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It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by $ A_1B_1^{g_1}C_1 + A_2B_2^{g_2}C_2 + \cdots + A_kB_k^{g_k}C_{k} = D $ and $ A_1B_1^{g_1}A_2B_2^{g_2} \cdots A_kB_k^{g_k}A_{k+1} = A $, where $ A_1 $, $ A_2 $, $ \ldots $, $ A_{k+1} $, $ C_1 $, $ C_2 $, $ \ldots $, $ C_{k} $ and $ A $ and $ D $ are given, and $ B_1^{g_1} $, $ B_2^{g_2} $, $ \ldots $, $ B_k^{g_k} $ are generalized inverses of matrices $ B_1 $, $ B_2 $, $ \ldots $, $ B_k $. These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of $ \{1\} $- and $ \{1, 2\} $-generalized inverses of matrices with $ k = 2, 3 $ by using some elementary methods, including a series of explicit rank equalities for block matrices.

It is common knowledge that matrix equalities involving ordinary algebraic operations of inverses or generalized inverses of given matrices can be constructed arbitrarily from theoretical and applied points of view because of the noncommutativity of the matrix algebra and singularity of given matrices. Two of such matrix equality examples are given by A 1 B 1 g 1 C 1 + A 2 B 2 g 2 C 2 + ⋯ + A k B k g k C k = D and A 1 B 1 g 1 A 2 B 2 g 2 ⋯ A k B k g k A k + 1 = A , where A 1 , A 2 , ... , A k + 1 , C 1 , C 2 , ... , C k and A and D are given, and B 1 g 1 , B 2 g 2 , ... , B k g k are generalized inverses of matrices B 1 , B 2 , ... , B k . These two matrix equalities include many concrete cases for different choices of the generalized inverses, and they have been attractive research topics in the area of generalized inverse theory. As an ongoing investigation of this subject, the present author presents in this article several groups of new results and facts on constructing and characterizing the above matrix equalities for the mixed combinations of { 1 } - and { 1 , 2 } -generalized inverses of matrices with k = 2 , 3 by using some elementary methods, including a series of explicit rank equalities for block matrices. [ABSTRACT FROM AUTHOR]

Our aim in this article is to establish several new norm and numerical radius inequalities for products and sums of operators acting on a complex Hilbert space (H , 〈 · , · 〉) . Some of the obtained inequalities improve well known ones. In addition, by using new techniques, we prove certain new inequalities related to ω A (·) and | | · | | A , where ω A (T) and | | T | | A denote the A-numerical radius and the A-operator seminorm of an operator T acting on the semi-Hilbert space (H , 〈 · , · 〉 A) , respectively. Here 〈 x , y 〉 A : = 〈 A x , y 〉 for every x , y ∈ H . [ABSTRACT FROM AUTHOR]

The purpose of this study is to extend the Fermatean fuzzy matrices to the theory of Hamacher operations. In this paper, the concept of Hamacher operations of Fermatean fuzzy matrices are introduced and some desirable properties of these operations, such as commutativity, idempotency, and monotonicity are discussed. Further, we prove DeMorgan’s laws over complement for these operations. Furthermore, the scalar multiplication and exponentiation operations of Fermatean fuzzy matrices are constructed and their algebraic properties are investigated. Finally, some properties of necessity and possibility operators of Fermatean fuzzy matrices are proved. [ABSTRACT FROM AUTHOR]

We investigate weak and strong structures for generalized topological spaces, among others products, sums, subspaces, quotients, and the complete lattice of generalized topologies on a given set. Also we introduce $${T_{3.5}}$$ generalized topological spaces and give a necessary and sufficient condition for a generalized topological space to be a $${T_{3.5}}$$ space: they are exactly the subspaces of powers of a certain natural generalized topology on [0,1]. For spaces with at least two points here we can have even dense subspaces. Also, $${T_{3.5}}$$ generalized topological spaces are exactly the dense subspaces of compact $${T_4}$$ generalized topological spaces. We show that normality is productive for generalized topological spaces. For compact generalized topological spaces we prove the analogue of the Tychonoff product theorem. We prove that also Lindelöfness (and $${\kappa}$$ -compactness) is productive for generalized topological spaces. On any ordered set we introduce a generalized topology and determine the continuous maps between two such generalized topological spaces: for $${|X|, |Y| \geqq 2}$$ they are the monotonous maps continuous between the respective order topologies. We investigate the relation of sums and subspaces of generalized topological spaces to ways of defining generalized topological spaces. [ABSTRACT FROM AUTHOR]

We give expressions for products of sinusoids as sums of sinusoids. [ABSTRACT FROM AUTHOR]

We believe that in any situation where the contribution to the effort is uneven, the higher contribution is given by the ratio of those who participate in the contribution. Most of the times, in most human actions, the contribution of those who bring their contribution to the achievement of a common effort is uneven. Therefore, in an unequal world, over 50% of effort shall be obtained by the intake of less than 50% of those who take part in it, and more than 50% of those who participate in such an effort contributes less than 50% of the mass of that contribution. It argues that managers favor complexity, which is responsible for the increased costs, and indirect principle of Pareto constitutes the fingertips therapy are obvious managerial philosophy of syllogisms. Any sequential analysis of firms with the best economic results may show that only about 20% of the products ensures that 80% of turnover. Considering the dimensions of the current economic activity, we are entitled to believe that such tips were followed scrupulously by policymakers who have taken over the leadership of the Romanian society immediately after the revolutionary year 1989. Since I was taught that businesses are prone to squandering and wastefulness is conditional upon the complexity and each other, it was not hard to be convinced of it decision-makers a simple affair will always be better than one. As such, he became an enthusiastic return to the property system had in the late ' 40s, he actively supported the idea of "resitutio in integrum", major State-owned firms were divided into dozens of smaller firms (see the national power system, unfortunately, Fls, etc.), what is each other, and particularly agriculture cooperative property and the State was divided into millions of private plots. What was the economy became complex in two decades very simple but painfully unsatisfying. We have but one consolation: everything is according to the Pareto principle For those who support the principle of Pareto as the only valid argument is enough to enact: wastefulness is favored by complexity; effectiveness would be different in terms of acceptance of simplicity; most of the activities are already useless, poorly designed, poorly co-ordinated, imperfectly executed and here are irrelevant, since 80% of them are interested in 20% of the products; the smallest part of the activities is appreciated by customers, and the greater part of the activities is uninteresting; the true productive force is already choked by vast productive reality; poor grades, a lot, which already covers 80% of the total, are saved by a small number of excellent results, etc. The wise man's advice: to lay at three-fourths of productive activities in order to focus on that which is now the most advanced. [ABSTRACT FROM AUTHOR]

In this article, I propose a new property (a) of functions f : X → Y , where X and Y are metric spaces. A function f : X → Y has the property (a) if for each real η > 0, the union … contains the graph of a continuous function g : X → Y and K(x, r) denotes the open ball {t ∈ X : ρx(t, x) < r} with center x and radius r > 0. The class of functions with the property (a) contains all functions almost continuous in the sense of Stallings and all functions graph continuous. Moreover, I examine the sums, the products, and the uniform and discrete limits of sequences of functions from this class [ABSTRACT FROM AUTHOR]