Abstract — A new approach for computing **the** inverses **of** **Vandermonde** **matrices** by using **the** cover-up technique **of** partial fraction expansion is presented. It provides insight **of** **the** relationship between **the** **Vandermonde** matrix and **the** coefficients **of** **the** associated partial fraction expansion. **The** method is suitable for either hand calculation or computer programming.

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ABSTRACT. In this paper one proposes to use a new approach **of** interval arithmetic, **the** so-called pseudo- intervals [1, 5, 13]. It uses a construction which is more canonical and based **on** **the** semi-group completion into **the** group, and it allows to build a Banach vector space. This is achieved by embedding **the** vector space into free algebra **of** dimensions higher than 4. It permits to perform linear algebra and differential calculus with pseudo-intervals. Some numerical applications for interval matrix eigenmode calculation, **inversion** and function minimization are exhibited for simple examples.

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fit into these formal definitions. Again, we stress that **the** definitions (35) and (36) are not unique, since other definitions are possible with different ordering or polarity **of** **the** variables (Sasao and Butler, 2007; Rushdi and Ghaleb, 2013; Rudeanu, 1974; Cheng et al., 2010). We call **the** Preparata Transformation in (1a) {T(n), n≥0} and give a novel proof for it by mathematical induction by proving **the** boundary and inductive cases as follows. **The** boundary case T(0) states Equation 37:

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to make **the** scheme more complex. (3) Using **the** technique **of** letter decomposition and composition to make **the** scheme more deceptive to a possible adversary. (4) Using mobile phone in **the** system to enhance **the** security. **The** rest **of** **the** paper is orga- nized as follows. In Section 2, we briefly introduce our original research in GF (p). In Section 3, we propose some novel ideas to improve our original scheme. In Section 4, we design **the** registration system. In Section 5, we give experimental results and analysis. We conclude **the** paper in Section 6.

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This study, which involves synthetic data suggests that there are some physical limita- tions to obtain a reliable velocity model for subbasalt zones in areas covered by high velocity rocks (like basalts and salts). In **the** case **of** thin basalt layers, **the** base basalt reflection is totally masked within **the** water-wave cone and it cannot be separated from **the** basalt refraction. There are several subjective factors that can affect and condition

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Many practical problems can be reduced to systems **of** linear equations Ax = b, where A, b are known **matrices** and x is a vector **of** unknowns. Systems **of** linear equations play a prominent role in economics, engineering, physics, chemistry, computer science and other fields **of** Pure and Applied Sciences [2]. A solution to a system **of** linear equations is a set **of** numerical values ….. that satisfies all **the** equations in a system [1]. There are two classes **of** iterative methods [3]: linear stationary and linear nonstationary. **The** stationary iterative methods are **the** Jacobi, Gauss-Seidel and SOR and Nonstationary include Krylov subspace methods: Conjugate Gradient, Minimal Residual, Quasi-Minimal Residual, Generalizes Minimal Residual and Biconjugate gradient methods. **The** choice **of** a method for solving linear systems will often depend **on** **the** structure **of** **the** matrix A. According to [8] ideally, iterative methods should have **the** property that for any starting vector , it converges to a solution Ax = b. [5] is **of** **the** view that examination **of** **the** Jacobi iterative method shows that in general one must save all **the** components **of** **the** vector while computing **the** components **of** **the** vector for an iterative method. According to Hadjidimos [6], **the** first step in **the** construction **of** solution **of** stationary iterative methods usually begins with splitting **of** matrix A. Thus, A = M – N where det M and M is easily invertible so that A = b is equivalent to = T + C , where T = and C = giving **the** iterative scheme = T + C , (k = 0,1,2……). [2] noted that for systems **of** linear equation A **the** splitting matrix may be chosen in a different way; that is, one can split matrix A as A = D L U where D is **the** diagonal matrix, L and U are strictly lower and strictly upper triangular **matrices** respectively. In solving **the** systems **of** linear equations Ax = b, therefore, we consider any convergent method which produces a sequence **of** iterates { [7] .Quite often **the** convergence is too slow and it has to be accelerated. According to [9] to improve **the** convergence rate **of** **the** basic iterative methods, one may transform **the** original system A = b into **the** preconditioned form PA = Pb, where � is called **the** preconditioned or a preconditioning matrix. Convergent numerical sequences occur quite often in natural Science and Engineering. Some **of** such sequences converge very slowly and their limits are not available without suitable convergent acceleration method. Some known acceleration schemes are: Chebyshev Extrapolation scheme [4] and residual Smoothing.

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Results. Generalizations **of** Hadamard and Belevitch **matrices** as a family **of** quasi-orthogonal **matrices** **of** odd orders are observed; they include, in particular, two-level Mersenne **matrices**. Definitions **of** section and layer **on** **the** set **of** generalized **matrices** are proposed. Calculation algorithms for **matrices** **of** adjacent layers and sections by **matrices** **of** lower orders are described. Approximation examples **of** **the** Belevitch matrix structures up to 22-nd critical order by Mersenne matrix **of** **the** third order are given. New formulation **of** **the** modified Scarpis method to approximate Hadamard **matrices** **of** high orders by lower order Mersenne **matrices** is proposed. Williamson method is described by example **of** one modular level **matrices** approximation by **matrices** with a small number **of** levels.

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ABSTRACT. Seismic **inversion** is routinely used in oil and gas exploration to estimate **the** elastic properties **of** **the** subsurface. However, most **of** **the** **inversion** methods used in **the** industry disregard an inherent phenomenon **of** wave propagation in elastic media: **the** conversion **of** compressional waves into shear waves and vice versa. In this paper we analyze **the** importance **of** **the** locally converted seismic waves in **the** results **of** compressional wave based elastic **inversion**. For this, **the** reflectivity method is used to model **the** seismic response **of** a layered elastic media and also as **the** base **of** a nonlinear **inversion** method. We show that **the** compressional waves generated by local conversion can hardly be identified and eliminated by moveout filters once their transit time are very close to that **of** primary reflections. To assess **the** impact **of** **the** locally converted waves, two versions **of** **the** **inversion** method were implemented: in **the** first one, all seismic events generated in a stratified medium were taken into account, including **the** effects **of** transmission, internal multiples and converted waves. In **the** second version, **the** converted waves were ignored. A series **of** synthetic data were generated using full reflectivity modeling and submitted to **the** two versions **of** **the** **inversion** methods, what allow us to evaluate **the** error made when these waves are ignored. We conclude that this error is proportional to **the** degree **of** contrast in elastic properties between layers and is greatly affected by **the** presence **of** thin layers.

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In this article we study **the** convexity and shape **of** **the** numerical range for nilpotent quaternionic **matrices**. To be more specific, we determine under what conditions W (A) has circular shape or, at least, is convex. In section 2 **of** this article we recall some definitions and fix notation. In section 3 we deal with **the** circularity **of** **the** numerical range. Theorem 3.1 shows that if **the** numerical range **of** a nilpotent matrix is a disk, its center must be located at **the** origin. This is **the** quaternionic analogue **of** [MM, proposition 1]. We conclude this section with theorem 3.5 which says that a sufficient condition for **the** numerical range **of** a nilpotent matrix A to be a disk is that **the** associated graph **of** A is a tree. This condition is not necessary as example 3.8 shows. In section 4 we extend **the** results **of** **the** previous section. Every matrix A ∈ M n (H) is, up to unitary equivalence, upper triangular and every

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decoration in Airtam "**the** sculpture with **the** busts **of** musicians, gift-bearers was subjected to **the** unified principle **of** division **of** **the** entire space into rhythmic segments" [47, 86].As an example, one may consider **the** case when **the** sculpture had a religious or symbolic significance, and was a crucial element in resolving **the** entire interior ("Hall **of** Warriors" in Toprak-kale). But **on** **the** whole, in contrast to a more constrained medieval sculpture, antique one differed in realistic nature (right arrangement **of** **the** figures), expressiveness (**the** types **of** faces and their emotions), naturalness (in different curves **of** **the** body), through which **the** aesthetic sides **of** **the** structures were emphasized. Monumental characters **of** **the** structures, their ideological concepts (for example, **the** idea **of** greatness) were emphasized by means **of** sculpture. And, **the** most important aspect, **the** sculpture differed in architectonic manner (Buddhas - under **the** arches, gandharvas– between **the** acanthuses), as **the** determinant factor was still **the** scale **of** **the** monument, **the** height **of** **the** walls, **the** conditions **of** observation, **the** very architectonics **of** **the** interior. A high quality performance, especially in **the** Greco-Bactrian period, testified to **the** high skills **of** **the** artists, in **the** best traditions **of** Hellenic arts. "**The** artist could be acourt master from Seleucid accompanying **the** king to a distant Bactria" [30, 190]. It is assumed that in Bactria existed at least three sculptural schools; their students were familiar with Asia Minor sculpture schools [48, 125p]. Smooth walls **of** buildings were divided not only horizontally –by friezes, zofors, but also vertically - through door and window openings, columns and pilasters. **The** synthesis **of** architecture and decorative plastic forms, generally typical for later Hellenistic states, could be seen in **the** ancient Bactrian capitals - Corinthian and composite ones, representing a complete architectural form. Professional masters "were widely using **the** approach **of** architectonic

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Corporations need to address their social obligations more consciously. It is important to understand what kinds **of** responsibilities construct CSR before involving in any CSR act ivities. One widely acknowledged theory **of** CSR‘s components is Carroll‘s four-part theory. Carroll (1991) developed his four-part theory **of** CSR, arguing that CSR is constituted by four kinds **of** social responsibilities: economic, legal, ethical, and philanthropic responsibilities. **The** economic responsibilities are **the** primary part **of** **the** four responsibilities. It is **the** fundamental layer **of** Carroll‘s CSR pyramid. All corporations are responsible for providing goods and services that are needed by **the** society. Consequently, profits from selling goods and services go to shareholders and other investors to keep a company survive and grow. Economic responsibilities **of** a company are **the** base for providing legal, ethical and philanthropic responsibilities. Legal responsibilities are **the** second layer **of** **the** CSR pyramid and are coexisting with economic responsibilities as fundamental precepts **of** **the** free enterprise system. Firms are expected to operate under **the** legal system and regulations while creating profits for shareholders. Firms are fulfilling **the** "social contract" between firms and **the** society by being legally responsible. Legally responsible also reflects **the** "codified ethics" **of** business operation, as well as **the** pursuit **of** economic responsibilities set by lawmakers (Carroll, 1991). Ethical responsibilities involve activities and practices that are expected by **the** society and done by firms voluntarily regarding fair, justice and **the** respect for or protection **of** stakeholders' moral rights. Ethical responsibilities are voluntary choices **of** firms, since they are not codified into any law or regulation. These responsibilities reflect social norms, expectations and concerns **of** consumers, employees, shareholders and **the** community. Ethical responsibilities go further than legal responsibilities because they involve newly emerging values and norms that **the** public expects a firm to comply with and are at a higher standard **of** business practices than that current legal system required. However, ethical responsibilities are not easy to deal with for firms because new expectations from **the** public keep emerging and this makes **the** legitimacy **of** ethical responsibilities continually under debate (Carroll, 1991). Philanthropic responsibilities involve firms‘ activities that are

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As we know that to find **the** area **of** sector **the** angle made by **the** chord (that is chord which divides **the** circle) is required. But in **the** below method we find **the** ratio **of** **the** segments **of** **the** circle. Thus by relating **the** area **of** segment to **the** area **of** sector **the** area **of** sector could be found. **The** ratio **of** area **of** segments is related to tangents that are drawn through diameter **on** either side.

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Intermetals are creep-resistant materials with unique properties intermediate between **the** properties **of** metals and ceramics. Their low density, high strength and resistance to oxidation, combined with sufficient ductility and fracture toughness, make them an attractive material for machine elements operating at elevated temperatures and in corrosive environments. Intermetallic phases based **on** Fe and Al (called intermetallics) are intermetallic compounds very interesting in terms **of** **the** technological properties, which combine **the** properties **of** metals and ceramics. From **the** Fe-Al phase equilibrium diagram it follows that **the** widest use could have two phases: FeAl and Fe 3 Al. **Of** these two phases, more useful seems be **the** FeAl

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health care, and opportunities for participation in public arenas, which worsens when men have moved to urban centers leaving their women in **the** rural areas. While it is recognized that women play fundamental roles in **the** socio-economic development **of** their communities, they are often excluded from local decision-making processes because their views are not solicited and their interests are not taken into consideration by their counterparts (Opare, 2005). Society as a whole would benefit greatly if women had adequate access to decision-making opportunities. Women have been and continue to be subordinated to men in several ways that adversely affect their well-being. **The** Beijing Platform for Action emphasized that “women’s equal participation in decision making is not only a demand for justice or democracy, but can also be seen as a necessary condition for women’s interests to be taken into account. Without **the**

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Moreover, increase in refinement **of** aluminium structure results from influence **of** rotate electromagnetic field **on** liquid metal in time **of** its solidification in mould. However in this case effectiveness **of** inoculation fundamentally depends **on** value **of** frequency **of** supply voltage feeding induction coil, which generates electromagnetic field. Application **of** frequency **of** supply voltage f 50 Hz does not guarantee favourable transformation **of** pure aluminium structure. Whereas induction coil fed with frequency **of** supply voltage larger than power network i.e. 75 Hz or mainly 100 Hz, generates rotate electromagnetic field, which guarantees favourable refinement **of** structure from point **of** view **of** assumed criterion **of** minimum value **of** factor R (Tab. 2 and Fig. 6c). In this case value **of** factor R is more favourable in comparison with value obtained for aluminium after Ti and B inoculation and after common influence **of** both modifying factors.

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