Brownian ratchet theory refers to the phenomenon that nonequilibrium fluctuations in an isothermal medium and anisotropic system can induce mechanical force and motion. This concept of noise-induced transport has motivated an abundance of theoretical and applied research. One of the exciting applications of the ratchet theory lies in the possible explanation of the operating mode of biological molecular motors. Biomolecular motors are proteins able of converting chemical energy into mechanical motion and force. Because of their dimension, the many small parts that make up molecular motors must operate at energies only a few times greater than those of the thermal baths. The description of molecular motors must be stochastic in nature. Here, we review the theoretical concepts of the Brownian ratchet theory and its possible link to the operation of biomolecular motors. We illustrate the principle of the ratchet theory with models of two molecular motors: a rotary motor (F0F1ATP synthase) and a linear motor (myosin II).

The notion of blossom in extended Chebyshev spaces offers adequate generalizations and extra-utilities to the tools for free-form design schemes. Unfortunately, such advantages are often overshadowed by the complexity of the resulting algorithms. In this work, we show that for the case of Müntz spaces with integer exponents, the notion of Chebyshev blossom leads to elegant algorithms whose complexities are embedded in the combinatorics of Schur functions. We express the blossom and the pseudo-affinity property in Müntz spaces in term of Schur functions. We derive an explicit expression of the Chebyshev-Bernstein basis via an inductive argument on nested Müntz spaces. We also reveal a simple algorithm for the dimension elevation process. Free-form design schemes in Müntz spaces with Young diagrams as shape parameter will be discussed.

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We provide explicit quadrature rules for spaces of C 1 quintic splines with uniform knot sequences over finite domains. The quadrature nodes and weights are derived via an explicit recursion that avoids numerical solvers. Each rule is optimal, that is, requires the minimal number of nodes, for a given function space. For each of n subintervals, generically, only two nodes are required which reduces the evaluation cost by 2/3 when compared to the classical Gaussian quadrature for polynomials over each knot span. Numerical experiments show fast convergence, as n grows, to the "two-third" quadrature rule of Hughes et al. [23] for infinite domains.

We provide explicit expressions for quadrature rules on the space of C 1 cubic splines with non-uniform, symmetrically stretched knot sequences. The quadrature nodes and weights are derived via an explicit recursion that avoids an intervention of any numerical solver and the rule is optimal, that is, it requires minimal number of nodes. Numerical experiments validating the theoretical results and the error estimates of the quadrature rules are also presented.

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