Moment curves and cyclic symmetry for positive Grassmannians

We show that for each k and n, the cyclic shift map on the complex Grassmannian Gr(k,n) has exactly $\binom{n}{k}$ fixed points. There is a unique totally nonnegative fixed point, given by taking n equally spaced points on the trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is even). We introduce a parameter q, and show that the fixed points of a q-deformation of the cyclic shift map are precisely the critical points of the mirror-symmetric superpotential $\mathcal{F}_q$ on Gr(k,n). This follows from results of Rietsch about the quantum cohomology ring of Gr(k,n). We survey many other diverse contexts which feature moment curves and the cyclic shift map.Comment: 18 pages. v2: Minor change

The second moment of the number of integral points on elliptic curves is bounded

In this paper, we show that the second moment of the number of integral points on elliptic curves over $\mathbb{Q}$ is bounded. In particular, we prove that, for any $0 < s < \log_2 5 = 2.3219 \ldots$, the $s$-th moment of the number of integral points is bounded for many families of elliptic curves --- e.g., for the family of all integral short Weierstrass curves ordered by naive height, for the family of only minimal such Weierstrass curves, for the family of semistable curves, or for subfamilies thereof defined by finitely many congruence conditions. For certain other families of elliptic curves, such as those with a marked point or a marked $2$-torsion point, the same methods show that for $0 < s < \log_2 3 = 1.5850\ldots$, the $s$-th moment of the number of integral points is bounded. The main new ingredient in our proof is an upper bound on the number of integral points on an affine integral Weierstrass model of an elliptic curve depending only on the rank of the curve and the number of square divisors of the discriminant. We obtain the bound by studying a bijection first observed by Mordell between integral points on these curves and certain types of binary quartic forms. The theorems on moments then follow from H\"older's inequality, analytic techniques, and results on bounds on the average sizes of Selmer groups in the families.Comment: 14 pages, comments welcome

Interaction curves for vibration and buckling of thin-walled composite box beams under axial loads and end moments

Interaction curves for vibration and buckling of thin-walled composite box beams with arbitrary lay-ups under constant axial loads and equal end moments are presented. This model is based on the classical lamination theory, and accounts for all the structural coupling coming from material anisotropy. The governing differential equations are derived from the Hamilton’s principle. The resulting coupling is referred to as triply flexural–torsional coupled vibration and buckling. A displacement-based one-dimensional finite element model with seven degrees of freedoms per node is developed to solve the problem. Numerical results are obtained for thin-walled composite box beams to investigate the effects of axial force, bending moment, fiber orientation on the buckling loads, buckling moments, natural frequencies and corresponding vibration mode shapes as well as axial-moment–frequency interaction curves

Moments of the critical values of families of elliptic curves, with applications

We make conjectures on the moments of the central values of the family of all elliptic curves and on the moments of the first derivative of the central values of a large family of positive rank curves. In both cases the order of magnitude is the same as that of the moments of the central values of an orthogonal family of L-functions. Notably, we predict that the critical values of all rank 1 elliptic curves is logarithmically larger than the rank 1 curves in the positive rank family. Furthermore, as arithmetical applications we make a conjecture on the distribution of a_p's amongst all rank 2 elliptic curves, and also show how the Riemann hypothesis can be deduced from sufficient knowledge of the first moment of the positive rank family (based on an idea of Iwaniec).Comment: 24 page

Moments of L'(1/2) in the Family of Quadratic Twists

We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to -1 and odd orthogonal symmetry. Assuming GRH we prove the asymptotic formulae for (1) the second moment with one secondary term, (2) the moment of two distinct modular forms f and g and (3) the first moment with controlled weight and level dependence. We also include some immediate corollaries to elliptic curves via the modularity theorem and the work of Gross and Zagier.Comment: Many minor typos fixed and improvements in the writing made in this revision. To appear in IMR