Fix an elliptic curve E/Q, and assume the Riemann Hypothesis for the Lfunction L(ED, s) for every quadratic twist ED of E by D ∈ Z. We combine Weil’s explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of ED. We derive from this an upper bound for the density of low-lying zeros of L(ED, s) which is compatible with the...

This paper is the second of a series devoted to the study of the rank of J0(q) (the Jacobian of the modular curve X0(q)), from the analytic point of view stemming from the Birch and Swinnerton-Dyer conjecture, which is tantamount to the study, on average, of the order of vanishing at the central critical point of the L-functions of primitive weight two forms f of level q (q prime). We prove tha...

A new approach to counterexamples to A new approach to counterexamples to L The derivation of counterexamples to L 1 estimates can be reduced to a geometric decomposition procedure along rank-one lines in matrix space. We illustrate this concept in two concrete applications. Firstly, we recover a celebrated, and rather complex, counterexample by Ornstein, proving the failure of Korn's inequalit...

We show that the skew-growth function of a dual Artin monoid of finite type P has exactly rank(P ) =: l simple real zeros on the interval (0, 1]. The proofs for types Al and Bl are based on an unexpected fact that the skewgrowth functions, up to a trivial factor, are expressed by Jacobi polynomials due to a Rodrigues type formula in the theory of orthogonal polynomials. The skew-growth function...

We present a supervised learning to rank algorithm that effectively orders images by exploiting the structure in image sequences. Most often in the supervised learning to rank literature, ranking is approached either by analyzing pairs of images or by optimizing a list-wise surrogate loss function on full sequences. In this work we propose MidRank, which learns from moderately sized sub-sequenc...

Distributive supermatroids generalize matroids to partially ordered sets. Completing earlier work of Barnabei, Nicoletti and Pezzoli we characterize the lattice of flats of a distributive supermatroid. For the prominent special case of a polymatroid the description of the flat lattice is particularly simple. Large portions of the proofs reduce to properties of weakly submodular rank functions. ...

A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the quasisymmetric functions associated to matroids by the Hopf algebra morphism F of Billera, Jia, and Reiner [3]. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as wel...

This paper presents a representation theory for permutation-valued functions, which in their general form can also be called listwise ranking functions. Pointwise ranking functions assign a score to each object independently, without taking into account the other objects under consideration; whereas listwise loss functions evaluate the set of scores assigned to all objects as a whole. In many s...

Let π be a partition. BG-rank(π) is defined as an alternating sum of parities of parts of π [1]. In [2], Berkovich and Garvan found theta series representations for the t-core generating functions P n≥0 at,j(n)q , where at,j(n) denotes the number of t-cores of n with BG-rank = j. In addition, they found positive eta-quotient representations for odd t-core generating functions with extreme value...