Algebra

Wakaf adalah amalan mulia yang sangat di galakkan oleh Islam. Sebagai salah satu mekanisme kebajikan, wakaf agak unik berbanding dengan sedekah, hadiah dan derma kerana harta yang diwakafkan itu akan di 'tahan' ia itu tidak dihabiskan tetapi sebaliknya apa-apa manfaat atau pulangan daripada penggunaan harta wakaf itu boleh dibelanjankan. Bermula dengan amalan wakaf harta tetap secara langsung, dewasa ini perkembangan wakaf adalah dalam bentuk wakaf tunai dan daripada tunai inilah sebahagiannya dimanfaatkan untuk pembangunan atau pembelian hartanah atau harta lain dan sebahagian pula akan dilaburkan dalam pasaran kewangan. Mengambil iktibar daripada sejarah amalan wakaf di dalam tamaddun Islam serta amalan 'endowmen' di dunia Barat, mekanisme atau kaedah wakaf tunai (aset cair) dalam berbagai aplikasi atau produk perlu dilaksanakan dengan segera. Pembangunan ini dapat dipercepatkan melalui badan khas diperingkat negeri dan kebangsaan dengan mengambil pelajaran daripada perkembangan institusi-institusi zakat. Penulis membuat saranan akan beberapa produk wakaf pendidikan yang berpotensi.

This study was conducted to develop a model of principal's instructional leadership and teaching competency. Research respondents were 481 teachers from 55 secondary day schools in Terengganu. The modified Principal Instructional Management Rating Skill (PIMRS) and Malaysian Teacher Standard (SGM) were applied as research instruments. Data were analyzed using structural equation modeling (SEM) by AMOS 18. The study successfully developed a relationship model between instructional leadership and teaching competency (χ2=12.495, χ2/df =1.562, p = 0.130, Bootstrap = 0.164). The findings also show construct of define school goals (0.81), curriculum and instructional management (0.92) and promote teaching and learning climate (0.92) are significant dimensions in instructional leadership. The findings also show construct of interaction with pupils (0.88), teaching strategy (0.89) and class control (0.88) are significant dimensions in teaching competency. Furthermore, the result also found that instructional leadership was a significant predictor of teaching competency (CR=7.635,0.349, p = 0.00).

Debate about teacher supply, demand, retention, and attrition has been renewed in recent years by an increased concern about the reduced numbers of prospective teachers entering teacher education programs, the high attrition rate of beginning teachers, and the resulting teacher shortages. U.S. schools are experiencing teacher shortages, especially in low-income urban areas, because of increased school enrollment, teacher retirement, reduction in class size, teacher attrition, and turnover related to low salaries, job dissatisfaction, and lack of administrative support and influence over decision-making. Recently, the increased interest in teacher quality has been the topic of debate for policy-makers, the public, and the educational community. The purpose of this study was to determine if a nontraditional teacher preparation program, the Transition To Teaching program, was a viable way to ease the teacher shortages in a high poverty, urban U.S. school district, and at the same time, to evaluate the impact of teacher training on students' academic achievement. The results of this study afford evidence that the students taught by 1st-year, alternatively prepared teachers achieved as well as or better than their peers taught by traditionally certified 1st-year teachers, according to student achievement in mathematics, specifically Algebra I.

Implementasi algoritma Particle Swarm Optimization (PSO) pada robot untuk pencarian sumber kebocoran gas memerlukan informasi posisi dari setiap robot. Penggunaan kamera memungkinkan untuk mendapatkan informasi posisi robot yang lebih absolut dan akurat dibandingkan dengan perangkat pencari informasi posisi lainnya. Paper ini memaparkan suatu metode yang akan digunakan untuk mencari posisi robot. Metode yang diajukan menggunakan beberapa teknik pengolahan citra, seperti, Color Filtering dan Blobs Filtering guna mengenali dan memperoleh posisi robot. Berdasarkan eksperimen yang telah dilakukan, metode ini mampu menentukan posisi absolut robot sehingga ia dapat menyediakan informasi posisi robot dalam implementasi algoritma PSO.

Use of spreadsheets in a beginning algebra course was investigated mainly with regard to their potential to promote generalization of patterns. Less is known about their use in promoting understanding and learning of transformational activities. The overall purpose of this paper is to consider the conceptual aspects of learning a transformational skill (use of the distributive law to produce equivalent algebraic expressions) in a learning sequence composed of both spreadsheets and paper-and-pencil activities. We conducted a sequence of classroom activities in several classes, and analyzed the students’ work on a spreadsheet activity and on an assessment activity by both qualitative and quantitative methods. The findings indicate both encouraging benefits and some potential sources of difficulties caused by the use of spreadsheets at initial stages of learning symbolic transformations.

Abstrak Inovasi teknologi otomotif khususnya sepeda motor terus dikembangkan untuk mendapatkan kestabilan dan kenyamanan dalam pengendalian. Saat ini produsen sepeda motor telah memproduksi kendaraan yang memakai sistem transmisi otomatik. Transmisi otomatik merupakan sistem transmisi yang hanya membutuhkan pengendalian kecepatan dan pengendalian pengereman. Dari konsep tersebut telah dikembangkan sistem transmisi otomatik secara variabel yang disebut dengan Continously Variable Transmission (CVT) sistem. CVT pada sepeda motor menggunakan speed governor yang mengatur kedudukan atau diameter puli primer untuk merubah ratio transmisi sesuai dengan putaran mesi.Penelitian ini bertujuan untuk mengetahui pengaruh berat roller sentrifugal yang terdapat dalam speed governor terhadap kinerja traksi. Kinerja traksi dianalisa dengan melakukan pemodelan matematik dengan kendaraan model sepeda motor Nouvo 115 cc, 4 tak. Sebagai parameter input pada perhitungan meliputi : kecepatan (V = 0-30 km / jam, V = 40-70 km / jam, dan V = 80-90 km / jam), torsi mesin, berat roller sentrifugal, dinamika kendaraan (gaya berat, gaya hambat, dsb). Untuk dapat menjawab permasalahan yang timbul dilakukan penelitian dengan mempergunakan metode simulasi yang dibandingkan dengan pengujian di jalan lurus datar. Dari hasil simulasi dan eksprimen dilapangan didapat : untuk berat roller sentrifugal 8 gr kinerja traksi terbesar terjadi pada kecepatan rendah sehingga akselerasi pada kecepatan rendah paling cepat dibandingkan dengan roller sentrifugal 10,2 (standar) atau 12 gr. Sedangkan dengan berat roller sentrifugal 12 gr akan didapat kinerja traksi terbesar pada kecepatan tinggi sehingga kendaraan akan mudah dipercepat pada kecepatan tinggi tersebut, dan untuk roller sentrifugal 10,2 gr (standar) memiliki kinerja traksi diantara roller sentrifugal 8 gr dan 12 gr. Jadi roller 8 gr menghasilkan kinerja traksi yang terbaik, karena pada kecepatan rendah dibutuhkan kemampuan akselarasi yang besar Kata kunci: Kompresor aksial, kaskade, sudut serang.

Proofs that the area of a circle is πr 2 can be found in mathematical literature dating as far back as the time of the Greeks. The early proofs, e.g. Archimedes, involved dividing the circle into wedges and then fitting the wedges together in a way to approximate a rectangle. Later more sophisticated proofs relied on arguments involving infinite sequences and calculus. Generally speaking, both of these approaches are difficult to explain to unsophisticated non-mathematics majors. This paper presents a less known but interesting and intuitive proof that was introduced in the twelfth century. It discusses challenges that were made to the proof and offers simple rebuttals to those challenges.

In this paper we generalize the shortest path algorithm to the shortest cycles in each homotopy class on a surface with arbitrary topology, by utilizing the universal covering space (UCS) in algebraic topology. In order to store and handle the UCS, we propose a two-level data structure which is efficient for storage and easy to process. For the shortest cycle algorithm and the UCS data structure, we showed some practical applications, such as topological denoise in geometric modeling, polygonal schema construction in computational topology and etc. This restricted problem can be solved more efficiently and gives direct intuition towards a solution to the general problem. Our algorithm is based on the universal covering space (see section 2.2); therefore we also investigated efficient data structures to handle the universal covering space.

In this paper, we present a view of kernels from a fuzzy set theoretical perspective. Indeed, it turns out that kernels which are positive-definite functions have to fulfill a consistency property given by the so-called T-transitivity of a fuzzy T-equivalence relation with respect to the triangular norm T. As a result, we introduce a triangular norm T cos which is characterized as being the greatest one for which all kernels mapping to the unit interval and having constant 1 in its diagonal are T cos-equivalences. Finally, some applied relevant aspects are discussed as a starting point for further research.

Abstract. Let k be a field and let F⊂ k [x1,..., xn] be a finite set of monomials whose exponents lie on a positive hyperplane. We give necessary conditions for the normality of both the Rees algebra R [Ft] and the subring k [F]. If the monomials in F have the same ...

We prove that the double shuffle Lie algebra ds, dual to the space of new formal multiple zeta values, injects into the Kashiwara-Vergne Lie algebra krv 2 defined and studied by Alekseev-Torossian. The proof is based on a reformulation of the definition of krv 2 , and uses a theorem of Ecalle on a property of elements of ds. §1. Definitions and main results Let Q x, y denote the ring of polynomials in non-commutative variables x and y, and Lie[x, y] the Lie algebra of Lie polynomials inside it. For each n ≥ 1, let Q n x, y (resp. Lie n [x, y]) denote the subspace of homogeneous polynomials (resp. Lie polynomials) of degree n. For k ≥ 1, let Q n≥k x, y (resp. Lie n≥k [x, y]) denote the space of polynomials (resp. Lie polynomials) all of whose monomials are of degree ≥ k, i.e. the direct sum of the Q n x, y (resp. Lie n [x, y]) for n ≥ k. The main theorem of this paper gives an injective map between two Lie algebras studied in the literature concerning formal multiple zeta values: the double shuffle Lie algebra ds, investigated in papers by Racinet and Ecalle amongst others (the associated graded of ds is also studied in papers by Zagier, Kaneko and others), and the Kashiwara-Vergne Lie algebra introduced in work of Alekseev and Torossian (cf. [AT]). We begin by recalling the definitions of these two Lie algebras. As vector spaces, both are subspaces of the free Lie algebra Lie[x, y]. For any non-trivial monomial w and polynomial f ∈ Q x, y , we use the notation (f |w) for the coefficient of the monomial w in the polynomial f , and extend it by linearity to polynomials w without constant term. Set y i = x i−1 y for all i ≥ 1; then all words ending in y can be written as words in the variables y i. The stuffle product st(u, v) ∈ Q x, y of two such words u and v is defined recursively by st(1, u) = st(u, 1) = u and st(y i u, y j v) = y i st(u, y j v) + y j st(y i u, v) + y i+j st(u, v). Definition 1.1. The double shuffle Lie algebra ds * is the vector space of elements f ∈ Lie n≥3 [x, y] such that f st(u, v) = 0 for all words u, v ∈ Q x, y ending in y but not both simultaneously powers of y. * The equivalence of the present definition with the usual definition introduced in [R] is proven in [CS], Theorem 2, which proves that if a polynomial f ∈ Lie n [x, y] has the property of the present definition, then f + (−1) n−1 n (f |x n−1 y)y n satisfies the stuffle relations for all pairs of words u, v ending in y. Since the words ending in x are not involved in this condition, this is equivalent to the assertion that π y (f) + (−1) n−1 n (f |x n−1 y)y n satisfies stuffle, where π y (f) denotes the projection of f onto just its words ending in y. This is the standard form of the defining property of elements of ds.

This work explores the deformation theory of algebraic structures in a very general setting. These structures include commutative, associative algebras, Lie algebras, and the infinity versions of these structures, the strongly homotopy associative and Lie algebras. In all these cases the algebra structure is determined by an element of a certain graded Lie algebra which plays the role of a differential on this algebra. We work out the deformation theory in terms of the Lie algebra of coderivations of an appropriate coalgebra structure and construct a universal infinitesimal deformation as well as a miniversal formal deformation. By working at this level of generality, the main ideas involved in deformation theory stand out more clearly.

The algebra of qualified relations proposed by Ceri and Pelagatti supports pruning of distributed database queries, minimizing the number of sites involved in evaluation. Domain interval typing in the framework of formal type inference subsumes this approach, pruning not only unsatisfiable subqueries involving fragmentation predicates, but also unsatisfiable subqueries mentioning any relation attribute

INTISARI Dalam tesis ini dianalisa mengenai model matematika peningkatan enzim hati akibat mono-infeksi HIV. Model ini dibangun atas dasar asumsi infeksi HIV hanya menyebabkan replikasi virus terjadi hanya pada sel CD4+, dan tidak terjadi pada sel hepatosit. Selain itu, tingginya enzim hati (alanine aminotransferase) merupakan indikator kuat dalam menentukan tingkat kerusakan hati. Model HIV dalam hati ini adalah pengembangan dari model infeksi viral HIV ke sel CD4+ yang diteliti oleh Perelson (1994), yang kemudian dimodifikasi oleh Nampala (2013) dengan penambahan model sel CTLs dalam tubuh. Berbeda dengan apa yang dilakukan dengan peneliti sebelumnya, dalam tesis ini dilakukan suatu koreksi syarat eksistensi ekuilibrium endemik dan dilakukan metode baru dalam menentukan kestabilan lokal dan global titik ekuilibrium bebas penyakit (infeksi). Analisis bifurkasi dilakukan disekitar untuk mengetahui kestabilan titik ekuilibrium endemik. Hasil dari penelitian ini adalah titik ekuilibrium bebas infeksi dan titik ekuilibrium endemik keduanya tidak stabil ketika Hal ini mengindikasikan bahwa HIV dengan berjalannya waktu mampu menyebabkan kerusakan hati yang ekstrim tanpa gejala lazim dalam waktu cukup singkat. Hasil simulasi numerik memberikan hubungan menarik antara peningkatan enzim hati, virus, sel CD4+, dan sel hepatosit. Kata kunci : Level alanine aminotransferase (ALT), HIV, endemik, bifurkasi. ABSTRACT In this thesis, we analyse the mathematical model of liver enzyme elevation due to mono-infection with HIV. This model is built on the assumption of HIV infection only causes viral replication only on CD4+ cells, and it doesn't occur in hepatocytes cell. In addition, high liver enzymes (alanine aminotransferase or ALT) is a strong indicator in determining the extent of liver damage. Models of HIV in the heart is the development of models viral infection of HIV to CD4+ cells were examined by Perelson (1994), and then modified by Nampala (2013) models with the addition of model CTLs cells in the body. In contrast to what is done by the previous researcher, in this thesis we do a correction of conditions existence of endemic equilibrium and perform a new method to determine the local and global stability of the disease-free equilibrium point (infection). Bifurcation analyse around for determine the stability of the endemic equilibrium point. The results of this research is the point of the infection-free equilibrium and endemic equilibrium point which is unstable when both. This indicates that HIV as time goes capable of causing extreme damage to the liver in short time. Numerical simulation results provide an interesting relationship between liver enzymes, viruses, CD4+ cells, and hepatocyte cells.

This paper discusses the case of one teacher, Jackie, whose instructional practices illuminate the importance of textbooks and student/parent expectations in shaping pedagogy. Jackie teaches in the Plainview district, which offers parents and students a choice between a reform-oriented, integrated curriculum (Core Plus) and a more conventional algebra sequence (the University of Chicago series). Each day, Jackie teaches two very different sections of accelerated eighth-grade mathematics using each of these curricular materials. Drawing from students' survey responses, classroom observations, and teacher interview data, we show ways in which Jackie's pedagogy differs considerably between the two courses and we shed light on reasons underlying this variation.By examining one teacher who enacts different practices in each of the two curricular contexts, this paper highlights factors that contribute to teachers' enacted curricula -factors that have been understated in previous mathematics education research on teacher development. The study establishes the importance of distinguishing between global and local teacher change, and suggests implications for future studies of teaching and reform.

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Asphalt is a small component in the flexible pavement. As a binder, it is commonly needed 4 %-10 % of total mixtures of the mix design. In the mixed design have to consider the technical aspect and another aspects including to obtain material easly on the site. There for, it is necessary to study the alternative material (asphalt) when asphalt not ready in the market any time. This study to compare both the asphalt Pertamina Pen 60/70 products with the asphalt Esso Pen 60/70 products base on the Bina Marga specifications. In order to have the best result, it is necessary to study in the laboratory to know Marshall characteristic and Structural Values (modulus elastisity and material strengthening coefficiens) on both of the asphalt Pertamina Pen 60/70 products with the asphalt Esso Pen 60/70 products on the asphalt optimum content. By using the VIM as a Marshall convensional 6 % find the asphalt optimum content value, based on it ± 0,5 % the VIM-PRD value on the 2 x 400 blow was determined. From the individual test result as well as in mixed result show that the stability value of the asphalt Pertamina Pen 60/70 products (906 kg) is lower than the stability value of asphalt Esso Pen 60/70 products (989 kg). Beside that, the Voids in Mix (VIM) as well as the PRD value indicate that the result for the asphalt Pertamina Pen 60/70 products is lower than the VIM value of the asphalt Esso Pen 60/70 products. Which caused impervious layer and the air can not filled in the mixing to result in oksidation and to be damage the asphalt. The result study show that Voids in Mineral Aggregate (VMA) Marshall value of the Pertamina Pen 60/70 products is lower than the VMA value of the asphalt Esso Pen 60/70 products. But for PRD value is invertion result. The Voids in Filled with Bitumen (VFB) Marshall as well as PRD value show that the asphalt Pertamina Pen 60/70 products bigger than the VFB value of the asphalt Esso Pen 60/70 products. The result study concerning the characteristics of both asphalt trade fulfill the requirement of Bina Marga specifications.

An efficient technique to uncover the relationship between the representations of a group G and the representations of a subgroup H is to Ž . consider gradings by GrH , the class of the left H-cosets of G. This l w x method of study was initiated by Dade in 3 , where he describes the structure of the simple objects in the category R᎐gr of G-graded R-modules for a G-graded ring R, as a generalization of the classical Clifford theory. A systematic investigation of modules graded by G-sets was started w x w x by Nastasescu et al. in 9 , and then continued in other papers 1, 8, 5 .˘Ž . Categorical properties of the category G, A, R ᎐gr of A-graded left R-modules, where R is a G-graded ring and A a left G-set, were studied first, and later a series of finiteness conditions for the objects of this category were considered.

In the first part of this paper we study prime ideals of the tensor product of the central closures of prime algebras. It is shown that some of the prime ideals are centrally generated. This extends a result of Nicholson and Watters [S], and is applied to obtain a simpler and more "visible" proof of Krempa's Lemma 3.4 in [2] which plays an important role in a description of radicals of tensor products. Using this lemma we extend, in the second part of the paper, the following recent result of Lawrence [3]: the tensor product of algebras, over an algebraically closed field, containing no nonzero algebraic ideals is Jacobson semisimple. Lawrence's proof is quite difficult; it is based on the "Units Theorem" [9] and several theorems of commutative algebra, in particular valuation theory. We present a simpler proof which is based only on the Units Theorem and avoids any use of the valuation machinery. Next we show the result remains valid for algebras over perfect fields. And then we prove that, for arbitrary fields, the Jacobson radical of the tensor product of algebras not containing nonzero algebraic ideals is equal to the prime radical.

The paper derives the CSS construction for quantum codes over noncommutative rings. As an application, optimal quantum codes are derived from generalized Reed-Solomon codes over commutative and noncommutative rings. For this purpose, the existence of extremal subtractive sets in commutative rings is established. Furthermore, the foundations for constacyclic quantum codes over finite chain rings are established.

Faculty members should determine the factors that contribute to poor performance of both teachers and students through regular monitoring and evaluation of the teaching and learning process. Teaching styles seem to vary as much as learning styles. Matching teaching style to learning style is not a panacea that solves all classroom conflicts. Other factors such as classroom climate, previous background, motivation, and gender will of course greatly influence the amount and quality of learning that takes place. Still, for math teachers, being self reflective and explicit about the role of learning styles can make teaching more rewarding and enhance student learning at the same time. Hence, it is deemed necessary to assess the students' learning styles and teachers' teaching styles. This research was conducted to identify and assess teaching and learning styles and its effect on performance of freshman engineering students of TIP Manila in algebra during the first semester of s...

Library are usually used by visitors as media to search a reference and obtain information. Problems at this time is that not many libraries have shaped the web information system for online services. This research aims to analyze and design web-based library information system and testing of the performance of library information systems. Results of research. Results of research shows that the library information system is web-based software built with PHP and MySQL, can work in the Internet network, with the ability to receive data on the WEB library visits, serving registration members get access to a wider, providing services to the availability of information needs click in the form of books, ordering and serving the order book from the members, to serve and return of rental transactions directly.

We classify irreducible representations of the special linear groups in positive characteristic with small weight multiplicities with respect to the group rank and give estimates for the maximal weight multiplicities. For the natural embeddings of the classical groups, inductive systems of representations with totally bounded weight multiplicities are classified. An analogue of the Steinberg tensor product theorem for arbitrary indecomposable inductive systems for such embeddings is proved.

Let L n (C) be the variety of complex n-dimensional Lie algebras. The group GL n (C) acts on it via change of basis. An orbit O(µ) under this action consists of all structures isomorphic to µ. The aim of this paper is to give a complete classification of orbit closures of 4-dimensional Lie algebras, i.e., determining all µ ∈ O(λ) where λ ∈ L 4 (C). Starting with a classification of complex Lie algebras of dimension n ≤ 4 , we study the behaviour of several Lie algebra invariants under degeneration, i.e. under transition to the orbit closure. As a corollary, we will show that all degenerations in L 3 (C) can be realized via a one-parameter subgroup, but this is not the case in L 4 (C).

We use a recent result concerning the eigenvalues of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices to prove that the asymptotic spectrum of the product of Toeplitz sequences, whose symbols have a real-valued essentially bounded product h, is described by the function h in the “Szegö way”. Then, using Mergelyan’s theorem, we extend the result

Taking as a point of departure the vision of school algebra as a formalized meta-discourse of arithmetic, we have been following six pairs of 7th-grade students (12-13 years old) as they gradually modify their spontaneous meta-arithmetic toward the "official" algebraic form of talk. In this paper we take a look at the very beginning of this process. Preliminary analyses of data have shown, unsurprisingly, that while reflecting on arithmetic processes and relations, the uninitiated 7th graders were employing colloquial means, which could not protect them against occasional ambiguities. More unexpectedly, this spontaneous meta-arithmetic, although not supported by any previous algebraic schooling, displayed some algebra-like features, not to be normally found in everyday discourses.

This paper addresses the problem of parametric representation and estimation of complex planar curves in 2-D, surfaces in 3-D and nonplanar space curves in 3-D. Curves and surfaces can be defined either parametrically or implicitly, and we use the latter representation. A planar curve is the set of zeros of a smooth function of two variables X-Y, a surface is the set of zeros of a smooth function of three variables X-~-Z, and a space curve is the intersection of two surfaces, which are the set of zeros of two linearly independent smooth functions of three variables X-!/-Z. For example, the surface of a complex object in 3-D can be represented as a subset of a single implicit surface, with similar results for planar and space curves. We show how this unified representation can be used for object recognition, object position estimation, and segmentation of objects into meaningful subobjects, that is, the detection of "interest regions" that are more complex than high curvature regions and, hence, more useful as features for object recognition. Fitting implicit curves and surfaces to data would be ideally based on minimizing the mean square distance from the data points to the curve or surface. Since the distance from a point to a curve or surface cannot be computed exactly by direct methods, the approximate distance, which is a first-order approximation of the real distance, is introduced, generalizing and unifying previous results. We fit implicit curves and surfaces to data minimizing the approximate mean square distance, which is a nonlinear least squares problem. We show that in certain cases, this problem reduces to the generalized eigenvector fit, which is the minimization of the sum of squares of the values of the functions that define the curves or surfaces under a quadratic constraint function of the data. This fit is computationally reasonable to compute, is readily parallelizable, and, hence, is easily computed in real time. In general, the generalized eigenvector lb provides a very good initial estimate for the iterative minimization of the approximate mean square distance. Although we are primarily interested in the 2-D and 3-D cases, the methods developed herein are dimension independent. We show that in the case of algebraic curves and surfaces, i.e., those defined by sets of zeros of polynomials, the minimizers of the approximate mean square distance and the generalized eigenvector fit are invariant with respect to similarity transformations. Thus, the generalized eigenvector lit is independent of the choice of coordinate system, which is a very desirable property for object recognition, position estimation, and the stereo matching problem. Finally, as applications of the previous techniques, we illustrate the concept of "interest regions"

In this article properties of the $(b, c)$-inverse, the inverse along an element, the outer inverse with prescribed range and null space $A^{(2)}_{T, S}$ and the Moore-Penrose inverse will be studied in the contexts of Banach spaces operators, Banach algebras and $C^*$-algebras. The main properties to be considered are the continuity, the differentiability and the openness of the sets of all invertible elements defined by all the aforementioned outer inverses but the Moore-Penrose inverse. The relationship between the $(b, c)$-inverse and the outer inverse $A^{(2)}_{T, S}$ will be also characterized