In [Lem2003b] we have sketched the historical development of problems related to Legendre’s equations ar−bs = 1 and the associated Pell equation x−dy = 1 with d = ab. In [Lem2003c] we discussed certain “non-standard” ideas to solve the Pell equation. Now we move from the historical to the modern part: below we will describe the theory of the first 2-descent on Pell conics and explain its connec...

Let D:= (X,B,I) be a 2-design. For each xeX set Bx:= {jeB:xIj}, and define Xh dually for each h e B. We call D reduced if Xh = Xt implies that h = i for all h,ieB. Assume that D is a reduced 2-design with parameters v, b, k, r, X. Let G be a flag transitive automorphism group of D. By F*(G) we mean (as usual) the generalized Fitting subgroup of G which is defined to be the inverse image in G of...

OBJECTIVE Endovascular robotics systems, now approved for clinical use in the United States and Europe, are seeing rapid growth in interest. Determining who has sufficient expertise for safe and effective clinical use remains elusive. Our aim was to analyze performance on a robotic platform to determine what defines an expert user. METHODS During three sessions, 21 subjects with a range of en...

Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank-2, odd-degree moduli space as a universal elliptic curve with a globally-defin...

We calculate limits on the trilinear soft-breaking parameter, At, in the Minimal Supersymmetric Standard model by requiring the absence of nonzero top squark vacuum expectation values. Assuming a low tan β, which implies a large top Yukawa coupling, we also calculate one-loop corrections to the effective potential. The resulting numerical calculations of the charge/color breaking limits are pre...

Co-Higgs bundles are Higgs bundles in the sense of Simpson, but with Higgs fields that take values in the tangent bundle instead of the cotangent bundle. Given a vector bundle on P, we find necessary and sufficient conditions on its Grothendieck splitting for it to admit a stable Higgs field. We characterize the rank-2, odd-degree moduli space as a universal elliptic curve with a globally-defin...

When participants studied a list of paired associates for several study-test cycles, their judgments of learning (JOLs) exhibited relatively good calibration on the 1st cycle, with a slight overconfidence. However, a shift toward marked underconfidence occurred from the 2nd cycle on. This underconfidence-with-practice (UWP) effect was very robust across several experimental manipulations, such ...

We present a heuristic that suggests that ranks of elliptic curves E over Q are bounded. In fact, it suggests that there are only finitely many E of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich–Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank. We also discuss analogues for ellipti...