Süleyman Şafak - Academia.edu (original) (raw)

Papers by Süleyman Şafak

Research paper thumbnail of The Coefficients of the Polynomial Interpolation in Terms of Finite Differences and Numerical Differentiations

Journal of Informatics and Mathematical Sciences, 2012

In this note, the polynomial interpolation of degree nnn passing through n+1n+1n+1 distinct points is... more In this note, the polynomial interpolation of degree nnn passing through n+1n+1n+1 distinct points is considered. The coefficients of the polynomial interpolation are investigated in terms of finite differences and numerical differentiations. The coefficients are formulated by the use of divided differences and correlated with forward, backward differences and numerical differentiations. It is seen that the coefficients of the polynomial interpolation can be found and computed by using finite differences, numerical differentiations and generating special formulae for equidistant points or not.

Research paper thumbnail of On the Trivariate Polynomial Interpolation

Abstract: This paper is concerned with the formulae for computing the coefficients of the trivari... more Abstract: This paper is concerned with the formulae for computing the coefficients of the trivariate polynomial interpolation (TPI) passing through ) 1 )( 1 )( 1 ( + + + r n m distinct points in the solid rectangular region. The TPI is formulated as a matrix equation using Kronecker product and Khatri-Rao product of the matrices and the coefficients of the TPI are computed using the generalized inverse of a matrix. In addition, the closed formulae of the coefficients of the bivariate and univariate polynomial interpolations are obtained by the use of the inverse of the Vandermonde matrix. It is seen that the trivariate polynomial interpolation can be investigated as the matrix equation and the coefficients of the TPI can be computed directly from the solution of the matrix equation. Also, it is shown that the bivariate polynomial interpolation (BPI) is the special case of the TPI when 0 = r . Numerical examples are represented.

Research paper thumbnail of Comparing Mohr Coulomb and Drucker Prager function in three dimensional analysis on rock

Rapid development is happening in the solution of engineering problems in recent years. The most ... more Rapid development is happening in the solution of engineering problems in recent years. The most important of all, develops in the area of computer software with no doubt. There are many programs that are finite element, finite different boundary element based. Some of these programmes use the Mohr-Coulomb failure criterion for the purpose of mining problems. This function is not

Research paper thumbnail of Determination of instantaneous breaking rate by Geological Strength Index, Block Punch Index and power of impact hammer for various rock mass conditions

Tunnelling and Underground Space Technology, 2011

... Aksoy a , Corresponding Author Contact Information , E-mail The Corresponding Author , V. Oza... more ... Aksoy a , Corresponding Author Contact Information , E-mail The Corresponding Author , V. Ozacar a , N. Demirel b , SC Ozer a and S ... Singh et al., 2002), Schmitd Hammer rebound values, rock strength, and excavation performance estimations (Goktan and Gunes, 2005), using ...

Research paper thumbnail of An updated formula and method to predict the performance of impact hammers

International Journal of Rock Mechanics and Mining Sciences, 2013

Research paper thumbnail of The Best Approximate Solution of the Inconsistent Linear System via a Quadratic Penalty Function

Bulletin of the Malaysian Mathematical Sciences Society, 2014

In this paper, the inconsistent linear system of m equations in n unknowns is formulated as a qua... more In this paper, the inconsistent linear system of m equations in n unknowns is formulated as a quadratic programming problem, and the best approximate solution with the minimum norm for the inconsistent system of the linear equations is investigated using the optimality conditions of the quadratic penalty function (QPF). In addition, several algebraic characterizations of the equivalent cases of the QPF are given using the orthogonal decompositions of the coefficient matrices obtained from optimality conditions, and analytic results we obtained are satisfied with numerical examples.

Research paper thumbnail of Further results on the algebraic characterizations of a minimum cost network flow problem equivalent to the transportation problem

International Journal of Computer Mathematics, 2005

In this study a minimum cost network flow problem with m+n+2 nodes and mn arcs, which is equivale... more In this study a minimum cost network flow problem with m+n+2 nodes and mn arcs, which is equivalent to the transportation problem with m sources and n destinations, is described as an axial four-index transportation problem of order 1×m×n×1. Several algebraic characterizations of this problem of order 1×m×n×1 are investigated using the singular value decomposition and generalized inverses of its

Research paper thumbnail of Some new results on the algebraic characterizations of an equality constrained optimization problem equivalent to the transportation problem

International Journal of Computer Mathematics, 2007

An equality constrained optimization problem equivalent to the transportation problem with m sour... more An equality constrained optimization problem equivalent to the transportation problem with m sources and n destinations is described. The optimality condition and some algebraic characterizations of the problem are investigated using its Hessian matrix. In addition, several algebraic characterizations of an equivalent case of the transportation problem are given using the spectral decomposition and generalized inverses of its coefficient matrix.

Research paper thumbnail of Yari Pozi̇ti̇f Tanimli Matri̇sleri̇n İz Eşi̇tsi̇zli̇kleri̇

... Dosya Biçimi: PDF. Dil: Türkçe. Konu(lar): TEKNOLOJİ,. Hakemli mi?: Evet. Yazar(lar): ÖZEL, M... more ... Dosya Biçimi: PDF. Dil: Türkçe. Konu(lar): TEKNOLOJİ,. Hakemli mi?: Evet. Yazar(lar): ÖZEL, MUSTAFA (Yazar), BULUT, HAMZA (Yazar), ŞAFAK, SÜLEYMAN (Yazar), Çeven, Yılmaz (Yazar),. Emeği Geçen(ler): URL: http://deu.mitosweb.com/browse/47650/. ...

Research paper thumbnail of Dağitim Problemi̇ni̇n Opti̇malli̇k Koşullarinin İncelenmesi̇

Bu calismada, m cikis ve n varisli bir dagitim probleminin optimallik kosullari, Lagrange fonksiy... more Bu calismada, m cikis ve n varisli bir dagitim probleminin optimallik kosullari, Lagrange fonksiyonu ve Hessian matrisinin ozellikleri kullanilarak incelenmistir. Problemin ve indirgenmis halinin ayni cebirsel ozelliklere sahip oldugu gorulmustur.

Research paper thumbnail of An analytical method for computing the coefficients of the trivariate polynomial approximation

This paper deals with computing the coefficients of the trivariate polynomial approximation (TPA)... more This paper deals with computing the coefficients of the trivariate polynomial approximation (TPA) of large distinct points given on . 3  The TPA is formulated as a matrix equation using Kronecker and Khatri-Rao products of the matrices. The coefficients of the TPA are computed using the generalized inverse of the matrix. It is seen that the trivariate polynomial approximation can be investigated as the matrix equation and the coefficients of the TPA can be computed directly from the solution of the matrix equation.

Research paper thumbnail of The Coefficients of the Polynomial Interpolation in Terms of Finite Differences and Numerical Differentiations

Journal of Informatics and Mathematical Sciences, 2012

In this note, the polynomial interpolation of degree nnn passing through n+1n+1n+1 distinct points is... more In this note, the polynomial interpolation of degree nnn passing through n+1n+1n+1 distinct points is considered. The coefficients of the polynomial interpolation are investigated in terms of finite differences and numerical differentiations. The coefficients are formulated by the use of divided differences and correlated with forward, backward differences and numerical differentiations. It is seen that the coefficients of the polynomial interpolation can be found and computed by using finite differences, numerical differentiations and generating special formulae for equidistant points or not.

Research paper thumbnail of On the Trivariate Polynomial Interpolation

Abstract: This paper is concerned with the formulae for computing the coefficients of the trivari... more Abstract: This paper is concerned with the formulae for computing the coefficients of the trivariate polynomial interpolation (TPI) passing through ) 1 )( 1 )( 1 ( + + + r n m distinct points in the solid rectangular region. The TPI is formulated as a matrix equation using Kronecker product and Khatri-Rao product of the matrices and the coefficients of the TPI are computed using the generalized inverse of a matrix. In addition, the closed formulae of the coefficients of the bivariate and univariate polynomial interpolations are obtained by the use of the inverse of the Vandermonde matrix. It is seen that the trivariate polynomial interpolation can be investigated as the matrix equation and the coefficients of the TPI can be computed directly from the solution of the matrix equation. Also, it is shown that the bivariate polynomial interpolation (BPI) is the special case of the TPI when 0 = r . Numerical examples are represented.

Research paper thumbnail of Comparing Mohr Coulomb and Drucker Prager function in three dimensional analysis on rock

Rapid development is happening in the solution of engineering problems in recent years. The most ... more Rapid development is happening in the solution of engineering problems in recent years. The most important of all, develops in the area of computer software with no doubt. There are many programs that are finite element, finite different boundary element based. Some of these programmes use the Mohr-Coulomb failure criterion for the purpose of mining problems. This function is not

Research paper thumbnail of Determination of instantaneous breaking rate by Geological Strength Index, Block Punch Index and power of impact hammer for various rock mass conditions

Tunnelling and Underground Space Technology, 2011

... Aksoy a , Corresponding Author Contact Information , E-mail The Corresponding Author , V. Oza... more ... Aksoy a , Corresponding Author Contact Information , E-mail The Corresponding Author , V. Ozacar a , N. Demirel b , SC Ozer a and S ... Singh et al., 2002), Schmitd Hammer rebound values, rock strength, and excavation performance estimations (Goktan and Gunes, 2005), using ...

Research paper thumbnail of An updated formula and method to predict the performance of impact hammers

International Journal of Rock Mechanics and Mining Sciences, 2013

Research paper thumbnail of The Best Approximate Solution of the Inconsistent Linear System via a Quadratic Penalty Function

Bulletin of the Malaysian Mathematical Sciences Society, 2014

In this paper, the inconsistent linear system of m equations in n unknowns is formulated as a qua... more In this paper, the inconsistent linear system of m equations in n unknowns is formulated as a quadratic programming problem, and the best approximate solution with the minimum norm for the inconsistent system of the linear equations is investigated using the optimality conditions of the quadratic penalty function (QPF). In addition, several algebraic characterizations of the equivalent cases of the QPF are given using the orthogonal decompositions of the coefficient matrices obtained from optimality conditions, and analytic results we obtained are satisfied with numerical examples.

Research paper thumbnail of Further results on the algebraic characterizations of a minimum cost network flow problem equivalent to the transportation problem

International Journal of Computer Mathematics, 2005

In this study a minimum cost network flow problem with m+n+2 nodes and mn arcs, which is equivale... more In this study a minimum cost network flow problem with m+n+2 nodes and mn arcs, which is equivalent to the transportation problem with m sources and n destinations, is described as an axial four-index transportation problem of order 1×m×n×1. Several algebraic characterizations of this problem of order 1×m×n×1 are investigated using the singular value decomposition and generalized inverses of its

Research paper thumbnail of Some new results on the algebraic characterizations of an equality constrained optimization problem equivalent to the transportation problem

International Journal of Computer Mathematics, 2007

An equality constrained optimization problem equivalent to the transportation problem with m sour... more An equality constrained optimization problem equivalent to the transportation problem with m sources and n destinations is described. The optimality condition and some algebraic characterizations of the problem are investigated using its Hessian matrix. In addition, several algebraic characterizations of an equivalent case of the transportation problem are given using the spectral decomposition and generalized inverses of its coefficient matrix.

Research paper thumbnail of Yari Pozi̇ti̇f Tanimli Matri̇sleri̇n İz Eşi̇tsi̇zli̇kleri̇

... Dosya Biçimi: PDF. Dil: Türkçe. Konu(lar): TEKNOLOJİ,. Hakemli mi?: Evet. Yazar(lar): ÖZEL, M... more ... Dosya Biçimi: PDF. Dil: Türkçe. Konu(lar): TEKNOLOJİ,. Hakemli mi?: Evet. Yazar(lar): ÖZEL, MUSTAFA (Yazar), BULUT, HAMZA (Yazar), ŞAFAK, SÜLEYMAN (Yazar), Çeven, Yılmaz (Yazar),. Emeği Geçen(ler): URL: http://deu.mitosweb.com/browse/47650/. ...

Research paper thumbnail of Dağitim Problemi̇ni̇n Opti̇malli̇k Koşullarinin İncelenmesi̇

Bu calismada, m cikis ve n varisli bir dagitim probleminin optimallik kosullari, Lagrange fonksiy... more Bu calismada, m cikis ve n varisli bir dagitim probleminin optimallik kosullari, Lagrange fonksiyonu ve Hessian matrisinin ozellikleri kullanilarak incelenmistir. Problemin ve indirgenmis halinin ayni cebirsel ozelliklere sahip oldugu gorulmustur.

Research paper thumbnail of An analytical method for computing the coefficients of the trivariate polynomial approximation

This paper deals with computing the coefficients of the trivariate polynomial approximation (TPA)... more This paper deals with computing the coefficients of the trivariate polynomial approximation (TPA) of large distinct points given on . 3  The TPA is formulated as a matrix equation using Kronecker and Khatri-Rao products of the matrices. The coefficients of the TPA are computed using the generalized inverse of the matrix. It is seen that the trivariate polynomial approximation can be investigated as the matrix equation and the coefficients of the TPA can be computed directly from the solution of the matrix equation.