Renaturation and hybridization reactions lead to the pairing of complementary single-stranded nucleic acids. We present here a theoretical investigation of the mechanism of these reactions in vitro under thermal conditions (dilute solutions of single-stranded chains, in the presence of molar concentrations of monovalent salts and at elevated temperatures). The mechanism follows a Kramers' proce...

The presence of a fractional-exponent growth law relating knot energy and knot topology is known to be an essential characteristic for the existence of ‘ideal’ knots. In this paper, we show that the energy infimum EN stratified at the Hopf charge N of the knot energy of the Faddeev type induced from the Hopf fibration S/S (nR1) in general dimensions obeys the sharp fractional-exponent growth la...

and Applied Analysis 3 Table 1: The coding sequences of the first exon of β-globin gene of different species. Species Coding sequence ATGGTGCACCTGACTCCTGAGGAGAAGTCTGCCGT Human TACTGCCCTGTGGGGCAAGGTGAACGTGGATTAAG TTGGTGGTGAGGCCCTGGGCAG ATGCTGACTGCTGAGGAGAAGGCTGCCGTCACCGG Goat CTTCTGGGGCAAGGTGAAAGTGGATGAAGTTGGTG CTGAGGCCCTGGGCAG ATGGTGCACTTGACTTCTGAGGAGAAGAACTGCA Opossum TCACTACCATCTGGTCTAAGGTGCA...

If F is a family of graphs containing a bipartite graph, then for some δ > 0, ex(n,F) < n2−δ by the Kövari-Sós-Turán Theorem. Erdős and Simonovits [5] conjectured that for every finite family graph F of graphs, there exists α = α(F) such that ex(n,F) = Θ(nα). This α is called the exponent of F . If F is a bipartite graph containing a cycle, then in general it is not known if F = {F} has an expo...

Based on the Chinese Remainder Theorem (CRT), Quisquater and Couvreur proposed an RSA variant, RSA-CRT, to speedup RSA decryption. According to RSA-CRT, Wiener suggested another RSA variant, Rebalanced RSA-CRT, to further speedup RSA-CRT decryption by shifting decryption cost to encryption cost. However, such an approach will make RSA encryption very time-consuming because the public exponent e...

Then choose an integer e with 1 < e < φ(n) and gcd (e, φ(n)) = 1. Then compute the integer d with 1 < d < φ(n) and de = 1modφ(n). Since gcd (e, φ(n)) = 1, such a number d exists. It can be computed by Extended Euclidean Algorithm. Public key is pair (n, e) and Private key is (d, p, q). n is called the RSA modulus ,e is encryption exponent and d is decryption exponent. Encryption: Let x ∈ Zn be ...

We say a number is flat if it can be written as a non-trivial power of 2 times an odd squarefree number. The power is the “exponent” and the number of odd primes the “length”. Let N be flat and 4-perfect with exponent a and length m. If a 6≡ 1 mod 12, then a is even. If a is even and 3 ∤ N then m is also even. If a ≡ 1 mod 12 then 3 | N and m is even. If N is flat and 3-perfect and 3 ∤ N , then...