scholarly journals A necessary and sufficient condition for the convexity of the one-parameter generalized inverse trigonometric sine function according to power mean

2021 ◽  
pp. 559-573
Author(s):  
Miao-Ying Hong ◽  
Miao-Kun Wang ◽  
Yuming Chu
Keyword(s):  
Generalized Inverse ◽  
Sufficient Condition ◽  
Sine Function ◽  
Necessary And Sufficient Condition ◽  
Power Mean ◽  
The One ◽  
Necessary And Sufficient

On generalized inverses of matrices

1966 ◽  
Vol 62 (4) ◽  
pp. 673-677 ◽  
Author(s):  
M. H. Pearl
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Maximal Rank ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Finite Dimensional ◽  
The Matrix ◽  
Generalized Inverses Of Matrices ◽  
Necessary And Sufficient

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

Russian ‘Liberal’ Opposition: The Divide Be-Tween ‘Radicals’ and ‘Moderates’?

2013 ◽  
Vol 9 (1) ◽  
pp. 63-70
Author(s):  
Anna Taitslin
Keyword(s):  
Rule Of Law ◽  
Private Property ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Social Consensus ◽  
The Rule Of Law ◽  
The One ◽  
The Right ◽  
Necessary And Sufficient ◽  
Party State

The paper reflects on the divide emerged amidst the liberal opposition in Russia between the left liberals and the right liberals. The divide is not just about split-up between the radicals and the moderates. It re-flects the crisis of liberal ideas as formed in the 1990s, when the tran-sition to economy based on private property was seen as necessary and sufficient condition for dismantling the command economy and the one-party state. The ultimate issue at hand is the notion of the rule of law and a possibility of wider social consensus on the minimal rule of law threshold.

On the stability of some solutions of the Stefan problem

1991 ◽  
Vol 2 (1) ◽  
pp. 1-15 ◽  
Author(s):  
Riccardo Ricci ◽  
Xie Weiqing
Keyword(s):  
Stefan Problem ◽  
Travelling Wave ◽  
Sufficient Condition ◽  
Travelling Wave Solutions ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
The Stability ◽  
The One ◽  
Necessary And Sufficient ◽  
Travelling Wave Front

We investigate the stability of travelling wave solutions of the one-dimensional under-cooled Stefan problem. We find a necessary and sufficient condition on the initial datum under which the free boundary is asymptotic to a travelling wave front. The method applies also to other types of solutions.

scholarly journals The Constructive Lift Monad

1995 ◽  
Vol 2 (20) ◽  
Author(s):  
Anders Kock
Keyword(s):  
The Netherlands ◽  
The Other ◽  
Constructive Mathematics ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bottom Element ◽  
The One ◽  
Gt Element ◽  
Necessary And Sufficient ◽  
Connected Spaces

The lift monad is the construction which to a poset freely adjoins a bottom<br />element to it, or equivalently (from the classical viewpoint), the construction which freely adjoins suprema for subsets with at most one element. In constructive mathematics (i.e. inside a topos), these two constructions are no longer equivalent, since the equivalence is based on the boolean reasoning that a set with at most one element either is a singleton {x}, or is empty.<br />Likewise based on boolean reasoning is the proof of two important properties of the lift monad T :<br />1) If a poset C has filtered suprema, then so does TC.<br />2) Every poset with a bottom element ? is "free", i.e. comes about by<br />applying T to some poset (namely the original poset less the bottom).<br />Both these properties fail to hold constructively, if the lift monad is interpreted<br />as "adding a bottom"; see Remark below. If, on the other hand,<br />we interpret the lift monad as the one which freely provides supremum for<br />each subset with at most one element (which is what we shall do), then the first property holds; and we give a necessary and sufficient condition that the second does. Finally, we shall investigate the lift monad in the context of (constructive) locale theory. I would like to thank Bart Jacobs for guiding me to the literature on Z-systems; to Gonzalo Reyes for calling my attention to Barr's work on totally connected spaces; to Steve Vickers for some pertinent correspondence.<br />I would like to thank the Netherlands Science Organization (NWO) for supporting my visit to Utrecht, where a part of the present research was carried out, and for various travel support from BRICS.

scholarly journals When Does a Dual Matrix Have a Dual Generalized Inverse?

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1386
Author(s):  
Firdaus E. Udwadia
Keyword(s):  
Explicit Expression ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Generalized Inverses ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Real Matrices ◽  
Explicit Expressions

This paper deals with the existence of various types of dual generalized inverses of dual matrices. New and foundational results on the necessary and sufficient conditions for various types of dual generalized inverses to exist are obtained. It is shown that unlike real matrices, dual matrices may not have {1}-dual generalized inverses. A necessary and sufficient condition for a dual matrix to have a {1}-dual generalized inverse is obtained. It is shown that a dual matrix always has a {1}-, {1,3}-, {1,4}-, {1,2,3}-, {1,2,4}-dual generalized inverse if and only if it has a {1}-dual generalized inverse and that every dual matrix has a {2}- and a {2,4}-dual generalized inverse. Explicit expressions, which have not been reported to date in the literature, for all these dual inverses are provided. It is shown that the Moore–Penrose dual generalized inverse of a dual matrix exists if and only if the dual matrix has a {1}-dual generalized inverse; an explicit expression for this dual inverse, when it exists, is obtained irrespective of the rank of its real part. Explicit expressions for the Moore–Penrose dual inverse of a dual matrix, in terms of {1}-dual generalized inverses of products, are also obtained. Several new results related to the determination of dual Moore-Penrose inverses using less restrictive dual inverses are also provided.

scholarly journals Q-Extension of Starlike Functions Subordinated with a Trigonometric Sine Function

Mathematics ◽  
2020 ◽  
Vol 8 (10) ◽  
pp. 1676
Author(s):  
Saeed Islam ◽  
Muhammad Ghaffar Khan ◽  
Bakhtiar Ahmad ◽  
Muhammad Arif ◽  
Ronnason Chinram
Keyword(s):  
Extreme Point ◽  
Point Theorem ◽  
Starlike Functions ◽  
Partial Sums ◽  
Sufficient Condition ◽  
Sine Function ◽  
Necessary And Sufficient Condition ◽  
Closure Theorem ◽  
Necessary And Sufficient ◽  
Distortion Bound

The main purpose of this article is to examine the q-analog of starlike functions connected with a trigonometric sine function. Further, we discuss some interesting geometric properties, such as the well-known problems of Fekete-Szegö, the necessary and sufficient condition, the growth and distortion bound, closure theorem, convolution results, radii of starlikeness, extreme point theorem and the problem with partial sums for this class.

Animalhood, interests, and rights*

2020 ◽  
Vol 11 (2) ◽  
pp. 156-172
Author(s):  
Juan Pablo Mañalich R.
Keyword(s):  
Legal Rights ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Moral Requirement ◽  
Intentional States ◽  
Nonhuman Animals ◽  
The Subject ◽  
The One ◽  
Necessary And Sufficient ◽  
Normative Status

A being to which intentional states – such as desires or preferences – may be ascribed is a being capable of having (actual) interests, whereas to be the subject of interests of some kind is both a necessary and sufficient condition to be the holder of individual rights. After clarifying the sense in which, according to the ‘interest-theory’, the notion of a rights-subject specifies a distinctive normative status, this article will highlight the importance of distinguishing between subjectivity-dependent interests capable of being attributed to conscious beings, on the one hand, and biologically structured needs of conscious and nonconscious living beings, on the other. This distinction allows one to see that the moral requirement of recognizing legal rights for (individual) animals ought not to be conflated with biocentric demands of ecological justice. However, the argument thus delineated will not, without more, answer the crucial question of which specific legal rights ought to be ascribed to nonhuman animals. The article closes with an exploration of the need for holding onto the distinction between rights-subjecthood and personhood by analyzing some implications of Tooley's ‘particular-interest principle’.

scholarly journals Some Condition for Scalar and Vector Measure Games to Be Lipschitz

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
F. Centrone ◽  
A. Martellotti
Keyword(s):  
Vector Measure ◽  
Probability Measures ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The One ◽  
Necessary And Sufficient

We provide a characterization for vector measure gamesν=f∘μinpNA∞, withμvector of nonatomic probability measures, analogous to the one of Tauman for games inpNA, and also provide a necessary and sufficient condition for a particular class of vector measure games to belong toAC∞.

Characterization of ℓ1 minimizer in one-bit compressed sensing

2019 ◽  
Vol 17 (06) ◽  
pp. 1005-1021 ◽  
Author(s):  
Ning Bi ◽  
Jun Tan
Keyword(s):  
Compressive Sensing ◽  
Information Science ◽  
Convex Sets ◽  
Sufficient Condition ◽  
Relaxation Model ◽  
Necessary And Sufficient Condition ◽  
The One ◽  
Necessary And Sufficient ◽  
Theoretical Results

The conception of one-bit compressive sensing (one-bit CS) was first introduced in 2008 by Boufounos and Baraniuk [1-Bit compressive sensing, in Proc. Conf. on Information Science and Systems (CISS, Princeton, NJ, 2008)]. Since then, many efficient algorithms have been developed for dealing with the one-bit CS problem. However, few theoretical results are available on one-bit CS. In this paper, we focus on one-bit CS theory with its relaxation model [Formula: see text] and present a necessary and sufficient condition such that the signal [Formula: see text] is the unique [Formula: see text] minimizer in noiseless one-bit CS (Theorem 3). Moreover, by using the improved separation theorem of convex sets (Theorem 6), we completely characterize the [Formula: see text] minimizer in one-bit CS (Theorem 2). Finally, as an application of Theorem 2, the [Formula: see text] minimizer for the considered model can, in general, be a non-sparse vector.

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