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Whole protein structural convergence is not thought to occur but some convergence of pockets and secondary structural elements have been documented. Some secondary structure convergence occurs due to some residues favouring being in α-helix (helical propensity) and for hydrophobic patches or pocket to be formed at the ends of the parallel sheets.
The methods applied to infer convergent evolution depend on whether pattern-based or process-based convergence is expected. Pattern-based convergence is the broader term, for when two or more lineages independently evolve patterns of similar traits. Process-based convergence is when the convergence is due to similar forces of natural selection ...
The different notions of convergence capture different properties about the sequence, with some notions of convergence being stronger than others. For example, convergence in distribution tells us about the limit distribution of a sequence of random variables. This is a weaker notion than convergence in probability, which tells us about the ...
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).
Convergence implies "Cauchy convergence", and Cauchy convergence, together with the existence of a convergent subsequence implies convergence. The concept of completeness of metric spaces, and its generalizations is defined in terms of Cauchy sequences.
Convergence proof techniques are canonical patterns of mathematical proofs that sequences or functions converge to a finite limit when the argument tends to infinity.. There are many types of sequences and modes of convergence, and different proof techniques may be more appropriate than others for proving each type of convergence of each type of sequence.
An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence. This is because a power series is absolutely convergent on the interior of its disk of convergence. [a]
There are many examples of countries that have converged with developed countries which validate the catch-up theory. [5] Based on case studies on Japan, Mexico and other countries, Nakaoka studied social capabilities for industrialization and clarified the features of human and social attitudes in the catching-up process of Japan in the Meiji period (1868-1912).