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We construct a broad class of solutions of the KP-I equation by using a reduced version of the Grammian form of the $\tau$-function. The basic solution is a linear periodic chain of lumps propagating with distinct group and wave velocities. More generally, our solutions are evolving linear arrangements of lump chains, and can be viewed as the KP-I analogues of the family of line-soliton solutions of KP-II. However, the linear arrangements that we construct for KP-I are more general, and allow degenerate configurations such as parallel or superimposed lump chains. We also construct solutions describing interactions between lump chains and individual lumps, and discuss the relationship between the solutions obtained using the reduced and regular Grammian forms.

We have utilized tau-assembled and tau-stabilized microtubules (MTs), in the absence of taxol, to investigate the effects of tau isoforms with three and four MT binding repeats upon kinesin-driven MT gliding. MTs were assembled in the presence of either 3-repeat tau (3R tau) or 4-repeat tau (4R tau) at tau:tubulin dimer molar ratios that approximate those found in neurons. MTs assembled with 3R tau glided at 31.1 μm/min versus 25.8 μm/min for 4R tau, a statistically significant 17% difference. Importantly, the gliding rates for either isoform did not change over a fourfold range of tau concentrations. Further, tau-assembled MTs underwent minimal dynamic instability behavior while gliding and moved with linear trajectories. In contrast, MTs assembled with taxol in the absence of tau displayed curved gliding trajectories. Interestingly, addition of 4R tau to taxol-stabilized MTs restored linear gliding, while addition of 3R tau did not. The data are consistent with the ideas that (i) 3R and 4R tau-assembled MTs possess at least some isoform-specific features that impact upon kinesin translocation, (ii) tau-assembled MTs possess different structural features than do taxol-assembled MTs, and (iii) some features of tau-assembled MTs can be masked by prior assembly by taxol. The differences in kinesin-driven gliding between 3R and 4R tau suggest important features of tau function related to the normal shift in tau isoform composition that occurs during neural development as well as in neurodegeneration caused by altered expression ratios of otherwise normal tau isoforms.

It is shown how many of the partial theta function identities in Ramanujan's lost notebook can be generalized to infinite families of such identities. Key in our construction is the Bailey lemma and a new generalization of the Jacobi triple product identity. By computing residues around the poles of our identities we find a surprising connection between partial theta function identities and Garrett?Ismail?Stanton-type extensions of multisum Rogers?Ramanujan identities.

On page 45 of his lost notebook, Ramanujan recorded two asymptotic formulas for two continued fractions involving the Riemann zeta-function and Dirichlet -functions. The paper proves a more general theorem and derives Ramanujan's claims as a corollary of the theorem.

On page 212 of his lost notebook, Ramanujan defined a new class invariant λn and constructed a table of values for λn. The paper constructs a new class of series for 1/π associated with λn. The new method also yields a new proof of the Borweins' general series for 1/π belonging to Ramanujan's ‘theory of q2’.

Letwhere φ is the notation used by Ramanujan in his notebooks [15], and is the familiar notation of Whittaker and Watson [20, p. 464]. It is well known that [1, p. 102] (with a misprint corrected)where denotes the ordinary or Gaussian hypergeometric function; k, 0 < k < 1, is the modulus; K is the complete elliptic integral of the first kind; andwhere K′=K(k′) and is the complementary modulus. Thus, an evaluation of any one of the functions φ, , or K yields an evaluation of the other two functions. However, such evaluations may not be very explicit. For example, if K(k) is known for a certain value of k, it may be difficult or impossible to explicitly determine K′, and so q cannot be explicitly determined. Conversely, it may be possible to evaluate φ(q) for a certain value of q, but it may be impossible to determine the corresponding value of k. (Recall that [1, p. 102].)