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In this paper, we introduce some explicit and recursive formulas for the n$$ n $$th q$$ q $$‐derivative of a quotient of two functions. We establish the connections between linear algebra, the homomorphisms between commutative algebras and the properties of q$$ q $$‐differential operator. One of the formulas involves the determinant of the functions and their q$$ q $$‐derivatives. Then, we show that the formulas in the classical mathematical analysis are their special cases. Finally, we used formulas to obtain some interesting q$$ q $$‐identities. [ABSTRACT FROM AUTHOR]

We study the star and minus partial orders on the set $ {\mathcal {B}}(H,K) $ B (H , K) of all bounded operators acting from a Hilbert space H to a Hilbert space K. We extend and strengthen some results from matrix case to the case of general operators, which may not possess generalized inverses. By means of some norm inequalities, we give necessary and sufficient condition under which two operators have orthogonal ranges, and thus, we give a characterization of the star partial order. When A = PB for some projection P, we prove the equivalence of $ A \lt ^*B $ A < ∗ B with $ f(AA^*)A \lt ^*f(BB^*)B $ f (A A ∗) A < ∗ f (B B ∗) B for a wide class of continuous functions f. Also, we prove that $ A \lt ^*B $ A < ∗ B if and only if $ A \lt ^-B $ A < − B and $ A^2 \lt ^{-}B^{2} $ A 2 < − B 2 when A is a weak EP operator and B is a self-adjoint operator. Finally, we consider the Moore–Penrose invertibility and the ordinary invertibility of a linear combination of operators when they are related with one of these two orders. [ABSTRACT FROM AUTHOR]

The paper presents the procedure for determining the factor of safety (FoS) using the strength reduction method (SRM) for the case of a concrete damage plasticity constitutive model. The SRM was originally used in a slope stability analysis and in its original form, this method was applied by reducing the shear strength of the material. Since damage in concrete occurs due to exceeding the normal stresses in the principal directions, and not due to exceeding the shear strength, this method was modified and adapted to the concrete damage plasticity constitutive model. Instead of reducing the failure surface, the parameters which describe the mechanical behavior in the case of uniaxial compression and uniaxial tension were reduced. In this way, the reduction of stress and the corresponding strain was carried out in the entire range of total strain, without changing the shape of the failure surface in the deviator plane. For the proposed methodology, a numerical algorithm was developed and implemented into the software PAK. The algorithm was verified through test examples and the obtained results were compared with analytically calculated FoS. The excellent agreement is observed between the FoS obtained by applying the proposed algorithm and the analytically calculated FoS. [ABSTRACT FROM AUTHOR]

A positive answer on a question posed in Eshkaftaki et al. [DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] is provided. Precisely, the concrete form of bounded linear operators which preserve DSS-weak majorization on discrete Lebesgue space ℓ 1 (I) , when I is an infinite set, is presented. Interestingly, it is shown that the class of DSS-weak majorization preservers is a proper subset of the class of majorization preservers on ℓ 1 (I) , despite the fact that these two classes of preservers coincide [Eshkaftaki et al. DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] when above relations are defined on ℓ p (I) when p ∈ (1 , ∞). In this way, the conjecture given at the end of the paper [Eshkaftaki et al. DSS-weak majorization and its linear preservers on ℓ p spaces. Linear Multilinear Algebra. 2018;66(10):2076–2088] is confirmed. [ABSTRACT FROM AUTHOR]

The aim of this short note is to indicate that Rao and Mitra's constrained matrix inverse A c r C R (introduced in 1972) is a direct precursor of recently introduced Drazin's ( b , c ) outer generalized inverse. Precisely, if one rewrites definition of A c r C R in purely multiplicative language, then it turns out that two inverses coincide. Recently introduced Mary's inverse along an element and Manjunatha Prasad and Mohana's core-EP inverse are the special cases of Rao and Mitra's constrained inverse. [ABSTRACT FROM AUTHOR]

The star, sharp, core and dual core partial orders are already known and investigated in the set of complex matrices. In this paper these orders are extensively studied in the context of arbitrary ring with involution. It is shown that the condition a < b , where < is one of the considered partial orders, defines appropriate idempotents and thus the suitable matrix representations of a and b are obtained. Using these representations we proved that a < b if and only if G ( b ) ⊆ G ( a ) , where G ( a ) is appropriate subset of the set of all generalized inverses of a . If < - is the minus partial order, the conditions under which a < - b implies a < b are determined. Also, several well-known results are generalized and some new results are proved. [ABSTRACT FROM AUTHOR]

The unified theory for matrix partial orders based on generalized inverses has already been done by Mitra. We consider a special kind of ring R , which generalizes the ring of linear operators on finite dimensional vector space, and extend Mitra's approach to it. Thus, we find necessary and sufficient condition for a G -based relation to be a partial order on R . Also, some known results are generalized and some new results are proved. [ABSTRACT FROM AUTHOR]

Letbe a unital ring admitting involution. We introduce an order onand show that in the case whenis a Rickart-ring, this order is equivalent to the well-known star partial order. The notion of the left-star and the right-star partial orders is extended to Rickart-rings. Properties of the star, the left-star and the right-star partial orders are studied in Rickart-rings and some known results are generalized. We found matrix forms of elementsandwhen, whereis one of these orders. Conditions under which these orders are equivalent to the minus partial order are obtained. The diamond partial order is also investigated. [ABSTRACT FROM AUTHOR]

Let R be a ring with involution. The recently introduced notions of the core and dual core inverse are extended from matrix to an arbitrary ⁎-ring case. It is shown that the group, Moore–Penrose, core and dual core inverse are closely related and they can be treated in the same manner using appropriate idempotents. The several characterizations of these inverses are given. Some new properties are obtained and some known results are generalized. A number of characterizations of EP elements in R are obtained. It is shown that core and dual core inverse belong to the class of inverses along an element and to the class of ( b , c ) -inverses. [ABSTRACT FROM AUTHOR]

The core inverse of matrix is generalized inverse which is in some sense in-between the group and Moore–Penrose inverse. In this paper a generalization of core inverse and core partial order to Hilbert space operator case is presented. Some properties are generalized and some new ones are proved. Connections with other generalized inverses are obtained. The useful matrix representations of operator and its core inverse are given. It is shown that A is less than B under the core partial order if and only if they have specific kind of simultaneous diagonalization induced by appropriate decompositions of Hilbert space. The relation is also characterized by the inclusion of appropriate sets of generalized inverses. The spectral properties of core inverse are also obtained. [ABSTRACT FROM AUTHOR]

The paper represents a solution example of a specific engineering problem using thermo-mechanical analysis. The paper provides theoretical basis of numerical solving the problem of heat conducting through continuum using the finite element method. Calculation of heat conducting using the finite element method determines the temperature field used as an input for thermo-mechanical calculation. The basic task of thermo-mechanical calculation was the identification of the cause of cracks on the powder material transport wagon. After the analysis of the wagon crack causes, repair of the cracks is suggested. Repeating of the FEM analysis on the reconstructed model confirms that the wagon satisfies the criteria of static and fatigue strength appointed by the standards. [ABSTRACT FROM AUTHOR]

The paper refers to a new method to quantify the energy losses due to frictional effects and imperfections in contacts in the case of real industrial tribomechanical systems. Whereby energy losses represent an integral indicator of quality of the real industrial tribomechanical system, in terms of the characteristics of the contact element materials, geometric accuracy, and manufacturing and assembly errors. This paper presents a complex theoretical model based on the differential equation of motion of a real tribomechanical system down a steep plane. The outputs of the theoretical model are exact mathematical expressions that define the current values of the coefficient of friction and the friction-caused energy losses. The measuring system enables the quantification of current values of the distance traveled per unit of time. Based on a series of experimentally determined values of distance traveled per unit of time, the values of energy losses of the real industrial tribomechanical system are determined using the developed theoretical model and the appropriate software support. The obtained results indicate a high reliability, a large potential, and a wide range of possible applications of the proposed method. [ABSTRACT FROM AUTHOR]

Abstract: This paper describes a methodology used to identify causes of cracking nearby the welded joint on the underframe of wagon type Sgmns for the transportation of containers and swap bodies. Because of complex mounting, bending and welding technology of underframe bottom part, it was not possible to solve this problem with a commercial software. The aim of this analysis is to identify and register the residual stresses that appear after the process of mounting of the underframe bottom part. According to the TSI standard and norm EN 12663:2000, the fatigue strength analysis of Sgmns wagon for containers transportation was done. The analysis was done to discover kind of loads which induce cracks in the welded joints. The obtained results for the case of maximum vertical load were shown. The analysis of obtained results identified that cause of cracking on the bottom part of underframe of wagon type Sgmns is fatigue. [Copyright &y& Elsevier]

This paper is the result of research and operation modeling of the new systems for cooling of cutting tools based on thermoelectric module. A copper inlay with thermoelectric module on the back side was added to a standard turning tool for metal processing. For modeling and simulating the operation of thermoelectric module, finite element method was used as a method for successful solving the problems of inhomogeneous transient temperature field on the cutting tip of lathe knives. Developed mathematical model is implemented in the software package PAK-T through which numerical results are obtained. Experimental research was done in different conditions of thermoelectric module operation. Cooling of the hot module side was done by a heat exchanger based on fluid using automatic temperature regulator. After the calculation is done, numerical results are in good agreement with experimental. It can be concluded that developed mathe-matical model can be used successfully for modeling of cooling of cutting tools. [ABSTRACT FROM AUTHOR]