We prove three conjectures concerning the evaluation of determinants, which are related to the counting of plane partitions and rhombus tilings. One of them has been posed by George Andrews in 1980, the other two are by Guoce Xin and Christian Krattenthaler. Our proofs employ computer algebra methods, namely the holonomic ansatz proposed by Doron Zeilberger and variations thereof. These variati...

We confirm a conjecture of Merel describing a certain relation between the jacobians of various quotients of X(p) in terms of specific correspondences.

We prove non-vanishing results for the central value of high derivatives of the complete L-function Λ(f, s) attached to primitive forms of weight 2 and prime level q. For fixed k ≥ 0 the proportion of primitive forms f such that Λ(f, 1/2) 6= 0 is ≥ pk +o(1) with pk > 0 and pk = 1/2 + O(k−2), as the level q goes to infinity. This result is (asymptotically in k) optimal and analogous to a result ...

Because QCD in four dimensions is renormalizable, not super-renormalizable, the fluctuations of the fermion determinant are significant at all physical scales. Fortunately the short distance behaviour of QCD is very well understood. In particular, we know that for sufficiently high momentum scales this physics should be accurately described by an improved gauge action. In the truncated determin...

We define the commuting algebra determinant of a finite group action on a finite set, a notion dual to the group determinant of Dedekind. We show that the following combinatorial example is a commuting algebra determinant. Let Bq(n) denote the set of all subspaces of an n-dimensional vector space over Fq. The type of an ordered pair (U, V ) of subspaces, where U, V ∈ Bq(n), is the ordered tripl...

In this brief note we study how the fractional mean curvature of order s ∈ (0, 1) varies with respect to C diffeomorphisms. We prove that, if α > s, then the variation under a C diffeomorphism Ψ of the s-mean curvature of a set E is controlled by the C norm of the Jacobian of Ψ. When α = 1 we discuss the stability of these estimates as s → 1− and comment on the consistency of our result with th...

We generalise our previous observation on the invariance of the Jarlskog determinant to matter effects in neutrino oscillations. Within the context of standard neutrino oscillation theory with matter effects, we present the complete set of (five) matter invariant observables for neutrino propagation in matter. We give some examples of their application. E-mail:[email protected] E-mail:w.g...

This article extends relations of Mumford’s UV W -expressions to those in subvariety in a hyperelliptic Jacobian using Baker’s method.