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We generalize Hermite interpolation with error correction, which is the methodology for multiplicity algebraic error correction codes, to Hermite interpolation of a rational function over a field K from function and function derivative values. We present an interpolation algorithm that can locate and correct <= E errors at distinct arguments y in K where at least one of the values or values of a derivative is incorrect. The upper bound E for the number of such y is input. Our algorithm sufficiently oversamples the rational function to guarantee a unique interpolant. We sample (f/g)^(j)(y[i]) for 0 <= j <= L[i], 1 <= i <= n, y[i] distinct, where (f/g)^(j) is the j-th derivative of the rational function f/g, f, g in K[x], GCD(f,g)=1, g <= 0, and where N = (L[1]+1)+...+(L[n]+1) >= C + D + 1 + 2(L[1]+1) + ... + 2(L[E]+1) where C is an upper bound for deg(f) and D an upper bound for deg(g), which are input to our algorithm. The arguments y[i] can be poles, which is truly or falsely indicated by a function value infinity with the corresponding L[i]=0. Our results remain valid for fields K of characteristic >= 1 + max L[i]. Our algorithm has the same asymptotic arithmetic complexity as that for classical Hermite interpolation, namely soft-O(N). For polynomials, that is, g=1, and a uniform derivative profile L[1] = ... = L[n], our algorithm specializes to the univariate multiplicity code decoder that is based on the 1986 Welch-Berlekamp algorithm.
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Molecular understanding for large deformations of soft bottlebrush polymer networks
Networks formed by crosslinking bottlebrush polymers are a class of soft materials with stiffnesses matching that of ‘watery’ hydrogels and biological tissues but contain no solvents. Because of their extreme softness, bottlebrush polymer networks are often subject to large deformations. However, it is poorly understood how molecular architecture determines the extensibility of the networks. Using a combination of experimental and theoretical approaches, we discover that the yield strain γ y of the network equals the ratio of the contour length L max to the end-to-end distance R of the bottlebrush between two neighboring crosslinks: γ y = L max / R − 1. This relation suggests two regimes: (1) for stiff bottlebrush polymers, γ y is inversely proportional to the network shear modulus G , γ y ∼ G −1 , which represents a previously unrecognized regime; (2) for flexible bottlebrush polymers, γ y ∼ G −1/2 , which recovers the behavior of conventional polymer networks. Our findings provide a new molecular understanding of the nonlinear mechanics for soft bottlebrush polymer networks.
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- Award ID(s):
- 1944625
- PAR ID:
- 10204958
- Date Published:
- Journal Name:
- Soft Matter
- Volume:
- 16
- Issue:
- 27
- ISSN:
- 1744-683X
- Page Range / eLocation ID:
- 6259 to 6264
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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<sc>A</sc>bstract We introduce a simple synthetic strategy to selectively degrade bottlebrush networks derived from well‐defined poly(4‐methylcaprolactone) (P4MCL) bottlebrush polymers. Functionalization of the hydroxyl groups present at the terminal ends of P4MCL side chains withα‐lipoic acid resulted in bottlebrush polymers having a range of molecular weights (Mn = 45–2200 kg mol−1) and a tunable number of reactive dithiolane chain ends. These functionalized chain ends act as efficient crosslinkers due to radical ring‐opening of the dithiolane rings under UV light. The resulting redox‐active disulfide crosslinks enable mild electrochemical or chemical degradation of the SS crosslinks to regenerate the starting bottlebrush polymer. P4MCL side chains and the disulfides can be degraded simultaneously using harsher reducing conditions. This combination of bottlebrush architecture with facile disulfide crosslinking presents a versatile platform for preparing highly tunable elastomers that undergo controlled degradation under mild conditions.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1039/D0SM00759E
