Applications of Differential and
Difference Equations
Dr. Nikunja Bihari Barik
Ph.D (IIT BBS)
Assistant Professor
Vellore Institute of Technology
Andhra Pradesh
February 24, 2023
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Applications of Differential and Difference Equations, Summaries of Computer Science
Various applications of differential and difference equations, including message encryption using the hill cipher, the pagerank algorithm for search engine ranking, the cayley-hamilton theorem, symmetric and skew-symmetric matrices, matrix diagonalization, and quadratic forms. It provides examples and explanations of these concepts, making it a valuable resource for students and researchers interested in the practical applications of these mathematical tools. Presented by dr. Nikunja bihari barik, an assistant professor at the vellore institute of technology in andhra pradesh, india.
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Applications of Differential and
Difference Equations
Dr. Nikunja Bihari Barik
Ph.D (IIT BBS) Assistant Professor Vellore Institute of Technology
Andhra Pradesh
Message Encryption wit Alice and Bob
Message Encryption
Lester Hill invented the Hill Cypher in 1929. A matrix is used to encrypt the messages is known as key matrix. The matrix must be invertible.
Message Encryption
Lester Hill invented the Hill Cypher in 1929. A matrix is used to encrypt the messages is known as key matrix. The matrix must be invertible.
Example Use the matrix
[ 4 1 3 1
] to obtain the Hill cipher encryption for
the plain text message ’UTES’.
Message Encryption
Lester Hill invented the Hill Cypher in 1929. A matrix is used to encrypt the messages is known as key matrix. The matrix must be invertible.
Example Use the matrix
[ 4 1 3 1
] to obtain the Hill cipher encryption for
the plain text message ’UTES’.
Answer UTES =
[ 20 19
] ,
[ 4 18
] . [ 4 1 3 1
] [ 20 19
]
[ 99 79
] mod26 =
[ 21 1
] [ V B
] .
Message Encryption
Lester Hill invented the Hill Cypher in 1929. A matrix is used to encrypt the messages is known as key matrix. The matrix must be invertible.
Example Use the matrix
[ 4 1 3 1
] to obtain the Hill cipher encryption for
the plain text message ’UTES’.
Answer UTES =
[ 20 19
] ,
[ 4 18
] . [ 4 1 3 1
] [ 20 19
]
[ 99 79
] mod26 =
[ 21 1
] [ V B
] . [ 4 1 3 1
] [ 4 18
]
[ 34 30
] mod26 =
[ 8 4
] [ I E
] .
The encryption message is ’VBIE’.
Page-rank algorithm
Example Use the matrix
[ 1 2 1 3
] to obtain the Hill cipher encryption for
the plain text message ’MATHEMATICS’. How does the search engine know which pages are most important?
Page-rank algorithm
Example Use the matrix
[ 1 2 1 3
] to obtain the Hill cipher encryption for
the plain text message ’MATHEMATICS’. How does the search engine know which pages are most important? The Page-rank algorithm was invented by Page and Brin around 1998 and used in the prototype of Google’s search engine.
Page-rank algorithm
Example Use the matrix
[ 1 2 1 3
] to obtain the Hill cipher encryption for
the plain text message ’MATHEMATICS’. How does the search engine know which pages are most important? The Page-rank algorithm was invented by Page and Brin around 1998 and used in the prototype of Google’s search engine. The objective is to estimate the popularity, or the importance, of a web-page, based on the interconnection of the web. (i) A page with more incoming links is more important than a page with less incoming links.
Page-rank algorithm
Example Use the matrix
[ 1 2 1 3
] to obtain the Hill cipher encryption for
the plain text message ’MATHEMATICS’. How does the search engine know which pages are most important? The Page-rank algorithm was invented by Page and Brin around 1998 and used in the prototype of Google’s search engine. The objective is to estimate the popularity, or the importance, of a web-page, based on the interconnection of the web. (i) A page with more incoming links is more important than a page with less incoming links. (ii) A page with a link from a page which is known to be of high importance is also important.
Page-rank algorithm
Let N be the total number of pages. The matrix A = [ aij ] where
aij =
1 L ( j ) ,^ if there is a link from^ j^ to^ i 0 , Otherwise
Page-rank algorithm
Let N be the total number of pages. The matrix A = [ aij ] where
aij =
1 L ( j ) ,^ if there is a link from^ j^ to^ i 0 , Otherwise Example Find the Page rank of the following pages.
Pagerank algorithm
Choose x =
Pagerank algorithm
Choose x =