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Application of RNA Aptamers in Nanotechnology and Therapeutics
Published in Peixuan Guo, Kirill A. Afonin, RNA Nanotechnology and Therapeutics, 2022
Evolution can be visualized in an abstract space of genotypes called sequence space (Maynard Smith 1970). For a population of RNA or DNA sequences with a particular length, the set of all possible sequences is represented in such a space, where each sequence is represented as a point next to a number of other sequences that differ from it at only one position. The capacity of each sequence to carry out a specified function allows the definition of a fitness landscape over sequence space with respect to this function. When the fitness values are plotted against all possible genotypes and their degree of similarity to form a “landscape,” the peaks represent either local or global optima of fitness possessed by the corresponding sequences. In the process of evolution, a population of sequences migrates on this landscape towards the peaks along paths of non-decreasing mean fitness. The notion of sequence space relates the biophysics of evolution to information theory and proves to be illustrative and useful in this context.
Artificial Enzymes
Published in Yubing Xie, The Nanobiotechnology Handbook, 2012
James A. Stapleton, Agustina Rodriguez-Granillo, Vikas Nanda
Multiple sequence alignments of homologous proteins can be used in library generation to provide information about which amino acids are allowed at each position, narrowing down the size of the sequence space to be searched. Bias from the evolutionary history of these sequences can be removed statistically (Halabi et al. 2009) or avoided entirely by selecting competent sequences from synthetic pools (Jäckel et al. 2010). Compatible diversity at each site is then built into synthetic degenerate oligonucleotides, which are assembled by PCR to yield a diverse collection of mutant genes. Designed libraries can also be constructed so as to preserve the correlations between amino acid identities at multiple positions. One such library (Lippow et al. 2010), which maintained the linkages between neighboring amino acid identities in a computationally redesigned active site, was shown to be enriched in active mutants relative to a control library with no interposition information.
Vector Valued Ideal Convergent Generalized Difference Sequence Spaces Associated with Multiplier Sequences
Published in S. A. Mohiuddine, Bipan Hazarika, Sequence Space Theory with Applications, 2023
A sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field K of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in K and can be turned into a vector space under the operations of point wise addition of functions and point wise scalar multiplication. All sequence spaces are linear subspaces of this space.
Bishop–Phelps cones given by an equation in Banach spaces
Published in Optimization, 2023
Truong Xuan Duc Ha, Johannes Jahn
The diagram in Figure 3 illustrates the approach of Proposition 4.4 in the following sense: The space () is renormed so that the Lorentz cone is a BP cone given by an equation. Recall that stands for the norm of ℓ in the dual space of , which equals the sequence space with , and stands for the norm of ℓ in the dual space of equipped with the new norm (see Figure 3).
Fuzzifying topology induced by Morsi fuzzy pseudo-norms
Published in International Journal of General Systems, 2022
Let be the sequence space, and for all . The mapping is defined as follows: It is easy to check that is a fuzzy normed vector space by routine. By Lemma 3.1, a family of left continuous and non-ascending norms may be given as follows: Now take the sequence in X, where . Then foe all b>0.5, , and . It means the sequence is convergent to θ in . On the other hand, for each b<0.5, , and . It implies that the sequence is convergent to θ in whenever b<0.5. By Theorem 4.7, .
Weighted statistical convergence through difference operator of sequences of fuzzy numbers with application to fuzzy approximation theorems
Published in International Journal of General Systems, 2019
S. A. Mohiuddine, Asim Asiri, Bipan Hazarika
Assume that ω is the space of all sequences of fuzzy numbers. Then, the operator (see Kızmaz 1981; Sarıgöl 1987; Et and Çolak 1995; Et, Mursaleen, and Işik 2013; Mohiuddine and Hazarika 2017) is given as and Let U be any sequence space. If then there exists one and only one sequence such that and for sufficiently large m, for instance m>2r, where (see Temizsu, Et, and Çinar 2016).