Top PDF On the Inversion of Vandermonde Matrices

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.

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.

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:

The first step of the wavelet estimation is purely based on seismic data (without any well-related information). The use of a large number of traces, with a larger time window than that usually involved in conventional inversion, allows for a more robust estimation. Figure 3 compares the average seismic spectrum, the seismic derived statistical wavelet spectrum, and a manually edited version of the wavelet spectrum to produce a smoother version. The next step comprises the least squares adjustment of phase and amplitude, using the edited statistical wavelet as a priori information with strong a priori weight. In this approach, the

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.

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

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.

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.

The second strategy uses all workers and firms in the sample. Since there is no direct within-group transformation that can make the two fixed effects vanish simultaneously, some other method must be used. Abowd et al. (1999) propose approximate statistical methods to the full least square solution, whereas Abowd et al. (2002) provide an exact solution through an interactive conjugate gradient technique that benefits from the existence of sparse matrices in the structure of the normal least square equations. Andrews et al.(2006) propose a method named FEiLSDVj in which workers’ fixed effects are firstly removed through time- demeaning of data over workers (this is the FEi part) and then the model is estimated via least squares with establishment dummies included in the equation (this is the LSDVj part). This method is appropriate for samples where the number of firms is not too large, which is not our case. Here, we follow the method proposed by Cornelissen (2008), which combines features of Andrews et al. (2006) and Abowd et al. (2002). Specifically, the method first uses the within transformation to eliminate the worker fixed effects and then explores the

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.

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

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

The essence of social economy is the inclusive function of the labor market through which the different forms of social economy that exist in the member states can play a role in the overcoming the crisis, especially in the creating of jobs, including in social services field Opinion of the European Economic and Social Committee on the post‐ 2010 Lisbon Strategy 9, p. .

The theoretical temperature of the sample during heat treatment and heat treatment of the characteristic parameters of the tested bronze samples shown in Figure 4. The heat treatment process consisted of the following steps: T1 - heating, T2 - annealing at 1000 °C, T3 - cooling in ambient air (sample temperature ≥λ50°C) at the time necessary to remove the sample from the oven and T4 - immersion in a tank with 10% NaCl in water at 19°C (hyperquenching and hardening). During all stages of heat treatment temperature of the sample was measured thermocouple Pt-PtRh10 using Crystaldigraph devices and recorded in computer memory.

The structure of the remelting zone of the steel C90 steel be- fore conventional tempering consitute cells, dendritic cells, sur- rounded with the cementite, inside of which there is a plate mar- tensite and retained austenite, whereas the structure HS 6-5-2 steel consititute cells, dendritic cells and dendrites surrounded with the eutectic, inside of which there is a plate martensite and retained austenite. Such a structure is characterized with the similar micro- hardness (790-800 HV0,065) and intensity of the tribilogical wear.

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

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.

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

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

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.