Sayyidati Khodijah | IAIN Walisongo (original) (raw)
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Papers by Sayyidati Khodijah
Linear Algebra and Its Applications, 2000
The operator trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary ... more The operator trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary invertible matrices A and to arbitrary invertible bounded operators A on a Hilbert space. Some background and motivation for these results is provided.
Il Nuovo Cimento B, 2003
By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way... more By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representation, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalize the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry.
Journal of Mathematical Analysis and Applications, 1993
In order to construct an extension of the complex numbers, we consider an n-dimensional commutati... more In order to construct an extension of the complex numbers, we consider an n-dimensional commutative algebra generated by the n vectors 1, e, ..., en−1 where the fundamental element satisfies the basic relation en = −1. These spaces can be classified according to the values of n: prime number, power of a prime number, general number. The question of the invertibility leads to the definition of a pseudo-norm for which the triangle inequality is not satisfied (the n = 1, 2 cases excepted). When one tries to pass from the polar form the cartesian one, one obtains functions generalizing the usual circular and hyperbolic functions and their inverse. The extended sine and hyperbolic sine functions thus constructed satisfy a determinantal-type relation and they lay the foundation of a new trigonometry for which summation and derivative formulas are given. An extended 2π quantity is defined as the periodicity of the generalized circular functions. This formalism is applied to solve the nth order differential equations (∑n−1i=1 (∂n/∂φni) ± ω) ƒ(φ) = 0. As a further application, the solutions of the n-laplacian operator are derived.
Artificial Intelligence, 1998
Due to the lack of exact quantitative information or the difficulty associated with obtaining or ... more Due to the lack of exact quantitative information or the difficulty associated with obtaining or processing such information, qualitative spatial knowledge representation and reasoning often become an essential means for solving spatial constraint problems as found in science and engineering. This paper presents a computational approach to representing and reasoning about spatial constraints in two-dimensional Euclidean space, where the a priori spatial information is not precisely expressed in quantitative terms. The spatial quantities considered in this work are qualitative distances and qualitative orientation angles. Here, we explicitly define the semantics of these quantities and thereafter formulate a representation of qualitative trigonometry (QTRIG). The resulting QTRIG formalism provides the necessary inference rules for qualitative spatial reasoning. In the paper, we illustrate how the QTRIG relationships can be employed in generating qualitative spatial descriptions in two-dimensional Euclidean geometric problems, and furthermore, how the derived qualitative spatial descriptions can be used to guide a simulated-annealing-based exact quantitative value assignment. Finally, we discuss an application of the proposed spatial reasoning method to the kinematic constraint analysis in computer-aided pre-parametric mechanism design.
Advances in Applied Clifford Algebras, 2001
In the recent monograph [8], G.L. Naber provides an interesting introduction to the special theor... more In the recent monograph [8], G.L. Naber provides an interesting introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. His mathematical model is based on a special indefinite inner product of index one and its associated group of orthogonal transformations (the Lorentz group). Also, in the same monograph, the Hawking, King and McCarthy’s topology [6] is presented. This topology is physically well motivated and has the remarkable property that its homeomorphism group is essentially just the Lorentz group. Starting from the remark that the inner product and the topology above can be generated by the so-called hyperbolic complex numbers, in this paper we introduce and study two-dimensional geometries and physics generated in a similar manner, by the more general so-called complex-type numbers, i.e. of the typez=x+qy,q ∉ ℝ, whereq 2=A+q(2B), A,A, B ∉ ℝ fixed.
... Differences in responses of experimental and control pupils The differences in conceptual dev... more ... Differences in responses of experimental and control pupils The differences in conceptual development are shown most clearly on the delayed post-test which consisted of 5 problems to test more standard trigonometric techniques and 6 to test ...
Geometriae Dedicata, 1990
We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, ... more We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces ℂP n and ℍP n (n≥2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces ℂH n and ℍH n (n≥2).
Mathematical Intelligencer, 2006
SYDNEY, WILD EGG, 2005 (http://wildegg.com) 300pp. AUD79.95 (about $60) HARDBACK ISBN 0-9757492-0-X
Absrrucr-A trigonometry for finite fields is introduced. In particular, the k-trigonometric funct... more Absrrucr-A trigonometry for finite fields is introduced. In particular, the k-trigonometric functions over the Galois Field GF(q) are defined and their main properties derived. This leads to the definition of the cq(.) function over GF(q), which in turn leads to a finite field Hartley Transform ...
Linear Algebra and Its Applications, 2000
The operator trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary ... more The operator trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary invertible matrices A and to arbitrary invertible bounded operators A on a Hilbert space. Some background and motivation for these results is provided.
Il Nuovo Cimento B, 2003
By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way... more By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representation, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalize the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry.
Journal of Mathematical Analysis and Applications, 1993
In order to construct an extension of the complex numbers, we consider an n-dimensional commutati... more In order to construct an extension of the complex numbers, we consider an n-dimensional commutative algebra generated by the n vectors 1, e, ..., en−1 where the fundamental element satisfies the basic relation en = −1. These spaces can be classified according to the values of n: prime number, power of a prime number, general number. The question of the invertibility leads to the definition of a pseudo-norm for which the triangle inequality is not satisfied (the n = 1, 2 cases excepted). When one tries to pass from the polar form the cartesian one, one obtains functions generalizing the usual circular and hyperbolic functions and their inverse. The extended sine and hyperbolic sine functions thus constructed satisfy a determinantal-type relation and they lay the foundation of a new trigonometry for which summation and derivative formulas are given. An extended 2π quantity is defined as the periodicity of the generalized circular functions. This formalism is applied to solve the nth order differential equations (∑n−1i=1 (∂n/∂φni) ± ω) ƒ(φ) = 0. As a further application, the solutions of the n-laplacian operator are derived.
Artificial Intelligence, 1998
Due to the lack of exact quantitative information or the difficulty associated with obtaining or ... more Due to the lack of exact quantitative information or the difficulty associated with obtaining or processing such information, qualitative spatial knowledge representation and reasoning often become an essential means for solving spatial constraint problems as found in science and engineering. This paper presents a computational approach to representing and reasoning about spatial constraints in two-dimensional Euclidean space, where the a priori spatial information is not precisely expressed in quantitative terms. The spatial quantities considered in this work are qualitative distances and qualitative orientation angles. Here, we explicitly define the semantics of these quantities and thereafter formulate a representation of qualitative trigonometry (QTRIG). The resulting QTRIG formalism provides the necessary inference rules for qualitative spatial reasoning. In the paper, we illustrate how the QTRIG relationships can be employed in generating qualitative spatial descriptions in two-dimensional Euclidean geometric problems, and furthermore, how the derived qualitative spatial descriptions can be used to guide a simulated-annealing-based exact quantitative value assignment. Finally, we discuss an application of the proposed spatial reasoning method to the kinematic constraint analysis in computer-aided pre-parametric mechanism design.
Advances in Applied Clifford Algebras, 2001
In the recent monograph [8], G.L. Naber provides an interesting introduction to the special theor... more In the recent monograph [8], G.L. Naber provides an interesting introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. His mathematical model is based on a special indefinite inner product of index one and its associated group of orthogonal transformations (the Lorentz group). Also, in the same monograph, the Hawking, King and McCarthy’s topology [6] is presented. This topology is physically well motivated and has the remarkable property that its homeomorphism group is essentially just the Lorentz group. Starting from the remark that the inner product and the topology above can be generated by the so-called hyperbolic complex numbers, in this paper we introduce and study two-dimensional geometries and physics generated in a similar manner, by the more general so-called complex-type numbers, i.e. of the typez=x+qy,q ∉ ℝ, whereq 2=A+q(2B), A,A, B ∉ ℝ fixed.
... Differences in responses of experimental and control pupils The differences in conceptual dev... more ... Differences in responses of experimental and control pupils The differences in conceptual development are shown most clearly on the delayed post-test which consisted of 5 problems to test more standard trigonometric techniques and 6 to test ...
Geometriae Dedicata, 1990
We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, ... more We define a fourth basic invariant, which, besides the lengths of the three sides of a triangle, determines a triangle in the complex and quaternion projective spaces ℂP n and ℍP n (n≥2) uniquely up to isometry. We give inequalities describing the exact range of the four basic invariants. We express the angular invariants of a triangle with our basic invariants, giving a new completely elementary proof of the laws of trigonometry. As a corollary we derive a large number of congruence theorems. Finally we get, in exactly the same way, the corresponding results for triangles in the complex and quaternion hyperbolic spaces ℂH n and ℍH n (n≥2).
Mathematical Intelligencer, 2006
SYDNEY, WILD EGG, 2005 (http://wildegg.com) 300pp. AUD79.95 (about $60) HARDBACK ISBN 0-9757492-0-X
Absrrucr-A trigonometry for finite fields is introduced. In particular, the k-trigonometric funct... more Absrrucr-A trigonometry for finite fields is introduced. In particular, the k-trigonometric functions over the Galois Field GF(q) are defined and their main properties derived. This leads to the definition of the cq(.) function over GF(q), which in turn leads to a finite field Hartley Transform ...