علم الجنین و تخلیق جنین کے مختلف ادوار و مراحل قرآن کریم اور سائنس کے تناظر میں

Embryology is such an academic discipline which was based upon the Qura’nic revelation purely and its details were provided by The Holy Prophet (SAW) whereas the scientists remained totally unaware of its intricate details until twentieth century. It was the time when they discovered various stages of the creation and development of fetus inside the mother’s womb and after examination of all these stages through various scientific instruments they openly admitted that Qura’nic verses about fetal development are absolutely accurate. Their acknowledgement is a strong evidence of the authenticity and veracity of The Holy Qura’n for scientific minded people. Not only this but these embryologists also admitted that the information provided in the Holy Qura’n and the A╒adith of The Holy Prophet (SAW) helped them a great deal in formulating the basic hypotheses of their research. So, these scientific discoveries are clear proofs of the miraculous character of Holy Qura’n. The current article is an effort to elaborate the link of modern embryology with the details given in the sacred literature of Islam.

Standard Bases and Primary Decomposition in Polynomial Ring With Coefficients in Rings

The theory of standard bases in polynomial rings with coefficients in a ring A with respect to local orderings is developed. A is a commutative Noetherian ring with 1 and we assume that linear equations are solvable in A. Then the generalization of Faug ́ere F4-algorithm for polynomial rings with coefficients in Euclidean rings is given. This algorithm computes successively a Gr ̈obner basis replacing the reduction of one single s-polynomial in Buchberger’s algorithm by the simultaneous reduction of several polynomials. And finally we present an algorithm to compute a primary decomposition of an ideal in a polynomial ring over the integers. For this purpose we use algorithms for primary decomposition in polynomial rings over the rationals resp. over finite fields, and the idea of Shimoyama–Yokoyama resp. Eisenbud–Hunecke–Vasconcelos to extract primary ideals from pseudo-primary ideals. A parallelized version of the algorithm is implemented in Singular.