Search results for: roots of polynomials

Authors: Nevena Jakovčević Stor, Ivan Slapničar

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Any polynomial can be expressed as a characteristic polynomial of a complex symmetric arrowhead matrix. This expression is not unique. If the polynomial is real with only real distinct roots, the matrix can be chosen as real. By using accurate forward stable algorithm for computing eigen values of real symmetric arrowhead matrices we derive a forward stable algorithm for computation of roots of such polynomials in O(n^2 ) operations. The algorithm computes each root to almost full accuracy. In some cases, the algorithm invokes extended precision routines, but only in the non-iterative part. Our examples include numerically difficult problems, like the well-known Wilkinson’s polynomials. Our algorithm compares favorably to other method for polynomial root-finding, like MPSolve or Newton’s method.

Keywords: roots of polynomials, eigenvalue decomposition, arrowhead matrix, high relative accuracy

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Authors: Mojtaba Hakimi-Moghaddam

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Routh-Hurwitz stability criterion is a powerful approach to determine stability of linear time invariant systems. On the other hand, applying this criterion to characteristic equation of a system, whose stability or marginal stability can be determined. Although the command roots (.) of MATLAB software can be easily used to determine the roots of a polynomial, the characteristic equation of closed loop system usually includes parameters, so software cannot handle it; however, Routh-Hurwitz stability criterion results the region of parameter changes where the stability is guaranteed. Moreover, this criterion has been extended to characterize the stability of interval polynomials as well as fractional-order polynomials. Furthermore, it can help us to design stable and minimum-phase controllers. In this paper, theory and application of this criterion will be reviewed. Also, several illustrative examples are given.

Keywords: Hurwitz polynomials, Routh-Hurwitz stability criterion, continued fraction expansion, pure imaginary roots

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Authors: M. Tiachachat, M. Mihoubi

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The Bell polynomials are special polynomials in combinatorial analysis that have a wide range of applications in mathematics. They have interested many authors. The exponential partial Bell polynomials have been well reduced to some special combinatorial sequences. Numerous researchers had already been interested in the above polynomials, as evidenced by many articles in the literature. Inspired by this work, in this work, we propose a family of special polynomials named after the 2-successive partial Bell polynomials. Using the combinatorial approach, we prove the properties of these numbers, derive several identities, and discuss some special cases. This family includes well-known numbers and polynomials such as Stirling numbers, Bell numbers and polynomials, and so on. We investigate their properties by employing generating functions

Keywords: 2-associated r-Stirling numbers, the exponential partial Bell polynomials, generating function, combinatorial interpretation

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Authors: Laskri Yamina, Rehouma Abdel Hamid

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In this paper we present a new class named polar of monic orthogonal polynomials with respect to the area measure supported on G, where G is a bounded simply-connected domain in the complex planeℂ. We analyze some open questions and discuss some ideas properties related to solving asymptotic behavior of polar Bergman polynomials over domains with corners and asymptotic behavior of modified Bergman polynomials by affine transforms in variable and polar modified Bergman polynomials by affine transforms in variable. We show that uniform asymptotic of Bergman polynomials over domains with corners and by Pritsker's theorem imply uniform asymptotic for all their derivatives.

Keywords: Bergman orthogonal polynomials, polar rthogonal polynomials, asymptotic behavior, Faber polynomials

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Authors: Vandana N. Purav

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Fibonacci and Lucas polynomials are special cases of Chebyshev polynomial. There are two types of Chebyshev polynomials, a Chebyshev polynomial of first kind and a Chebyshev polynomial of second kind. Chebyshev polynomial of second kind can be derived from the Chebyshev polynomial of first kind. Chebyshev polynomial is a polynomial of degree n and satisfies a second order homogenous differential equation. We consider the difference equations which are related with Chebyshev, Fibonacci and Lucas polynomias. Thus Chebyshev polynomial of second kind play an important role in finding the recurrence relations with Fibonacci and Lucas polynomials.

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Authors: Sanju Vaidya, Aihua Li

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Vulnerability measures and topological indices are crucial in solving various problems such as the stability of the communication networks and development of mathematical models for chemical compounds. In 1947, Harry Wiener introduced a topological index related to molecular branching. Now there are more than 100 topological indices for graphs. For example, Hosoya polynomials (also called Wiener polynomials) were introduced to derive formulas for certain vulnerability measures and topological indices for various graphs. In this paper, we will find a relation between the Hosoya polynomials of any graph and its Mycielskian graph. Additionally, using this we will compute vulnerability measures, closeness and betweenness centrality, and extended Wiener indices. It is fascinating to see how Hosoya polynomials are useful in the two diverse fields, cybersecurity and chemistry.

Keywords: hosoya polynomial, mycielskian graph, graph vulnerability measure, topological index

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Authors: Yilmaz Simsek

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In this paper, we study the Bernstein-type basis functions with their generating functions. We give various properties of these polynomials with the aid of their generating functions. These polynomials and generating functions have many valuable applications in mathematics, in probability, in statistics and also in mathematical physics. By using the Bernstein-Galerkin and the Bernstein-Petrov-Galerkin methods, we give some applications of the Bernstein-type polynomials for solving high even-order differential equations with their numerical computations. We also give Bezier-type curves related to the Bernstein-type basis functions. We investigate fundamental properties of these curves. These curves have many applications in mathematics, in computer geometric design and other related areas. Moreover, we simulate these polynomials with their plots for some selected numerical values.

Keywords: generating functions, Bernstein basis functions, Bernstein polynomials, Bezier curves, differential equations

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Authors: Mohammad Moradi ‎

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In this article, propagation of a laser beam through turbulent ‎atmosphere is evaluated. At first the laser beam is simulated and then ‎turbulent atmosphere will be simulated by using Zernike polynomials. ‎Some parameter like intensity, PSF will be measured for four ‎wavelengths in different Cn2.

Keywords: laser beam propagation, phase screen, turbulent atmosphere, Zernike ‎polynomials

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Authors: Taweechai Nuntawisuttiwong, Natasha Dejdumrong

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Newton-Lagrange Interpolations are widely used in numerical analysis. However, it requires a quadratic computational time for their constructions. In computer aided geometric design (CAGD), there are some polynomial curves: Wang-Ball, DP and Dejdumrong curves, which have linear time complexity algorithms. Thus, the computational time for Newton-Lagrange Interpolations can be reduced by applying the algorithms of Wang-Ball, DP and Dejdumrong curves. In order to use Wang-Ball, DP and Dejdumrong algorithms, first, it is necessary to convert Newton-Lagrange polynomials into Wang-Ball, DP or Dejdumrong polynomials. In this work, the algorithms for converting from both uniform and non-uniform Newton-Lagrange polynomials into Wang-Ball, DP and Dejdumrong polynomials are investigated. Thus, the computational time for representing Newton-Lagrange polynomials can be reduced into linear complexity. In addition, the other utilizations of using CAGD curves to modify the Newton-Lagrange curves can be taken.

Keywords: Lagrange interpolation, linear complexity, monomial matrix, Newton interpolation

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Authors: Mohsen Zahraei

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‎In this paper, ‎the notion of ‎rank-k numerical range of rectangular complex matrix polynomials‎ ‎are introduced. ‎Some algebraic and geometrical properties are investigated. ‎Moreover, ‎for ε>0 the notion of Birkhoff-James approximate orthogonality sets for ε-higher ‎rank numerical ranges of rectangular matrix polynomials is also introduced and studied. ‎The proposed definitions yield a natural generalization of the standard higher rank numerical ranges.

Keywords: ‎‎Rank-k numerical range‎, ‎isometry‎, ‎numerical range‎, ‎rectangular matrix polynomials

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Authors: Koushlendra Kumar Singh, Manish Kumar Bajpai, Rajesh Kumar Pandey

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A discrete time fractional orderdifferentiator has been modeled for estimating the fractional order derivatives of contaminated signal. The proposed approach is based on Chebyshev’s polynomials. We use the Riemann-Liouville fractional order derivative definition for designing the fractional order SG differentiator. In first step we calculate the window weight corresponding to the required fractional order. Then signal is convoluted with this calculated window’s weight for finding the fractional order derivatives of signals. Several signals are considered for evaluating the accuracy of the proposed method.

Keywords: fractional order derivative, chebyshev polynomials, signals, S-G differentiator

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Authors: Negadi Kheira, Arab Ahmed, Kamal Elbokl Mohamed, Setti Fatima

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In order to evaluate influences of roots on soil shear strength, monotonic drained and undrained triaxial laboratory tests were carried out on reconstituted specimens at various confining pressure (σc’=50, 100, 200, 300, 400 kPa) and a constant relative density (Dr = 50%). Reinforcement of soil by fibrous roots is crucial for preventing soil erosion and degradation. Therefore, we investigated soil reinforcement by roots of acacia planted in the area of Chlef where shallow landslides and slope instability are frequent. These roots were distributed in soil in two forms: vertically and horizontally. The monotonic test results showed that roots have more impacts on the soil shear strength than the friction angle, and the presence of roots in soil substantially increased the soil shear strength. Also, the results showed that the contribution of roots on the shear strength mobilized increases with increase in the confining pressure.

Keywords: soil, monotonic, triaxial test, root fiber, undrained

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Authors: Hari O. Saxena, Ganesh

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In the present investigation, chemical fingerprints of methanolic extracts of roots, stem and leaves of Uraria picta were developed using HPTLC technique. These fingerprints will be useful for authentication as well as in differentiating the species from adulterants. These will also serve as a biochemical marker for this valuable species in pharmaceutical industries and plant systemic studies. Roots, stem and leaves of Uraria picta were further evaluated for quantification of an active ingredient lupeol to find out alternatives to roots. Results showed more content of lupeol in stem (0.048%, dry wt.) as compare to roots (0.017%, dry wt.) suggesting the utilization of stem in place of roots. It will avoid uprooting of this prestigious plant which ultimately will promote its conservation.

Keywords: chemical fingerprints, lupeol, quantification, Uraria picta

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Authors: S. R. Nayaka, Putta Swamy

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Graph polynomial is one of the algebraic representations of the Graph. The degree polynomial is one of the simple algebraic representations of graphs. The degree polynomial of a graph G of order n is the polynomial Deg(G, x) with the coefficients deg(G,i) where deg(G,i) denotes the number of vertices of degree i in G. In this article, we investigate the behavior of the roots of some families of Graphs in the complex field. We investigate for the graphs having only integral roots. Further, we characterize the graphs having single roots or having real roots and behavior of the polynomial at the particular value is also obtained.

Keywords: degree polynomial, regular graph, minimum and maximum degree, graph operations

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Authors: Kleche Myriam, Ziane Nadia, Berrebbah Houria, Djebar Mohammed Reda

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The Phytoremediation has now become a necessity. Thus, in our work, we are interested in the biological wastewater treatment of Oued Meboudja. The physicochemical analysis of water after treatment showed a significant reduction of suspended matter, COD and BOD5 and rate of metals in roots for example iron and zinc. We also highlighted some significant changes in biometric and physiological parameters such as increasing the number of roots and increased respiratory metabolism through the oxygen consumption in isolated roots of Phragmites australis, placed in a polluted environment.

Keywords: phragmites australis, roots, phytoremediation, iron, zinc

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Authors: Soner Nandappa D., Ahmed Mohammed Naji

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In a graph G = (V, E), the distance from a vertex v to a vertex u is the length of shortest v to u path. The eccentricity e(v) of v is the distance to a farthest vertex from v. The diameter diam(G) is the maximum eccentricity. The k-distance neighborhood of v, for 0 ≤ k ≤ e(v), is Nk(v) = {u ϵ V (G) : d(v, u) = k}. In this paper, we introduce a new distance degree based topological polynomial of a graph G is called a k- distance neighborhood polynomial, denoted Nk(G, x). It is a polynomial with the coefficient of the term k, for 0 ≤ k ≤ e(v), is the sum of the cardinalities of Nk(v) for every v ϵ V (G). Some properties of k- distance neighborhood polynomials are obtained. Exact formulas of the k- distance neighborhood polynomial for some well-known graphs, Cartesian product and join of graphs are presented.

Keywords: vertex degrees, distance in graphs, graph operation, Nk-polynomials

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Authors: P. K. Telengech, K. Monjero, J. Maling’a, A. Nyende, S. Gichuki

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There is need to identify an efficient protocol for use in extraction of high quality DNA for purposes of molecular work. Cassava roots are known for their high starch content, polyphenols and other secondary metabolites which interfere with the quality of the DNA. These factors have negative interference on the various methodologies for DNA extraction. There is need to develop a simple, fast and inexpensive protocol that yields high quality DNA. In this improved Dellaporta method, the storage roots are lyophilized to reduce the water content; the extraction buffer is modified to eliminate the high polyphenols, starch and wax. This simple protocol was compared to other protocols intended for plants with similar secondary metabolites. The method gave high yield (300-950ng) and pure DNA for use in PCR analysis. This improved Dellaporta protocol allows isolation of pure DNA from starchy cassava storage roots.

Keywords: cassava storage roots, dellaporta, DNA extraction, lyophilisation, polyphenols secondary metabolites

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Authors: Ali Dorostkar

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In this paper, a basic schematic of fractional dimensional optimization problem is presented. As will be shown, a method is performed based on a relation between roots and tangent lines of function in fractional dimensions for an arbitrary initial point. It is shown that for each polynomial function with order N at least N tangent lines must be existed in fractional dimensions of 0 < α < N+1 which pass exactly through the all roots of the proposed function. Geometrical analysis of tangent lines in fractional dimensions is also presented to clarify more intuitively the proposed method. Results show that with an appropriate selection of fractional dimensions, we can directly find the roots. Method is presented for giving a different direction of optimization problems by the use of fractional dimensions.

Keywords: tangent line, fractional dimension, root, optimization problem

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Authors: Ivan Baravy

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Interpolation problem, as it was initially posed in terms of polynomials, is well researched. However, further mathematical developments extended it significantly. Trigonometric interpolation is widely used in Fourier analysis, while its generalized representation as exponential interpolation is applicable to such problem of mathematical physics as modelling of Ziegler-Biersack-Littmark repulsive interatomic potentials. Formulated for finite fields, this problem arises in decoding Reed--Solomon codes. This paper shows the relation between different interpretations of the problem through the class of matrices of special structure - Hankel matrices.

Keywords: Berlekamp-Massey algorithm, exponential interpolation, finite fields, Hankel matrices, Hankel polynomials

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Authors: Satyanarayana Engu, Ahmed Mohd, V. Murugan

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We study the large time asymptotics of solutions to the Cauchy problem for a forced Burgers equation (FBE) with the initial data, which is continuous and summable on R. For which, we first derive explicit solutions of FBE assuming a different class of initial data in terms of Hermite polynomials. Later, by violating this assumption we prove the existence of a solution to the considered Cauchy problem. Finally, we give an asymptotic approximate solution and establish that the error will be of order O(t^(-1/2)) with respect to L^p -norm, where 1≤p≤∞, for large time.

Keywords: Burgers equation, Cole-Hopf transformation, Hermite polynomials, large time asymptotics

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Authors: M. W. Sun, Y. Zhang, L. M. Zhang, Z. H. Wang, Z. Q. Chen

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In the realm of flight control, the Proportional- Derivative (PD) control is still widely used for the attitude control in practice, particularly for the pitch control, and the attitude dynamics using PD controller should be investigated deeply. According to the empirical knowledge about the unstable flight dynamics, the control parameter combination conditions to generate sole or finite number of closed-loop oscillations, which is a quite smooth response and is more preferred by practitioners, are presented in analytical or numerical manners. To analyze the effects of the combination conditions of the control parameters, the roots of several polynomials are sought to obtain feasible solutions. These conditions can also be plotted in a 2-D plane which makes the conditions be more explicit by using multiple interval operations. Finally, numerical examples are used to validate the proposed methods and some comparisons are also performed.

Keywords: attitude control, dynamic performance, numerical solving method, interval, unstable flight dynamics

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Authors: S. Mohajer , R. M. Taha, M. Adel

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Developing our knowledge of when pineapple roots grow can lead to improved water, fertilizer applications, and more precise culture management. This paper presents current understanding of morphological traits in pineapple roots, highlighting studies using incubation periods and various solid MS media treated with different sucrose concentrations and pH, which directly assess in vitro environmental factors. Rooting parameters had different optimal sucrose concentrations and incubation periods. All shoots failed to root in medium supplemented with sucrose at 5 g/L and no roots formed within the first 45 days in medium enriched with sucrose at 10 g/L. After 75 days, all shoots rooted in medium enriched with 10 and 20 g/L sucrose. Moreover, MS medium supplied with 20 g/L sucrose resulted in the longest and the highest number of roots with 27.3 mm and 4.7, respectively. Root function, such as capacity for P and N uptake, declined rapidly with root length. As a result, the longer the incubation period, the better the rooting responses would be.

Keywords: environmental factors, in vitro rooting, pineapple, tissue culture

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Authors: Malika Yaici, Kamel Hariche, Tim Clarke

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The relationship between eigenstructure (eigenvalues and eigenvectors) and latent structure (latent roots and latent vectors) is established. In control theory eigenstructure is associated with the state space description of a dynamic multi-variable system and a latent structure is associated with its matrix fraction description. Beginning with block controller and block observer state space forms and moving on to any general state space form, we develop the identities that relate eigenvectors and latent vectors in either direction. Numerical examples illustrate this result. A brief discussion of the potential of these identities in linear control system design follows. Additionally, we present a consequent result: a quick and easy method to solve the polynomial eigenvalue problem for regular matrix polynomials.

Keywords: eigenvalues/eigenvectors, latent values/vectors, matrix fraction description, state space description

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Authors: Abdulali Bashir Ben Saleh

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The powdered beet red roots (PBRR) were used as an adsorbent to remove dyes namely methylene blue dye (as a typical cationic or basic dye) from aqueous solutions. The present study shows that used beet red roots powder exhibit adsorption trend for the dye. The adsorption processes were carried out at various conditions of concentrations, processing time and a wide range of pH between 2.5-11. Adsorption isotherm equations such as Freundlich, and Langmuir were applied to calculate the values of respective constants. Adsorption study was found that the currently introduced adsorbent can be used to remove cationic dyes such as methylene blue from aqueous solutions.

Keywords: beet red root, removal of deys, methylene blue, adsorption

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Authors: Roihatul Mutiah, Sukardiman, Aty Widyawaruyanti

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Calotropis gigantiea (L.) R. Br (Apocynaceae) commonly called as “Biduri” or “giant milk weed” is a well-known weed to many cultures for treating various disorders. Several studies reported that C.gigantea roots has anticancer activity. The main aim of this research was to isolate and identify an active marker compound of C.gigantea roots for quality control purpose of its extract in the development as anticancer natural product. The isolation methods was bioactivity guided column chromatography, TLC, and HPLC. Evaluated anticancer activity of there substances using MTT assay methods. Identification structure active compound by UV, 1HNMR, 13CNMR, HMBC, HMQC spectral and other references. The result showed that the marker active compound was identical as Calotropin.

Keywords: calotropin, Calotropis gigantea, anticancer, marker active

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Authors: Bharti Gupta, V. K. Kukreja

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A comprehensive numerical study is presented for the solution of time-dependent advection diffusion problems by using cubic B-spline collocation method. The linear combination of cubic B-spline basis, taken as approximating function, is evaluated using the zeros of shifted Chebyshev polynomials as collocation points in each element to obtain the best approximation. A comparison, on the basis of efficiency and accuracy, with the previous techniques is made which confirms the superiority of the proposed method. An asymptotic convergence analysis of technique is also discussed, and the method is found to be of order two. The theoretical analysis is supported with suitable examples to show second order convergence of technique. Different numerical examples are simulated using MATLAB in which the 3-D graphical presentation has taken at different time steps as well as different domain of interest.

Keywords: cubic B-spline basis, spectral norms, shifted Chebyshev polynomials, collocation points, error estimates

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Authors: Vorya Shabrandi

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The study factors of poverty and inequality causes in countries is the subject of many scholars and economists in the last century, theorists in various areas of economic science know different factors as the roots of poverty and inequality in Iran after the 1979 revolution. Economists have emphasized political elements and political scientists on political elements. This research reviews the political economy of poverty and corruption in Iran after the revolution. The findings of this research, based on AcemOgluand Robinson theory, show how the institutional structural dependence of Iran's economy to raw has led to the growth of its non-economic economic institutions and its consequence of the continuity of the release and looting economy and poverty and inequality in Iran's political economy Is. This research was carried out using descriptive-analytical and comparative methods. Many economists try to justify the conditions of the country based on war, sanctions; And the external factors, and ... knows. In this study, we tried to examine the roots of poverty and the looting economy of Iran by implementing Research AcemOgluand Robinson on the institutions and roots of poverty. Looking for a framework for understanding why countries, such as Iran, the reason for the difference in revenue in different countries, as well as the poor or wealth of countries, regardless of the non-effective and non-professional institutions, and why inefficient institutions in some countries, such as Iran, such as Iran It remains and does not have a voluntary political powers to change these institutions. Findings The research shows that institutions are broadly the main reason for the roots of the robust and looting economy (poverty and inequality) in Iran.

Keywords: Iran, plunderable (Loot) economy, raw shopping, poverty and inequality, acem oglu and robinson, non-inclusive institutions

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Authors: Natalia D. Vaysfel’d, Zinaida Y. Zhuravlova

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The loaded plane elastic semistrip, the lateral boundaries of which are fixed, is considered. The integral transformations are applied directly to Lame’s equations. It leads to one dimensional boundary value problem in the transformations’ domain which is formulated as a vector one. With the help of the matrix differential calculation’s apparatus and apparatus of Green matrix function the exact solution of a vector problem is constructed. After the satisfying the boundary condition at the semi strip’s edge the problem is reduced to the solving of the integral singular equation with regard of the unknown stress at the semis trip’s edge. The equation is solved with the orthogonal polynomials method that takes into consideration the real singularities of the solution at the ends of integration interval. The normal stress at the edge of the semis trip were calculated and analyzed.

Keywords: semi strip, Green's Matrix, fourier transformation, orthogonal polynomials method

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Authors: M. Fakharian, M. I. Khodakarami

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In this paper, a new trend for improvement in semi-analytical method based on scale boundaries in order to solve the 2D elastodynamic problems is provided. In this regard, only the boundaries of the problem domain discretization are by specific sub-parametric elements. Mapping functions are uses as a class of higher-order Lagrange polynomials, special shape functions, Gauss-Lobatto -Legendre numerical integration, and the integral form of the weighted residual method, the matrix is diagonal coefficients in the equations of elastodynamic issues. Differences between study conducted and prior research in this paper is in geometry production procedure of the interpolation function and integration of the different is selected. Validity and accuracy of the present method are fully demonstrated through two benchmark problems which are successfully modeled using a few numbers of DOFs. The numerical results agree very well with the analytical solutions and the results from other numerical methods.

Keywords: 2D elastodynamic problems, lagrange polynomials, G-L-Lquadrature, decoupled SBFEM

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Authors: Amelyn M. Ambal, Jose Hermis Patricio

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A macro-somatic clonal propagation study was undertaken to determine the effects of method of propagation, rooting hormone, and level of rooting hormone concentration of TBOS (Calophyllum inophyllum Mer. and Pongamia pinnata L.). A factorial experiment in SSSPD with three replications was used in the study and analyzed using ANOVA and LSD. Open mist propagation is effective for rooting Calophyllum inophyllum and Pongamia pinnata cuttings as it gave statistically higher number of adventitious roots, longer length of roots, and higher rooting percentage. C. inophyllum cuttings exhibit statistically higher rooting percentage compared to P. pinnata cuttings when subjected to open mist method and treated with 600 ppm of NAA. NAA is more effective than IBA in terms of number and length of roots, and rooting percentage produced. However, levels of hormone concentration were not generally effective on the rooting performance and shoot production of both species.

Keywords: adventitious roots, Calophyllum, close-mist, macro-somatic clonal propagation, Pongamia, open-mist

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