====== Dr. Junaid Alam Khan ======
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This is a personal web page of \\
**Dr. Junaid Alam Khan**\\
Associate Professor\\
Institute of Business Administration, Karachi - PAKISTAN.

IBA Profile: https://oric.iba.edu.pk/profile.php?id=jakhan \\
ResearchGate Profile: https://www.researchgate.net/scientific-contributions/2141923316_Junaid_Alam_Khan \\
Facebook: https://www.facebook.com/junaid.a.khan.58511

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{{ :dr-junaid-alam-khan.jpg?nolink |Dr Junaid Alam Khan}}

==== Procedure to Compute Sasbi-Standard Bases====
Let A=B_> be a localization of polynomial subalgebra B with respect to a local monomial ordering >. For a polynomial vector f in (R_>)^n (R_> is a localization of ring R with respect to >) and a finite set of polynomials vectors I in a module (A)^n, the following procedure computes a Sasbi-Standard weak normal form of f with respect to I over A.

  * [[mathcity>files/junaid/Sasbi-Standard_Bases_of_Modules-Library.txt|Download Code in Text file]]

====Procedure to Classify the Hypersurface Singularities of Corank 3 in Positive Characteristics====

  * [[mathcity>files/junaid/Classify-Procedure.txt|Download Code in Text file]]

====Contact Map Germs====

  * [[mathcity>files/junaid/Contact-Map-Germs.txt|Download Code in Text file]]

====Procedure to classify the right unimodal and bimodal Hypersurface singularities in positive characteristic by invariants====

  * [[mathcity>files/junaid/Right-uni-and-bimodal-in-+ve-char.txt|Download Code in Text file]]

====Procedure to classify the stably simple curve singularities====
Remarks: Compute the Sagbi- basis of the Module. Compute the Semi-Group of the Algebra provided the input is Sagbi Bases of the Algebra. Compute the Semi-Module provided that the inputs are the Sagbi Bases of the Algebra resp. Module.

  * [[mathcity>files/junaid/ClassifierSS.txt|Download Code in Text file]]

====Procedures to Compute SH-bases of subalgebra ====

  * [[mathcity>files/junaid/SH-basis_procedures.txt|Download Code in Text file]]
==== Procedure to Compute Sasbi Bases =====
Let A=B_> be a localization of a polynomial subalgebra B with respect to a local monomial ordering >. For a polynomial f of R_> (a localization of ring R with respect to >) and a finite set of polynomials I in A,  the following procedure computes a weak Sasbi normal form of f with respect to A.

  * [[mathcity>files/junaid/Sasbi_Bases-Library.txt|Download Code in Text file]]

==== Further on Sagbi Basis Under Composition ====

  * [[mathcity>files/junaid/Further_on_Sagbi_Basis_Under_Composition.txt|Download Code in TXT file]]