parts, has been studied by a large number of authors inthe recent past. This paper brings forth the important role played by SVEP,the {\it single-valued extension property,} in the study of some of these relations. In this work, we prove necessary and sufficient conditions of implication of the type $M_0$ satisfies property $(w)$ $\Leftrightarrow$ $M_C$ satisfies property $(w)$ to hold. Moreover, we explore certain conditions on $T\in\LB(\hh)$ and $S\in\LB(\K)$ so that the direct sum $T\oplus S$ obeys property $(w)$, where $\hh$ and $\K$ are Hilbert spaces.

Insensitivity and reversed Markov processes

1983 ◽  
Vol 15 (4) ◽  
pp. 752-768 ◽  
Author(s):  
W. Henderson
Keyword(s):  
Markov Processes ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
The Relationship

This paper is concerned with the relationship between insensitivity in a certain class of Markov processes and properties of that process when time is reversed. Necessary and sufficient conditions for insensitivity are established and linked to already proved results. A number of examples of insensitive systems are given.

Semicritical Rings and the Quotient Problem

1984 ◽  
Vol 27 (2) ◽  
pp. 160-170
Author(s):  
Karl A. Kosler
Keyword(s):  
Quotient Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Prime Ideals ◽  
Regular Elements ◽  
The Right ◽  
Minimal Prime ◽  
Necessary And Sufficient ◽  
Quotient Rings ◽  
The Relationship

AbstractThe purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

scholarly journals On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables

2020 ◽  
Vol 24 ◽  
pp. 39-55
Author(s):  
Julyan Arbel ◽  
Olivier Marchal ◽  
Hien D. Nguyen
Keyword(s):  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Random Variables ◽  
The Other ◽  
Bounded Support ◽  
Uniform Distributions ◽  
Standard Variance ◽  
Per Se ◽  
The One ◽  
The Relationship

We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaussianity and symmetry of the distribution. In particular, we demonstrate that symmetry is neither sufficient nor necessary for strict sub-Gaussianity. In contrast, simple necessary conditions on the one hand, and simple sufficient conditions on the other hand, for strict sub-Gaussianity are provided. These results are illustrated via various applications to a number of bounded random variables, including Bernoulli, beta, binomial, Kumaraswamy, triangular, and uniform distributions.

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