67 research outputs found
On the non-holonomic character of logarithms, powers, and the n-th prime function
We establish that the sequences formed by logarithms and by "fractional"
powers of integers, as well as the sequence of prime numbers, are
non-holonomic, thereby answering three open problems of Gerhold [Electronic
Journal of Combinatorics 11 (2004), R87]. Our proofs depend on basic complex
analysis, namely a conjunction of the Structure Theorem for singularities of
solutions to linear differential equations and of an Abelian theorem. A brief
discussion is offered regarding the scope of singularity-based methods and
several naturally occurring sequences are proved to be non-holonomic.Comment: 13 page
Codes for DNA Storage Channels
We consider the problem of assembling a sequence based on a collection of its
substrings observed through a noisy channel. The mathematical basis of the
problem is the construction and design of sequences that may be discriminated
based on a collection of their substrings observed through a noisy channel. We
explain the connection between the sequence reconstruction problem and the
problem of DNA synthesis and sequencing, and introduce the notion of a DNA
storage channel. We analyze the number of sequence equivalence classes under
the channel mapping and propose new asymmetric coding techniques to combat the
effects of synthesis and sequencing noise. In our analysis, we make use of
restricted de Bruijn graphs and Ehrhart theory for rational polytopes.Comment: 32 pages, 5 figure
Restricting SBH Ambiguity via Restriction Enzymes
Abstract. The expected number of n-base long sequences consistent with a given SBH spectrum grows exponentially with n, which severely limits the potential range of applicability of SBH even in an error-free setting. Restriction enzymes (RE) recognize specific patterns and cut the DNA molecule at all locations of that pattern. The output of a restriction assay is the set of lengths of the resulting fragments. By augmenting the SBH spectrum with the target string’s RE spectrum, we can eliminate much of the ambiguity of SBH. In this paper, we build on [20] to enhance the resolving power of restriction enzymes. We give a hardness result for the SBH+RE problem, and supply improved heuristics for the existing backtracking algorithm. We prove a lower bound on the number restric-tion enzymes required for unique reconstruction, and show experimental results that are not far from this bound.
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