A significant part of everyday learning occurs incidentally — a process typically described as implicit learning. A central issue in this domain and others, such as language acquisition, is the extent to which performance depends on the acquisition and deployment of abstract rules. Shanks and colleagues [22], [11] have suggested (1) that discrimination between grammatical and ungrammatical inst...

Let Pn and Qn be the polynomials obtained by repeated differentiation of the tangent and secant functions respectively. From the exponential generating functions of these polynomials we develop relations among their values, which are then applied to various numerical sequences which occur as values of the Pn and Qn. For example, Pn(0) and Qn(0) are respectively the nth tangent and secant number...

Let P n and Q n be the polynomials obtained by repeated diierentiation of the tangent and secant functions respectively. From the exponential generating functions of these polynomials we develop relations among their values, which are then applied to various numerical sequences which occur as values of the P n and Q n. For example, P n (0) and Q n (0) are respectively the nth tangent and secant...

A cardinal feature of freezing of gait (FOG) is high frequency (3-8 Hz) oscillation of the legs, and this study aimed to quantify the temporal pattern of lower-body motion prior to and during FOG. Acceleration data was obtained from sensors attached to the back, thighs, shanks, and feet in 14 Parkinson's disease patients performing timed-up-and-go tasks, and clinical assessment of FOG was perfo...

In this note we consider an autonomous Tonelli Lagrangian L on a closed manifold M , that is, a C2 function L : TM −→ R such that L is fiberwise strictly convex and superlinear. Then the Euler-Lagrange equation associated with L defines a complete flow φt on TM . Define Minv to be the set of Φt-invariant, compactly supported, Borel probability measures on TM . Mather showed that the function (c...

We consider a smooth Tonelli Lagrangian L : TTn → R and its viscosity solutions u(x, P ) characterized by the cell equation H(x, P + Dxu(x, P )) = H(P ), where H : T ∗Tn → R is the Hamiltonian associated with L. We will show that if P0 corresponds to a quasi-periodic Lagrangian invariant torus, then Dxu(x, P ) is Hölder continuous in P at P0 with Hölder exponent arbitrarily close to 1, and if b...

Let L be an autonomous Tonelli Lagrangian on a closed manifold of dimension two. Let C be the set of cohomology classes whose Mather set consists of periodic orbits, none of which is a fixed point. Then for almost all c in C, the Aubry set of c equals the Mather set of c.

Like all spiders, tarantulas (family Theraphosidae) synthesize silk in specialized glands and extrude it from spinnerets on their abdomen. In one species of large tarantula, Aphonopelma seemanni, it has been suggested that silk can also be secreted from the tarsi but this claim was later refuted. We provide evidence of silk secretion directly from spigots (nozzles) on the tarsi of three distant...

Let [Sn] n=0 be such that Sn tS+ j=1 aj* n j as n , with 1>|*1 |> |*2 |> } } } , such that limj *j=0. A well-known result by Wynn states that when the Shanks transformation or its equivalent =-algorithm is applied to [Sn] n=0 , then = (n) 2k &Stak+1[> k i=1 (*k+1&*i) (1&*i)] 2 *k+1 as n . In the present work we extend this result (i) by allowing some of the *j to have the same modulus and (ii) ...

We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or, equivalently, the left derivative of a function is the dual of the right derivative of the dual function. Such duality between left and right fractional operat...